# LaTeX2HTML 2002-2-1 (1.71)
# Associate images original text with physical files.


$key = q/displaystyle-sum_{i=1,N}x_i,y_i,m_i=-intx,y,dm,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1250.png"
 ALT="$\displaystyle - \sum_{i=1,N}x_i y_i  m_i=- \int x y dm,$">|; 

$key = q/displaystylefrac{partialL}{partialr}=m,r,dot{theta}^{,2}-frac{dU}{dr},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1610.png"
 ALT="$\displaystyle \frac{\partial L}{\partial r} = m r \dot{\theta}^{ 2} - \frac{dU}{dr},$">|; 

$key = q/p_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1668.png"
 ALT="$p_\theta$">|; 

$key = q/I_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img189.png"
 ALT="$I_0$">|; 

$key = q/1slashr^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1091.png"
 ALT="$1/r^3$">|; 

$key = q/displaystyleint_P^Qx,y^2,dl;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3380.png"
 ALT="$\displaystyle \int_P^Q x y^2 dl$">|; 

$key = q/{{bf{a}times{{bf{b}={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3316.png"
 ALT="${\bf a}\times{\bf b} = {\bf0}$">|; 

$key = q/{displaymath}{{bf{f}=-frac{G,M,m}{r^3},{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img662.png"
 ALT="\begin{displaymath}
{\bf f} = - \frac{G M m}{r^3} {\bf r}.
\end{displaymath}">|; 

$key = q/displaystylefrac{15}{4},m,e.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2937.png"
 ALT="$\displaystyle \frac{15}{4} m e.$">|; 

$key = q/{displaymath}V(x,y,z)=C,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2605.png"
 ALT="\begin{displaymath}
V(x,y,z) = C,
\end{displaymath}">|; 

$key = q/225^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3002.png"
 ALT="$225^\circ$">|; 

$key = q/displaystyle{{bf{x}_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1314.png"
 ALT="$\displaystyle {\bf x}_i,$">|; 

$key = q/displaystyleddot{Z}+m^2,Z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2767.png"
 ALT="$\displaystyle \ddot{Z} + m^2 Z$">|; 

$key = q/phi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img296.png"
 ALT="$\phi_0$">|; 

$key = q/displaystylefrac{partialU}{partialrho_1},frac{partialrho_1}{partialy}+frac{partialU}{partialrho_2},frac{partialrho_2}{partialy}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2569.png"
 ALT="$\displaystyle \frac{\partial U}{\partial \rho_1} \frac{\partial \rho_1}{\partial y}+\frac{\partial U}{\partial \rho_2} \frac{\partial \rho_2}{\partial y}=0,$">|; 

$key = q/r_1=sqrt{x^2+y^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img705.png"
 ALT="$r_1=\sqrt{x^2+y^2}$">|; 

$key = q/{displaymath}omega,sinalpha=sintheta,dot{phi}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1458.png"
 ALT="\begin{displaymath}
\omega \sin\alpha = \sin\theta \dot{\phi}.
\end{displaymath}">|; 

$key = q/y(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1716.png"
 ALT="$y(x)$">|; 

$key = q/C=C_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$C =C_3$">|; 

$key = q/{{bf{L}=(I_{x'x'},omega_{x'},,I_{y'y'},omega_{y'},,I_{z'z'},omega_{z'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1358.png"
 ALT="${\bf L} = (I_{x'x'} \omega_{x'}, I_{y'y'} \omega_{y'}, I_{z'z'} \omega_{z'})$">|; 

$key = q/L_parallel+{mitOmega},I_parallelslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img201.png"
 ALT="$L_\parallel + {\mit\Omega} I_\parallel/2$">|; 

$key = q/{displaymath}u(theta)=u_c,(1-e,costheta).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img774.png"
 ALT="\begin{displaymath}
u(\theta) = u_c (1-e \cos\theta).
\end{displaymath}">|; 

$key = q/mu_1=1-mu_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2600.png"
 ALT="$\mu_1=1-\mu_2$">|; 

$key = q/({{bf{a}times{{bf{b})times({{bf{c}times{{bf{b})=[{{bf{b}cdot({{bf{c}times{{bf{a})],{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3524.png"
 ALT="$({\bf a}\times {\bf b})\times ({\bf c}\times {\bf b}) = [{\bf b}\cdot({\bf c}\times {\bf a})] {\bf b}$">|; 

$key = q/displaystyledeltaZ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2811.png"
 ALT="$\displaystyle \delta Z$">|; 

$key = q/P={{bf{f}cdot{{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3286.png"
 ALT="$P={\bf f}\cdot {\bf v}$">|; 

$key = q/{displaymath}frac{dp_k}{dt}=frac{partialL}{partialq_k}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1665.png"
 ALT="\begin{displaymath}
\frac{d p_k}{dt} = \frac{\partial L}{\partial q_k}=0.
\end{displaymath}">|; 

$key = q/i=1,8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2392.png"
 ALT="$i=1,8$">|; 

$key = q/omega_perp=omega,sinalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1392.png"
 ALT="$\omega_\perp = \omega \sin\alpha$">|; 

$key = q/I_{z'z'}=I_parallel;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1376.png"
 ALT="$I_{z'z'} = I_\parallel$">|; 

$key = q/{displaymath}frac{d}{dt}left({{bf{a}times{{bf{b}right)=frac{d{{bf{a}}{dt}times{{bf{b}+{{bf{a}timesfrac{d{{bf{b}}{dt}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img3371.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\left({\bf a}\times{\bf b}\right) = \frac{d{\bf a}}{dt}\times{\bf b} + {\bf a}\times
\frac{d{\bf b}}{dt}.
\end{displaymath}">|; 

$key = q/{displaymath}ddot{lambda}-left[frac{(I_{y'y'}-I_{x'x'}),(I_{z'z'}-I_{y'y'})}{I_{x'x'},I_{z'z'}}right]omega_{y'}^{,2},lambda=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1550.png"
 ALT="\begin{displaymath}
\ddot{\lambda} - \left[\frac{(I_{y'y'}-I_{x'x'}) (I_{z'z'}-...
...})}{I_{x'x'} I_{z'z'}}\right]\omega_{y'}^{ 2}  \lambda = 0.
\end{displaymath}">|; 

$key = q/OPequiv|{{bf{r}|=r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3227.png"
 ALT="$OP\equiv \vert{\bf r}\vert=r$">|; 

$key = q/frightarrowpminfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1507.png"
 ALT="$f\rightarrow\pm\infty$">|; 

$key = q/displaystyletheta_0+deltatheta,cos(Omega_n,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2353.png"
 ALT="$\displaystyle \theta_0 + \delta\theta \cos(\Omega_n t),$">|; 

$key = q/displaystyle(a_y,b_z-a_z,b_y)^2+(a_z,b_x-a_x,b_z)^2+(a_x,b_y-a_y,b_x)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="440" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3310.png"
 ALT="$\displaystyle (a_y  b_z-a_z  b_y)^2 +(a_z  b_x - a_x  b_z)^2 +(a_x  b_y
-a_y  b_x)^2$">|; 

$key = q/{displaymath}int_0^{vslashv_t}frac{dy}{1-y^2}=frac{g}{v_t},t.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img257.png"
 ALT="\begin{displaymath}
\int_0^{v/v_t}\frac{dy}{1-y^2} = \frac{g}{v_t} t.
\end{displaymath}">|; 

$key = q/{displaymath}int_{v_0}^vfrac{dv'}{f(v')}=frac{t}{m},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img244.png"
 ALT="\begin{displaymath}
\int_{v_0}^v \frac{dv'}{f(v')} = \frac{t}{m},
\end{displaymath}">|; 

$key = q/{{bf{a}cdot{{bf{b}=a_x,b_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3359.png"
 ALT="${\bf a}\cdot {\bf b} = a_x  b_x$">|; 

$key = q/{displaymath}{{bf{v}=dot{r},,{{bf{e}_r+r,dot{theta},,{{bf{e}_theta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img688.png"
 ALT="\begin{displaymath}
{\bf v} = \dot{r}  {\bf e}_r + r \dot{\theta}  {\bf e}_\theta.
\end{displaymath}">|; 

$key = q/k^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img999.png"
 ALT="$k^2$">|; 

$key = q/sin^{-4}(thetaslash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1002.png"
 ALT="$\sin^{-4}(\theta/2)$">|; 

$key = q/{displaymath}frac{d}{dt}!left(frac{partialx_j}{partialq_i}right)=sum_{k=1,{calF}ot{q}_k+frac{partial^2x_j}{partialq_i,partialt}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1592.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial x_j}{\partial q_i}\right)...
...,\dot{q}_k +
\frac{\partial^2 x_j}{\partial q_i \partial t}.
\end{displaymath}">|; 

$key = q/displaystylesum_{j>0}c_j,sin(Omega_j,T-gamma_j).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2835.png"
 ALT="$\displaystyle \sum_{j&gt;0} c_j \sin(\Omega_j T-\gamma_j).$">|; 

$key = q/1times{calF};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1918.png"
 ALT="$1\times {\cal F}$">|; 

$key = q/lambda,mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1536.png"
 ALT="$\lambda \mu$">|; 

$key = q/omega_{z'}=omega,cosalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1391.png"
 ALT="$\omega_{z'} = \omega \cos\alpha$">|; 

$key = q/displaystyle-dot{x}',sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1647.png"
 ALT="$\displaystyle -\dot{x}' \sin\theta,$">|; 

$key = q/{displaymath}frac{d^2u}{dtheta^2}+u=frac{G,M}{h^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img747.png"
 ALT="\begin{displaymath}
\frac{d^2 u}{d\theta^2} + u = \frac{G M}{h^2}.
\end{displaymath}">|; 

$key = q/{displaymath}f(x,phi)=x-a,phi=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1818.png"
 ALT="\begin{displaymath}
f(x,\phi) = x - a \phi = 0.
\end{displaymath}">|; 

$key = q/(partialhslashpartialy)_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3442.png"
 ALT="$(\partial h/\partial y)_x$">|; 

$key = q/x(0)=dx(0)slashdt=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img441.png"
 ALT="$x(0)=dx(0)/dt=0$">|; 

$key = q/[(k_1+k_2)slashm]^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img503.png"
 ALT="$[(k_1+k_2)/m]^{1/2}$">|; 

$key = q/{displaymath}{{bf{g}=-nabla(Phi+chi).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2171.png"
 ALT="\begin{displaymath}
{\bf g} = - \nabla(\Phi+\chi).
\end{displaymath}">|; 

$key = q/lambda,mu,nu=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3540.png"
 ALT="$\lambda \mu \nu=-1$">|; 

$key = q/{displaymath}{{bf{a}=(ddot{r}-r,dot{theta}^{,2}),{{bf{e}_r+(r,ddot{theta}+2,dot{theta}),{{bf{e}_theta=-frac{G,M}{r^2},{{bf{e}_r.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img692.png"
 ALT="\begin{displaymath}
{\bf a} = (\ddot{r}-r \dot{\theta}^{ 2}) {\bf e}_r + (r ...
...\dot{\theta}) {\bf e}_\theta = - \frac{G M}{r^2} {\bf e}_r.
\end{displaymath}">|; 

$key = q/{displaymath}E=E,b^2left[left(frac{du}{dvartheta}right)^2+u^2right]+U(u).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img964.png"
 ALT="\begin{displaymath}
E = E b^2\left[\left(\frac{du}{d\vartheta}\right)^2 + u^2\right] + U(u).
\end{displaymath}">|; 

$key = q/{displaymath}x=r-r_c.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img850.png"
 ALT="\begin{displaymath}
x = r - r_c.
\end{displaymath}">|; 

$key = q/{displaymath}dT={{bf{grad},Tcdotd{{bf{r}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img3460.png"
 ALT="\begin{displaymath}
dT = {\bf grad} T \cdot d{\bf r} = 0.
\end{displaymath}">|; 

$key = q/{{bf{e}_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3500.png"
 ALT="${\bf e}_\phi$">|; 

$key = q/theta=theta_0={{rm{const};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1688.png"
 ALT="$\theta=\theta_0 = {\rm const}$">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{AppendixAslashfigA.26.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="308" BORDER="0"
 SRC="|."$dir".q|img3503.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{AppendixA/figA.26.eps}}
\end{figure}">|; 

$key = q/displaystylelangleFrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2094.png"
 ALT="$\displaystyle \langle F\rangle$">|; 

$key = q/{displaymath}aleft[-omega^2-{{rm{i},2,nu,omega+omega_0^{,2}right]{{rm{e}^{-{{rm{,t}=omega_0^{,2},X_1,{{rm{e}^{-{{rm{i},omega,t}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="368" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img367.png"
 ALT="\begin{displaymath}
a\left[-\omega^2-{\rm i} 2 \nu \omega + \omega_0^{ 2}\ri...
...mega t}
= \omega_0^{ 2} X_1 {\rm e}^{-{\rm i} \omega t}.
\end{displaymath}">|; 

$key = q/sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img465.png"
 ALT="$\sin\theta$">|; 

$key = q/r=a,theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img901.png"
 ALT="$r=a \theta$">|; 

$key = q/displaystylelambda_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1999.png"
 ALT="$\displaystyle \lambda_1$">|; 

$key = q/{displaymath}Phi(r)=-frac{4pi,G,gamma}{3},frac{a^3}{r}=-frac{G,M}{r}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2136.png"
 ALT="\begin{displaymath}
\Phi(r) = -\frac{4\pi G \gamma}{3} \frac{a^3}{r} = - \frac{G M}{r}
\end{displaymath}">|; 

$key = q/M=m+m';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2203.png"
 ALT="$M=m+m'$">|; 

$key = q/{displaymath}y_i',frac{partialF}{partialy_i'}-F={{rm{const},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1786.png"
 ALT="\begin{displaymath}
y_i' \frac{\partial F}{\partial y_i'} - F = {\rm const},
\end{displaymath}">|; 

$key = q/{displaymath}a_c=a_g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2207.png"
 ALT="\begin{displaymath}
a_c=a_g.
\end{displaymath}">|; 

$key = q/displaystyle-{{bf{r}_E,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2710.png"
 ALT="$\displaystyle -{\bf r}_E,$">|; 

$key = q/{{bf{R};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1127.png"
 ALT="${\bf R}$">|; 

$key = q/displaystyle-frac{G,M_0}{R_i}-Gsum_{j<i}frac{M_j}{R_i}left[1+frac{1}{4}left(fraci}right)^2+frac{9}{64}left(frac{R_j}{R_i}right)^4+cdotsright];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="445" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2395.png"
 ALT="$\displaystyle -\frac{G M_0}{R_i}-G \sum_{j&lt; i}\frac{M_j}{R_i}\left[1+\frac{1}{...
...c{R_j}{R_i}\right)^2 + \frac{9}{64}\left(\frac{R_j}{R_i}\right)^4+\cdots\right]$">|; 

$key = q/B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$B$">|; 

$key = q/{displaymath}{{bf{L}=sum_{i=1,N}{{bf{l}_i{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img155.png"
 ALT="\begin{displaymath}
{\bf L} = \sum_{i=1,N} {\bf l}_i
\end{displaymath}">|; 

$key = q/T({{rm{yr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2419.png"
 ALT="$T({\rm yr})$">|; 

$key = q/displaystyleV_0,cos(theta+2,{mitOmega},sinlambda,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1174.png"
 ALT="$\displaystyle V_0 \cos(\theta +2 {\mit\Omega} \sin\lambda t),$">|; 

$key = q/displaystyleleft[a^2,z-a,z^2+z^3slash3right]_0^a=frac{1}{3},a^3.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3426.png"
 ALT="$\displaystyle \left[a^2 z-a z^2+z^3/3\right]_0^a= \frac{1}{3} a^3.$">|; 

$key = q/displaystylefrac{r_p}{1-e},e^2,sin^2E.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img805.png"
 ALT="$\displaystyle \frac{r_p}{1-e} e^2 \sin^2E.$">|; 

$key = q/{figure}{epsfysize{=1.2in{centerline{epsffile{Chapter12slashfig12.06.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="549" HEIGHT="136" BORDER="0"
 SRC="|."$dir".q|img2240.png"
 ALT="\begin{figure}
\epsfysize =1.2in
\centerline{\epsffile{Chapter12/fig12.06.eps}}
\end{figure}">|; 

$key = q/{{bf{r}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img912.png"
 ALT="${\bf r}_2$">|; 

$key = q/t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$t$">|; 

$key = q/{displaymath}{calE}=left(frac{m_1+m_2}{m_2}right)E.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1045.png"
 ALT="\begin{displaymath}
{\cal E} = \left(\frac{m_1+m_2}{m_2}\right)E.
\end{displaymath}">|; 

$key = q/omega_{x'}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1455.png"
 ALT="$\omega_{x'}=0$">|; 

$key = q/partialFslashpartialy=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1737.png"
 ALT="$\partial F/\partial y = 0$">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{Chapter15slashfig15.37.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="513" HEIGHT="219" BORDER="0"
 SRC="|."$dir".q|img3175.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{Chapter15/fig15.37.eps}}
\end{figure}">|; 

$key = q/{displaymath}V(r)=-h^2left(frac{2,r_0}{r^3}+frac{1}{2,r^2}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img844.png"
 ALT="\begin{displaymath}
V(r) = - h^2\left(\frac{2 r_0}{r^3} + \frac{1}{2 r^2}\right).
\end{displaymath}">|; 

$key = q/{displaymath}K(x)=E-U(x).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img208.png"
 ALT="\begin{displaymath}
K(x) = E - U(x).
\end{displaymath}">|; 

$key = q/{displaymath}dot{u}^{,2}=f(u),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1503.png"
 ALT="\begin{displaymath}
\dot{u}^{ 2} = f(u),
\end{displaymath}">|; 

$key = q/displaystylefrac{g,sintheta}{1-m,cos^2thetaslash(m+M)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1655.png"
 ALT="$\displaystyle \frac{g \sin\theta}{1 - m \cos^2\theta/(m+M)}.$">|; 

$key = q/epsilon=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2149.png"
 ALT="$\epsilon=0$">|; 

$key = q/displaystyle-e,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2947.png"
 ALT="$\displaystyle -e,$">|; 

$key = q/{displaymath}eta_k(t)=alpha_k,{{rm{e}^{+sqrt{lambda_k},t}+beta_k,{{rm{e}^{-sqrt{lambda_k},t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1983.png"
 ALT="\begin{displaymath}
\eta_k(t) = \alpha_k  {\rm e}^{+\sqrt{\lambda_k} t}+\beta_k {\rm e}^{-\sqrt{\lambda_k} t},
\end{displaymath}">|; 

$key = q/{displaymath}{calE}equivfrac{1}{2}left(dot{xi}^{,2}+dot{eta}^{,2}+dot{zeta}^{,2}right)-frac{1}{r}=-frac{1}{2,a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2525.png"
 ALT="\begin{displaymath}
{\cal E} \equiv \frac{1}{2}\left(\dot{\xi}^{ 2}+\dot{\eta}^...
...} + \dot{\zeta}^{ 2}\right) - \frac{1}{r} = - \frac{1}{2 a}.
\end{displaymath}">|; 

$key = q/{displaymath}B=-frac{x_1,(nu,cosphi_1+omega,sinphi_1)}{omega_r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img445.png"
 ALT="\begin{displaymath}
B = -\frac{x_1 (\nu \cos\phi_1+\omega \sin\phi_1)}{\omega_r}.
\end{displaymath}">|; 

$key = q/{displaymath}U=U_0''-epsilon,alpha,cos(2,theta)+epsilon,beta_m,sin(2,theta),cos(varpi_n-phi),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="431" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2331.png"
 ALT="\begin{displaymath}
U = U_0'' -\epsilon \alpha \cos(2 \theta) +\epsilon \beta_m \sin(2 \theta) \cos(\varpi_n-\phi),
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{grad},f=nablafequivleft(frac{partialf}{partialx},,frac{partialf}{partialy},,frac{partialf}{partialz}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3477.png"
 ALT="\begin{displaymath}
{\bf grad} f = \nabla f \equiv \left(\frac{\partial f}{\par...
...partial f}{\partial y}, \frac{\partial f}{\partial z}\right).
\end{displaymath}">|; 

$key = q/displaystyle-frac{2,beta_m,costheta_0}{I_parallel,omega},sin(Omega_n,t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2351.png"
 ALT="$\displaystyle -\frac{2 \beta_m \cos\theta_0}{I_\parallel \omega} \sin(\Omega_n t).$">|; 

$key = q/{figure}{epsfysize{=1in{centerline{epsffile{Chapter04slashfig4.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="522" HEIGHT="110" BORDER="0"
 SRC="|."$dir".q|img644.png"
 ALT="\begin{figure}
\epsfysize =1in
\centerline{\epsffile{Chapter04/fig4.03.eps}}
\end{figure}">|; 

$key = q/a_p=a,(1-2,epsilonslash3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2175.png"
 ALT="$a_p=a (1-2 \epsilon/3)$">|; 

$key = q/displaystylefrac{51}{16},m^2,e,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2929.png"
 ALT="$\displaystyle \frac{51}{16} m^2 e,$">|; 

$key = q/omega=(omega_{x'}^2+omega_{y'}^2+omega_{z'}^2)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1390.png"
 ALT="$\omega=(\omega_{x'}^2+\omega_{y'}^2+\omega_{z'}^2)^{1/2}$">|; 

$key = q/{displaymath}{{bf{x}_k^{ast,T},{{bf{G},{{bf{x}_k=lambda_k,{{bf{x}_k^{ast,T},{{bf{M},{{bf{x}_k.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1930.png"
 ALT="\begin{displaymath}
{\bf x}_k^{\ast T} {\bf G} {\bf x}_k = \lambda_k 
{\bf x}_k^{\ast T} {\bf M} {\bf x}_k.
\end{displaymath}">|; 

$key = q/displaystyle{{bf{v}_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1027.png"
 ALT="$\displaystyle {\bf v}_1'$">|; 

$key = q/{displaymath}y'=frac{c}{sqrt{1-c^2}}={{rm{const}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1747.png"
 ALT="\begin{displaymath}
y' = \frac{c}{\sqrt{1-c^2}} = {\rm const}.
\end{displaymath}">|; 

$key = q/{displaymath}dot{phi}=1.00327,omega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1480.png"
 ALT="\begin{displaymath}
\dot{\phi} = 1.00327 \omega.
\end{displaymath}">|; 

$key = q/z_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2887.png"
 ALT="$z_j$">|; 

$key = q/lambda_k=gamma_k^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1959.png"
 ALT="$\lambda_k = \gamma_k^{ 2}$">|; 

$key = q/{displaymath}J_n=-2pi,G,gammaint_0^piP_n(costheta),int_0^{a_theta(theta)}r^{,2+n},dr,sintheta,dtheta{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="396" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2153.png"
 ALT="\begin{displaymath}
J_n = - 2\pi G \gamma\int_0^\pi P_n(\cos\theta) \int_0^{a_\theta(\theta)}r^{ 2+n} dr \sin\theta d\theta
\end{displaymath}">|; 

$key = q/displaystyle-(xi_2,dot{xi}+eta_2,dot{eta})+omega,(xi,eta_2-eta,xi_2)+xi,dot{xi}+eta,dot{eta}+zeta,dot{zeta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="411" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2518.png"
 ALT="$\displaystyle -(\xi_2 \dot{\xi}+\eta_2 \dot{\eta}) + \omega (\xi \eta_2-\eta \xi_2) + \xi \dot{\xi}
+ \eta \dot{\eta} + \zeta \dot{\zeta}.$">|; 

$key = q/displaystyle-n^2,a^3,frac{z_1}{r^3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2750.png"
 ALT="$\displaystyle - n^2 a^3 \frac{z_1}{r^3},$">|; 

$key = q/dF=3,x^2,y,dx+x^3,dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3402.png"
 ALT="$dF= 3 x^2 y dx + x^3 dy$">|; 

$key = q/omega_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2870.png"
 ALT="$\omega_4$">|; 

$key = q/P_2(x)=(1slash2),(3,x^2-1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2273.png"
 ALT="$P_2(x)=(1/2) (3 x^2-1)$">|; 

$key = q/{{bf{a}times({{bf{b}times{{bf{c})equiv(a_y,b_x,c_y,,-a_x,b_x,c_y,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3357.png"
 ALT="${\bf a}\times({\bf b}\times{\bf c}) \equiv (a_y  b_x  c_y,  -a_x  b_x  c_y,  0)$">|; 

$key = q/{displaymath}r=frac{r_p,(1+e)}{1+e,costheta},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img790.png"
 ALT="\begin{displaymath}
r = \frac{r_p (1+e)}{1+e \cos\theta},
\end{displaymath}">|; 

$key = q/displaystylex;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img533.png"
 ALT="$\displaystyle x$">|; 

$key = q/displaystyleR_Z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2829.png"
 ALT="$\displaystyle R_Z,$">|; 

$key = q/9.54;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2437.png"
 ALT="$9.54$">|; 

$key = q/s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1197.png"
 ALT="$s$">|; 

$key = q/tan(2,alpha)equiv2,tanalphaslash(1-tan^2alpha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1341.png"
 ALT="$\tan( 2 \alpha) \equiv 2 \tan\alpha/(1-\tan^2\alpha)$">|; 

$key = q/displaystylesqrt{-lambda_1}=sqrt{frac{k}{m}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2041.png"
 ALT="$\displaystyle \sqrt{-\lambda_1} = \sqrt{\frac{k}{m}},$">|; 

$key = q/Q=1.3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$Q=1.3$">|; 

$key = q/displaystyledeltaY-deltaX,deltaY,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2801.png"
 ALT="$\displaystyle \delta Y - \delta X \delta Y,$">|; 

$key = q/{displaymath}{{bf{g}'=-nabla(chi+Phi'),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2252.png"
 ALT="\begin{displaymath}
{\bf g}' = - \nabla(\chi+\Phi'),
\end{displaymath}">|; 

$key = q/f(u)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1506.png"
 ALT="$f(u)=0$">|; 

$key = q/{displaymath}x=frac{v_0,v_t,costheta}{g}left(1-{{rm{e}^{-g,tslashv_t}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img591.png"
 ALT="\begin{displaymath}
x = \frac{v_0 v_t \cos\theta}{g}\left(1-{\rm e}^{-g t/v_t}\right).
\end{displaymath}">|; 

$key = q/{displaymath}deltaint_{t_1}^{t_2}L(q_1,q_2,cdots,q_{calF},dot{q}_1,dot{q}_2,cdots,dot{q}_{calF},t),dt=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1792.png"
 ALT="\begin{displaymath}
\delta\int_{t_1}^{t_2} L(q_1,q_2,\cdots,q_{\cal F}, \dot{q}_1,\dot{q}_2,\cdots,\dot{q}_{\cal F},t) dt = 0.
\end{displaymath}">|; 

$key = q/displaystyleleft[frac{2,m_1,m_2,(1-costheta)}{(m_1+m_2)^2}right]{calE}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1053.png"
 ALT="$\displaystyle \left[\frac{2 m_1 m_2 (1-\cos\theta)}{(m_1+m_2)^2}\right] {\cal E}.$">|; 

$key = q/epsilonslash(k,r_c^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img879.png"
 ALT="$\epsilon/(k r_c^{ 2})$">|; 

$key = q/alpha=pislash4,,3pislash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1352.png"
 ALT="$\alpha=\pi/4, 
3\pi/4$">|; 

$key = q/{displaymath}K=frac{1}{2,m}left(p_r^{,2}+frac{p_theta^{,2}}{r^2}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1872.png"
 ALT="\begin{displaymath}
K = \frac{1}{2 m}\left(p_r^{ 2} + \frac{p_\theta^{ 2}}{r^2}\right).
\end{displaymath}">|; 

$key = q/{displaymath}r,ddot{theta}+2,dot{r},dot{theta}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img694.png"
 ALT="\begin{displaymath}
r \ddot{\theta} + 2 \dot{r} \dot{\theta} = 0.
\end{displaymath}">|; 

$key = q/phi_1simeqpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img383.png"
 ALT="$\phi_1\simeq \pi$">|; 

$key = q/displaystyle-frac{dV}{dr},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1618.png"
 ALT="$\displaystyle - \frac{d V}{dr},$">|; 

$key = q/v(0)=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$v(0)=-1$">|; 

$key = q/{displaymath}frac{d^2theta}{dt^2}+omega_0^{,2},thetasimeq0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img461.png"
 ALT="\begin{displaymath}
\frac{d^2\theta}{dt^2} + \omega_0^{ 2} \theta\simeq 0,
\end{displaymath}">|; 

$key = q/{displaymath}frac{dtheta}{dt}=frac{h}{r^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img936.png"
 ALT="\begin{displaymath}
\frac{d\theta}{dt} = \frac{h}{r^2},
\end{displaymath}">|; 

$key = q/0.43;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2473.png"
 ALT="$0.43$">|; 

$key = q/{displaymath}|{{bf{a}+{{bf{b}|leq|{{bf{a}|+|{{bf{b}|.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3512.png"
 ALT="\begin{displaymath}
\vert{\bf a} + {\bf b}\vert\leq \vert{\bf a}\vert+\vert{\bf b}\vert.
\end{displaymath}">|; 

$key = q/omega=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2493.png"
 ALT="$\omega =1$">|; 

$key = q/{displaymath}int_{v_{z,0}}^{v_z}frac{dv_z}{v_t+v_z}=-frac{g}{v_t},t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img587.png"
 ALT="\begin{displaymath}
\int_{v_{z 0}}^{v_z} \frac{dv_z}{v_t+v_z} = - \frac{g}{v_t} t,
\end{displaymath}">|; 

$key = q/n>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3103.png"
 ALT="$n&gt;1$">|; 

$key = q/displaystylerho_1^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2509.png"
 ALT="$\displaystyle \rho_1^{ 2}$">|; 

$key = q/displaystyleU_{yy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2663.png"
 ALT="$\displaystyle U_{yy}$">|; 

$key = q/{displaymath}dot{psi}=left(frac{I_perp-I_parallel}{I_parallel}right)costheta,dot{phi}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1712.png"
 ALT="\begin{displaymath}
\dot{\psi} = \left(\frac{I_\perp-I_\parallel}{I_\parallel}\right)\cos\theta \dot{\phi}.
\end{displaymath}">|; 

$key = q/omega_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1274.png"
 ALT="$\omega_x$">|; 

$key = q/{{bf{c}equiv(c_x,,c_y,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3355.png"
 ALT="${\bf c} \equiv (c_x,  c_y,  0)$">|; 

$key = q/cos^2theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img356.png"
 ALT="$\cos^2\theta$">|; 

$key = q/q_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1793.png"
 ALT="$q_1$">|; 

$key = q/{{bf{F};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3186.png"
 ALT="${\bf F}$">|; 

$key = q/1.66times10^{-7};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2420.png"
 ALT="$1.66\times 10^{-7}$">|; 

$key = q/{displaymath}v=pmleft(frac{2,[E-U(x)]}{m}right)^{1slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img229.png"
 ALT="\begin{displaymath}
v = \pm\left(\frac{2 [E-U(x)]}{m}\right)^{1/2},
\end{displaymath}">|; 

$key = q/X=xslash(v_0^{,2}slashg);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img612.png"
 ALT="$X=x/(v_0^{ 2}/g)$">|; 

$key = q/{displaymath}Phi(a_theta,theta)simeq-frac{G,M}{a}left[1+frac{4}{15},epsilon,P_2(costheta)+{calO}(epsilon^2)right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="396" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2163.png"
 ALT="\begin{displaymath}
\Phi(a_\theta,\theta) \simeq - \frac{G M}{a} \left[1+\frac{4}{15} \epsilon P_2(\cos\theta) + {\cal O}(\epsilon^2)\right],
\end{displaymath}">|; 

$key = q/V(Q);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3468.png"
 ALT="$V(Q)$">|; 

$key = q/{{bf{b}={{bf{r}+mu,{{bf{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3538.png"
 ALT="${\bf b} = {\bf r}+\mu {\bf p}$">|; 

$key = q/{displaymath}frac{x'^{,2}}{a^2}+frac{y'^{,2}}{b^2}=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img560.png"
 ALT="\begin{displaymath}
\frac{x'^{ 2}}{a^2} + \frac{y'^{ 2}}{b^2} = 1,
\end{displaymath}">|; 

$key = q/a,exp(-{{rm{i},omega,t)neq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img317.png"
 ALT="$a \exp(-{\rm i} \omega t)\neq 0$">|; 

$key = q/displaystylee,sinhH-H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img831.png"
 ALT="$\displaystyle e \sinh H - H$">|; 

$key = q/{{bf{r}_i-{{bf{r}_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img160.png"
 ALT="${\bf r}_i-{\bf r}_j$">|; 

$key = q/{displaymath}R=frac{1}{2}sum_ik_i,dot{x}_i^{,2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1715.png"
 ALT="\begin{displaymath}
R = \frac{1}{2} \sum_i k_i \dot{x}_i^{ 2}
\end{displaymath}">|; 

$key = q/{displaymath}A=-x_1,cosphi_1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img443.png"
 ALT="\begin{displaymath}
A = - x_1 \cos\phi_1.
\end{displaymath}">|; 

$key = q/displaystyleI_{y'y'},dot{lambda}-(I_{z'z'}-I_{x'x'}),omega_{x'},mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1534.png"
 ALT="$\displaystyle I_{y'y'} \dot{\lambda} - (I_{z'z'}-I_{x'x'}) \omega_{x'} \mu$">|; 

$key = q/{displaymath}T_J=frac{1}{a}+2,sqrt{a,(1-e^2)},cosI.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2531.png"
 ALT="\begin{displaymath}
T_J = \frac{1}{a} + 2 \sqrt{a (1-e^2)} \cos I.
\end{displaymath}">|; 

$key = q/I_{yy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1256.png"
 ALT="$I_{yy}$">|; 

$key = q/displaystyleL_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1398.png"
 ALT="$\displaystyle L_{x'}$">|; 

$key = q/h^2=G,M_0,r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2469.png"
 ALT="$h^2=G M_0 r$">|; 

$key = q/displaystylev,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3039.png"
 ALT="$\displaystyle v,$">|; 

$key = q/displaystyle-int_0^{T_r}frac{dE}{dt},dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img349.png"
 ALT="$\displaystyle -\int_0^{T_r} \frac{dE}{dt} dt$">|; 

$key = q/displaystyley;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img535.png"
 ALT="$\displaystyle y$">|; 

$key = q/phi';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2080.png"
 ALT="$\phi'$">|; 

$key = q/1+g,m^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2853.png"
 ALT="$1+g m^2$">|; 

$key = q/-frac{9}{4},m^2,e+3,e,x_4+3,e,y_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2899.png"
 ALT="$-\frac{9}{4} m^2 e + 3 e x_4+3 e y_4$">|; 

$key = q/{displaymath}{{bf{f}=-frac{k,{{rm{e}^{-beta,t}}{r^2},{{bf{e}_r,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1886.png"
 ALT="\begin{displaymath}
{\bf f} = - \frac{k {\rm e}^{-\beta t}}{r^2} {\bf e}_r,
\end{displaymath}">|; 

$key = q/I=sum_{i=1,N}m_i,r_i^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img182.png"
 ALT="$I=\sum_{i=1,N} m_i  r_i^{ 2}$">|; 

$key = q/0<e<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img796.png"
 ALT="$0&lt;e &lt; 1$">|; 

$key = q/displaystyle-left[frac{(I_{y'y'}-I_{x'x'}),omega_{x'}}{I_{z'z'}}right]lambda.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1541.png"
 ALT="$\displaystyle - \left[\frac{(I_{y'y'}-I_{x'x'}) \omega_{x'}}{I_{z'z'}}\right]\lambda.$">|; 

$key = q/{displaymath}f(0)=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img286.png"
 ALT="\begin{displaymath}
f(0) = 0,
\end{displaymath}">|; 

$key = q/dot{~}equivdslashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1790.png"
 ALT="$\dot{&nbsp;}\equiv d/dt$">|; 

$key = q/{displaymath}{{bf{a}+(-{{bf{a})={{bf{0}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3202.png"
 ALT="\begin{displaymath}
{\bf a} + (-{\bf a}) = {\bf0}.
\end{displaymath}">|; 

$key = q/displaystylefrac{partialU}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2566.png"
 ALT="$\displaystyle \frac{\partial U}{\partial x}$">|; 

$key = q/m_1>m_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1060.png"
 ALT="$m_1&gt;m_2$">|; 

$key = q/{displaymath}t=int_{u_0}^ufrac{du'}{sqrt{f(u')}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img1505.png"
 ALT="\begin{displaymath}
t = \int_{u_0}^u\frac{du'}{\sqrt{f(u')}}.
\end{displaymath}">|; 

$key = q/{displaymath}Omega_phi=frac{3}{2},frac{epsilon,n_s^{,2}}{omega},costheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2297.png"
 ALT="\begin{displaymath}
\Omega_\phi = \frac{3}{2} \frac{\epsilon n_s^{ 2}}{\omega} \cos\theta,
\end{displaymath}">|; 

$key = q/displaystylex_1,cos(n,t)-y_1,sin(n,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2726.png"
 ALT="$\displaystyle x_1 \cos(n t) -y_1 \sin(n t),$">|; 

$key = q/v_{z,0}=v_0,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img588.png"
 ALT="$v_{z 0}=v_0 \sin\theta$">|; 

$key = q/displaystyleL_{z'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1402.png"
 ALT="$\displaystyle L_{z'}$">|; 

$key = q/{{bf{r}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2050.png"
 ALT="${\bf r}'$">|; 

$key = q/v(0)=-3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$v(0)=-3$">|; 

$key = q/G_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1908.png"
 ALT="$G_{ij}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter13slashfig13.10.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="273" BORDER="0"
 SRC="|."$dir".q|img2679.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter13/fig13.10.eps}}
\end{figure}">|; 

$key = q/displaystylep_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1868.png"
 ALT="$\displaystyle p_r$">|; 

$key = q/displaystyle(1-e^2)^{1slash2},n,a^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img766.png"
 ALT="$\displaystyle (1-e^2)^{1/2} n a^2,$">|; 

$key = q/{displaymath}lambda=-mleft(3,g,costheta-2,g+frac{v_0^{,2}}{a}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1839.png"
 ALT="\begin{displaymath}
\lambda = - m \left(3 g \cos\theta - 2 g + \frac{v_0^{ 2}}{a}\right).
\end{displaymath}">|; 

$key = q/displaystylev_y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="$\displaystyle v_y ,$">|; 

$key = q/{{bf{f}_{12}=-{{bf{f}_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img914.png"
 ALT="${\bf f}_{12} = -{\bf f}_{21}$">|; 

$key = q/(xi,,eta,,zeta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2484.png"
 ALT="$(\xi, \eta, \zeta)$">|; 

$key = q/{displaymath}l,frac{d^2{theta}}{dt^2}+g,sintheta=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img457.png"
 ALT="\begin{displaymath}
l \frac{d^2{\theta}}{dt^2} +g \sin\theta=0.
\end{displaymath}">|; 

$key = q/{displaymath}dot{phi}=omegaleft[1+3,cos^2alpharight]^{1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img1467.png"
 ALT="\begin{displaymath}
\dot{\phi} = \omega\left[1+ 3 \cos^2\alpha\right]^{1/2}.
\end{displaymath}">|; 

$key = q/1.37555;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3114.png"
 ALT="$1.37555$">|; 

$key = q/Tsimeqm,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1192.png"
 ALT="$T\simeq m g$">|; 

$key = q/displaystyledot{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1862.png"
 ALT="$\displaystyle \dot{x}$">|; 

$key = q/displaystylev_x-u,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="$\displaystyle v_x - u,$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter12slashfig12.07.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="254" BORDER="0"
 SRC="|."$dir".q|img2269.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter12/fig12.07.eps}}
\end{figure}">|; 

$key = q/displaystylea',cos(n',t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2719.png"
 ALT="$\displaystyle a' \cos(n' t),$">|; 

$key = q/deltay;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2637.png"
 ALT="$\delta y$">|; 

$key = q/{displaymath}U(x)simeqfrac{1}{2},m,omega_0^{,2},x^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img303.png"
 ALT="\begin{displaymath}
U(x) \simeq \frac{1}{2} m \omega_0^{ 2} x^2.
\end{displaymath}">|; 

$key = q/Q_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1576.png"
 ALT="$Q_i$">|; 

$key = q/dsigmaslashd{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1006.png"
 ALT="$d\sigma/d{\mit\Omega}$">|; 

$key = q/displaystyledeltaZ,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2774.png"
 ALT="$\displaystyle \delta Z,$">|; 

$key = q/4.669;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3126.png"
 ALT="$4.669$">|; 

$key = q/x=pma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img240.png"
 ALT="$x=\pm a$">|; 

$key = q/y_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2730.png"
 ALT="$y_1'$">|; 

$key = q/{{bf{e}_{y'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1435.png"
 ALT="${\bf e}_{y'}$">|; 

$key = q/omega_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img327.png"
 ALT="$\omega_r$">|; 

$key = q/{displaymath}x_1=frac{omega_0^{,2},X_1}{left[(omega_0^{,2}-omega^2)^2+4,nu^2,omega^2right]^{1slash2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img373.png"
 ALT="\begin{displaymath}
x_1 = \frac{\omega_0^{ 2} X_1}{\left[(\omega_0^{ 2}-\omega^2)^2
+ 4 \nu^2 \omega^2\right]^{1/2}},
\end{displaymath}">|; 

$key = q/displaystylefrac{3}{2},m^2,cos[2,(1-m),T]+frac{3}{2},m^2,cos[2,(1-m),T],deltaX;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2788.png"
 ALT="$\displaystyle \frac{3}{2} m^2 \cos[2 (1-m) T]+\frac{3}{2} m^2 \cos[2 (1-m) T] \delta X$">|; 

$key = q/4.9pm0.20;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3117.png"
 ALT="$4.9\pm 0.20$">|; 

$key = q/{{bf{g}_c=-nablachi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2169.png"
 ALT="${\bf g}_c = - \nabla\chi$">|; 

$key = q/phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img658.png"
 ALT="$\phi$">|; 

$key = q/e=0.0068;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2463.png"
 ALT="$e=0.0068$">|; 

$key = q/Phi';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2228.png"
 ALT="$\Phi'$">|; 

$key = q/19.19;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2440.png"
 ALT="$19.19$">|; 

$key = q/L_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1495.png"
 ALT="$L_\phi$">|; 

$key = q/displaystyleddot{z},dot{z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2553.png"
 ALT="$\displaystyle \ddot{z} \dot{z}$">|; 

$key = q/a_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2237.png"
 ALT="$a_+$">|; 

$key = q/displaystylelambda_l,{{bf{M},{{bf{x}_l.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1939.png"
 ALT="$\displaystyle \lambda_l {\bf M} {\bf x}_l.$">|; 

$key = q/{displaymath}epsilon=frac{I_parallel-I_perp}{I_0}simeqfrac{I_parallel-I_perp}{I_parallel}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2199.png"
 ALT="\begin{displaymath}
\epsilon = \frac{I_\parallel-I_\perp}{I_0}\simeq \frac{I_\parallel-I_\perp
}{I_\parallel}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=1.25in{centerline{epsffile{AppendixAslashfigA.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="137" BORDER="0"
 SRC="|."$dir".q|img3184.png"
 ALT="\begin{figure}
\epsfysize =1.25in
\centerline{\epsffile{AppendixA/figA.01.eps}}
\end{figure}">|; 

$key = q/displaystylesinpsi,sintheta,{{bf{e}_{x'}+cospsi,sintheta,{{bf{e}_{y'}+costheta,{{bf{e}_{z'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="350" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1442.png"
 ALT="$\displaystyle \sin\psi \sin\theta {\bf e}_{x'} + \cos\psi \sin\theta {\bf e}_{y'}
+ \cos\theta {\bf e}_{z'},$">|; 

$key = q/q_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2027.png"
 ALT="$q_3$">|; 

$key = q/60^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2631.png"
 ALT="$60^\circ$">|; 

$key = q/displaystylea_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3240.png"
 ALT="$\displaystyle a_{x'}$">|; 

$key = q/sin^2theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img357.png"
 ALT="$\sin^2\theta$">|; 

$key = q/{{bf{a}cdot{{bf{b}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3271.png"
 ALT="${\bf a} \cdot {\bf b} =0$">|; 

$key = q/displaystyleI_{x'x'},dot{omega}_{x'}-(I_{y'y'}-I_{z'z'}),lambda,mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1533.png"
 ALT="$\displaystyle I_{x'x'} \dot{\omega}_{x'} - (I_{y'y'}-I_{z'z'}) \lambda \mu$">|; 

$key = q/{displaymath}{{bf{x}_i^{ast,T},{{bf{A},{{bf{x}_i=lambda_i^{,ast},{{bf{x}_i^{ast,T}{{bf{x}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1294.png"
 ALT="\begin{displaymath}
{\bf x}_i^{\ast T} {\bf A} {\bf x}_i = \lambda_i^{ \ast} {\bf x}_i^{\ast T}{\bf x}_i.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d{{bf{r}}{dt}=frac{d{{bf{r}}{dt'}+mbox{{boldmath{{Omega{}times{{bf{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1110.png"
 ALT="\begin{displaymath}
\frac{d{\bf r}}{dt} = \frac{d{\bf r}}{dt'}+\mbox{\boldmath$\Omega$}\times{\bf r},
\end{displaymath}">|; 

$key = q/Q_{i,0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1904.png"
 ALT="$Q_{i 0}$">|; 

$key = q/lrightarrowl+{calO}(deltay^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1724.png"
 ALT="$l\rightarrow l + {\cal O}(\delta y^{ 2})$">|; 

$key = q/{displaymath}tilde{{bf{I},hat{mbox{{boldmath{{omega{}}=lambda,hat{mbox{{boldmath{{omega{}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1328.png"
 ALT="\begin{displaymath}
\tilde{\bf I} \hat{\mbox{\boldmath$\omega$}} = \lambda \hat{\mbox{\boldmath$\omega$}},
\end{displaymath}">|; 

$key = q/{displaymath}f_{drag}=-2,m,nu,frac{dx}{dt},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img307.png"
 ALT="\begin{displaymath}
f_{drag} = - 2 m \nu \frac{dx}{dt},
\end{displaymath}">|; 

$key = q/displaystyle(,[a_x,c_x+a_y,c_y],b_x,,0,,0)-(a_x,b_x,c_x,,a_x,b_x,c_y,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="400" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3361.png"
 ALT="$\displaystyle ( [a_x  c_x + a_y  c_y]  b_x,  0,  0) - (a_x  b_x  c_x,  a_x  b_x  c_y,  0)$">|; 

$key = q/{mitOmega}^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1203.png"
 ALT="${\mit\Omega}^{ 2}$">|; 

$key = q/{{bf{grad},T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3453.png"
 ALT="${\bf grad} T$">|; 

$key = q/displaystylebar{theta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2973.png"
 ALT="$\displaystyle \bar{\theta}$">|; 

$key = q/displaystyle-n'^{,2},a'^{,3},frac{{{bf{r}'}{|{{bf{r}'|^{,3}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2715.png"
 ALT="$\displaystyle - n'^{ 2} a'^{ 3} \frac{{\bf r}'}{\vert{\bf r}'\vert^{ 3}},$">|; 

$key = q/{displaymath}sin^2theta,dot{phi}=h,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1680.png"
 ALT="\begin{displaymath}
\sin^2\theta \dot{\phi} = h,
\end{displaymath}">|; 

$key = q/T_m({{rm{yr})=n_sslashn_m=0.081;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2362.png"
 ALT="$T_m({\rm yr})= n_s/n_m=0.081$">|; 

$key = q/rho_a,A,v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img276.png"
 ALT="$\rho_a A v$">|; 

$key = q/displaystylefrac{1}{2}-mu_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2601.png"
 ALT="$\displaystyle \frac{1}{2}-\mu_2,$">|; 

$key = q/displaystylev_1,(1,,0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1024.png"
 ALT="$\displaystyle v_1 (1,  0),$">|; 

$key = q/{displaymath}1-x_{max}^{,2}-alpha,x_{max}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img993.png"
 ALT="\begin{displaymath}
1 - x_{max}^{ 2} - \alpha x_{max}= 0.
\end{displaymath}">|; 

$key = q/L(q_1,q_2,dot{q}_1,dot{q}_2,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1797.png"
 ALT="$L(q_1,q_2,\dot{q}_1,\dot{q}_2,t)$">|; 

$key = q/G,M=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2492.png"
 ALT="$G M=1$">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{Chapter15slashfig15.26.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="517" HEIGHT="219" BORDER="0"
 SRC="|."$dir".q|img3121.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{Chapter15/fig15.26.eps}}
\end{figure}">|; 

$key = q/displaystyle-2,{mitOmega},V_0,sinlambda,costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1169.png"
 ALT="$\displaystyle -2 {\mit\Omega} V_0 \sin\lambda \cos\theta,$">|; 

$key = q/{{bf{a}={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="${\bf a}={\bf0}$">|; 

$key = q/{displaymath}l=int_{-aslash2}^{aslash2}[1+y'^{,2}]^{1slash2},dx=int_{-aslash2}^{aslash2}cosh(xslashh),dx=2,h,sinh(aslash2,h).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="523" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1779.png"
 ALT="\begin{displaymath}
l= \int_{-a/2}^{a/2}[1+y'^{ 2}]^{1/2} dx = \int_{-a/2}^{a/2} \cosh(x/h)  dx = 2 h \sinh(a/2 h).
\end{displaymath}">|; 

$key = q/displaystylen,{{bf{a}+m,{{bf{a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3211.png"
 ALT="$\displaystyle n {\bf a} + m {\bf a},$">|; 

$key = q/{displaymath}{{bf{x}_k^{T},{{bf{M},deltadot{{bf{q}^{(0)}=sqrt{lambda}_k,(alpha_k-beta_k).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1965.png"
 ALT="\begin{displaymath}
{\bf x}_k^{T} {\bf M} \delta \dot{\bf q}^{(0)} =\sqrt{\lambda}_k (\alpha_k - \beta_k).
\end{displaymath}">|; 

$key = q/d^3{{bf{r}=r,dr,dtheta,dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3492.png"
 ALT="$d^3{\bf r} = r dr d\theta dz$">|; 

$key = q/{displaymath}g'simeqg-{mitOmega}^2,R,cos^2lambdasimeq9.8-0.034,cos^2lambda,,,{{rm{mslashs^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1141.png"
 ALT="\begin{displaymath}
g' \simeq g - {\mit\Omega}^2 R \cos^2\lambda \simeq 9.8 - 0.034 \cos^2\lambda    {\rm m/s^2}.
\end{displaymath}">|; 

$key = q/V<C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2608.png"
 ALT="$V&lt; C$">|; 

$key = q/displaystylefrac{1}{3}+frac{2}{3}left(frac{3}{2},cos^2theta-frac{1}{2}right)left(frac{3}{2},cos^2theta'-frac{1}{2}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="333" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2098.png"
 ALT="$\displaystyle \frac{1}{3}+ \frac{2}{3}\left(\frac{3}{2} \cos^2\theta-\frac{1}{2}\right)\left(\frac{3}{2} \cos^2\theta'-\frac{1}{2}\right).$">|; 

$key = q/f(x,y)=x,y^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3376.png"
 ALT="$f(x,y)= x y^2$">|; 

$key = q/2,Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3053.png"
 ALT="$2 Q$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter15slashfig15.39.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="345" BORDER="0"
 SRC="|."$dir".q|img3182.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter15/fig15.39.eps}}
\end{figure}">|; 

$key = q/{displaymath}tanpsi=frac{sintheta}{costheta+1}=tan(thetaslash2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1067.png"
 ALT="\begin{displaymath}
\tan\psi = \frac{\sin\theta}{\cos\theta+1} =\tan(\theta/2).
\end{displaymath}">|; 

$key = q/E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="$E$">|; 

$key = q/y'equivdyslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1718.png"
 ALT="$y'\equiv dy/dx$">|; 

$key = q/displaystylesqrt{frac{m,(1+2,alpha)}{k}},(q_1+q_2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2048.png"
 ALT="$\displaystyle \sqrt{\frac{m (1+2 \alpha)}{k}} (q_1+q_2).$">|; 

$key = q/a=6.92times10^7,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2186.png"
 ALT="$a=6.92\times 10^7 {\rm m}$">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{Chapter15slashfig15.34.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="517" HEIGHT="219" BORDER="0"
 SRC="|."$dir".q|img3168.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{Chapter15/fig15.34.eps}}
\end{figure}">|; 

$key = q/{displaymath}int_0^E(1-e,cosE),dE=left(frac{G,M}{a^3}right)^{1slash2}t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img806.png"
 ALT="\begin{displaymath}
\int_0^E (1-e \cos E) dE = \left(\frac{G M}{a^3}\right)^{1/2} t,
\end{displaymath}">|; 

$key = q/displaystylefrac{partialH}{partialp_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1854.png"
 ALT="$\displaystyle \frac{\partial H}{\partial p_i},$">|; 

$key = q/Omega_3=omega_1+Omega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2875.png"
 ALT="$\Omega_3=\omega_1+\Omega_1$">|; 

$key = q/4pi,(Aslashg)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img661.png"
 ALT="$4\pi (A/g)^{1/2}$">|; 

$key = q/displaystylefrac{domega_{y'}}{dt'}-{mitOmega},omega_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1383.png"
 ALT="$\displaystyle \frac{d\omega_{y'}}{dt'} - {\mit\Omega} \omega_{x'}$">|; 

$key = q/displaystyleI_{xy},cosalpha+(I_{yy}-lambda),sinalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1339.png"
 ALT="$\displaystyle I_{xy} \cos\alpha + (I_{yy}-\lambda) \sin\alpha$">|; 

$key = q/{displaymath}(1+2,alpha),left[m^2,lambda^2+2,m,k,(1+alpha),lambda+k^2,(1+2alpha)right]=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="435" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img2038.png"
 ALT="\begin{displaymath}
(1+2 \alpha) \left[m^2 \lambda^2 + 2 m k (1+\alpha) \lambda + k^2 (1+2\alpha)\right] = 0.
\end{displaymath}">|; 

$key = q/-frac{9}{2},e^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2892.png"
 ALT="$-\frac{9}{2} e^2$">|; 

$key = q/a=R_1+R_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img984.png"
 ALT="$a=R_1+R_2$">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}cdot{{bf{h}=omega,h,cosI=sqrt{a,(1-e^2)},cosI,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img2527.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$}\cdot {\bf h} = \omega h \cos I = \sqrt{a (1-e^2)} \cos I,
\end{displaymath}">|; 

$key = q/{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3192.png"
 ALT="${\bf c}$">|; 

$key = q/displaystyle-frac{4}{15},frac{G,m}{a},epsilon,P_2(costheta)-frac{G,m',a^2}{R^3},P_2(costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2234.png"
 ALT="$\displaystyle -\frac{4}{15} \frac{G m}{a} \epsilon P_2(\cos\theta) - \frac{G m' a^2}{R^3} P_2(\cos\theta),$">|; 

$key = q/{displaymath}lambda'+frac{y}{h}=-coshz.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1769.png"
 ALT="\begin{displaymath}
\lambda' + \frac{y}{h} = -\cosh z.
\end{displaymath}">|; 

$key = q/{displaymath}y=Bleft[cos(omega_0,t'),cos{mitDelta}+sin(omega_0,t'),sin{mitDelta}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img545.png"
 ALT="\begin{displaymath}
y = B\left[\cos(\omega_0 t') \cos{\mit\Delta} + \sin(\omega_0 t') \sin{\mit\Delta}\right].
\end{displaymath}">|; 

$key = q/n'=0.98560025^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2698.png"
 ALT="$n'=0.98560025^\circ$">|; 

$key = q/phi_1=pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img382.png"
 ALT="$\phi_1=\pi/2$">|; 

$key = q/{displaymath}E=frac{m}{2}left(frac{ds}{dt}right)^{2}+frac{1}{2},frac{m,g}{4,A},s^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img659.png"
 ALT="\begin{displaymath}
E = \frac{m}{2}\left(\frac{ds}{dt}\right)^{2} + \frac{1}{2} \frac{m g}{4 A} s^2,
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}cdot{{bf{b}=|{{bf{a}|,|{{bf{b}|,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3270.png"
 ALT="\begin{displaymath}
{\bf a} \cdot {\bf b} = \vert{\bf a}\vert  \vert{\bf b}\vert  \cos\theta.
\end{displaymath}">|; 

$key = q/displaystylefrac{partialK}{partialdot{r}}=m,dot{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1869.png"
 ALT="$\displaystyle \frac{\partial K}{\partial \dot{r}} = m \dot{r},$">|; 

$key = q/Omega_5simeq1-2,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2938.png"
 ALT="$\Omega_5\simeq 1-2 m$">|; 

$key = q/{displaymath}{{bf{a}=({{bf{B}times{{bf{C})slash({{bf{A}cdot{{bf{B}times{{bf{C}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3543.png"
 ALT="\begin{displaymath}
{\bf a} = ({\bf B}\times {\bf C})/({\bf A}\cdot{\bf B}\times {\bf C}),
\end{displaymath}">|; 

$key = q/23.44^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2367.png"
 ALT="$23.44^\circ$">|; 

$key = q/displaystyle3-mu_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2620.png"
 ALT="$\displaystyle 3 - \mu_2,$">|; 

$key = q/sin(a,theta)neqa,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img458.png"
 ALT="$\sin(a \theta)\neq a \sin\theta$">|; 

$key = q/{displaymath}dot{r}=-frac{dot{u}}{u^2}=-r^2,frac{du}{dtheta},frac{dtheta}{dt}=-h,frac{du}{dtheta}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img745.png"
 ALT="\begin{displaymath}
\dot{r} = - \frac{\dot{u}}{u^2} = - r^2 \frac{du}{d\theta} \frac{d\theta}{dt} = - h \frac{du}{d\theta}.
\end{displaymath}">|; 

$key = q/displaystylecospsi,{{bf{e}_{x'}-sinpsi,{{bf{e}_{y'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1444.png"
 ALT="$\displaystyle \cos\psi {\bf e}_{x'} -\sin\psi {\bf e}_{y'}.$">|; 

$key = q/displaystyleiota,sin[(1+g,m^2),T-gamma_0]+frac{3}{2},e,iota,sin[(c-g),m^2,T-alpha_0+gamma_0];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="481" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2962.png"
 ALT="$\displaystyle \iota \sin[(1+g m^2) T-\gamma_0] + \frac{3}{2} e \iota \sin[(c-g) m^2 T-\alpha_0+\gamma_0]$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.07.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="274" BORDER="0"
 SRC="|."$dir".q|img3069.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.07.eps}}
\end{figure}">|; 

$key = q/{displaymath}Phi(r)=-frac{G,M}{a}sum_{n=0,infty}[P_n(0)]^2left(frac{a}{r}right)^{n+1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2380.png"
 ALT="\begin{displaymath}
\Phi(r) = - \frac{G M}{a}\sum_{n=0,\infty} [P_n(0)]^2\left(\frac{a}{r}\right)^{n+1}.
\end{displaymath}">|; 

$key = q/-lambda(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1805.png"
 ALT="$-\lambda(t)$">|; 

$key = q/b_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2883.png"
 ALT="$b_j$">|; 

$key = q/315^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3003.png"
 ALT="$315^\circ$">|; 

$key = q/displaystyle-frac{g,sintheta,costheta}{(m+M)slashm-cos^2theta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1654.png"
 ALT="$\displaystyle - \frac{g \sin\theta \cos\theta}{(m+M)/m-\cos^2\theta},$">|; 

$key = q/BP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1700.png"
 ALT="$BP$">|; 

$key = q/{displaymath}{{bf{L}=M,{{bf{r}_{cm}timesdot{{bf{r}_{cm}+mu,{{bf{r}timesdot{{bf{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1096.png"
 ALT="\begin{displaymath}
{\bf L} = M {\bf r}_{cm}\times \dot{\bf r}_{cm} + \mu {\bf r}\times \dot{\bf r},
\end{displaymath}">|; 

$key = q/Omega_2=omega_1-Omega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2874.png"
 ALT="$\Omega_2=\omega_1-\Omega_1$">|; 

$key = q/{displaymath}frac{d}{dt}left(frac{partialL}{partialdot{q}_i}right)-frac{partialL}{partialq_i}+frac{partialR}{partialdot{q}_i}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1714.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\righ...
...al L}{\partial q_i} + \frac{\partial R}{\partial \dot{q}_i}=0,
\end{displaymath}">|; 

$key = q/C=pmsqrt{27slash16},(1-2,mu_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2683.png"
 ALT="$C=\pm\sqrt{27/16} (1-2 \mu_2)$">|; 

$key = q/-e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img645.png"
 ALT="$-e$">|; 

$key = q/dW={{bf{F}cdotd{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3410.png"
 ALT="$dW={\bf F}\cdot d{\bf r}$">|; 

$key = q/dot{q}_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1893.png"
 ALT="$\dot{q}_j$">|; 

$key = q/displaystyledeltaddot{Z}+deltaZ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2806.png"
 ALT="$\displaystyle \delta\ddot{Z} + \delta Z$">|; 

$key = q/vsimeqg,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img261.png"
 ALT="$v\simeq g t$">|; 

$key = q/{displaymath}U=-frac{1}{2},delta{{bf{q}^T,{{bf{G},delta{{bf{q},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1978.png"
 ALT="\begin{displaymath}
U=- \frac{1}{2} \delta {\bf q}^T {\bf G} \delta{\bf q},
\end{displaymath}">|; 

$key = q/lambda_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3154.png"
 ALT="$\lambda_1$">|; 

$key = q/tsimeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1214.png"
 ALT="$t\simeq 0$">|; 

$key = q/displaystyleI_{z'z'},dot{mu}-(I_{x'x'}-I_{y'y'}),omega_{x'},lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1535.png"
 ALT="$\displaystyle I_{z'z'}  \dot{\mu} - (I_{x'x'}-I_{y'y'}) \omega_{x'} \lambda$">|; 

$key = q/2.72;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2459.png"
 ALT="$2.72$">|; 

$key = q/{displaymath}omega=left(frac{5-alpha}{5-2,alpha},frac{G,M}{R^{,3}}right)^{1slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img194.png"
 ALT="\begin{displaymath}
\omega = \left(\frac{5-\alpha}{5-2 \alpha} \frac{G M}{R^{ 3}}\right)^{1/2},
\end{displaymath}">|; 

$key = q/I_perpneqI_parallel;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1377.png"
 ALT="$I_\perp\neq I_\parallel$">|; 

$key = q/{{bf{P};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img151.png"
 ALT="${\bf P}$">|; 

$key = q/{displaymath}z=v_0,sintheta,t-frac{g}{2},t^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img595.png"
 ALT="\begin{displaymath}
z = v_0 \sin\theta t - \frac{g}{2} t^2,
\end{displaymath}">|; 

$key = q/{displaymath}r^2,ddot{theta}+2,r,dot{r},dot{theta}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img734.png"
 ALT="\begin{displaymath}
r^2 \ddot{\theta} + 2 r \dot{r} \dot{\theta} = 0.
\end{displaymath}">|; 

$key = q/{{bf{a}times({{bf{b}times{{bf{c})=({{bf{a}cdot{{bf{c}),{{bf{b}-({{bf{a}cdot{{bf{b}),{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3519.png"
 ALT="${\bf a}\times ({\bf b}\times {\bf c}) = ({\bf a}\cdot{\bf c}) {\bf b} - ({\bf a}\cdot{\bf b}) {\bf c}$">|; 

$key = q/d^3{{bf{r}=r^2,sintheta,dr,dtheta,dphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3501.png"
 ALT="$d^3{\bf r} = r^2 \sin\theta dr d\theta d\phi$">|; 

$key = q/displaystyle(xi-xi_2)^2+(eta-eta_2)^2+zeta^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2512.png"
 ALT="$\displaystyle (\xi-\xi_2)^2+(\eta-\eta_2)^2 + \zeta^2.$">|; 

$key = q/{displaymath}Phi_n(r)=-frac{2pi,G}{(n+1slash2),r^{n+1}}int_0^rr'^{,n+2},rho_n(r'n}{n+1slash2}int_r^inftyr'^{,1-n},rho_n(r'),dr'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="598" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2126.png"
 ALT="\begin{displaymath}
\Phi_n(r) = -\frac{2\pi G}{(n+1/2) r^{n+1}}\int_0^r r'^{ ...
...,\pi G r^n}{n+1/2}\int_r^\infty r'^{ 1-n} \rho_n(r') dr'.
\end{displaymath}">|; 

$key = q/k<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1000.png"
 ALT="$k&lt;0$">|; 

$key = q/displaystyle(1+m_1slashm_2),v_1,(1,,0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1032.png"
 ALT="$\displaystyle (1+m_1/m_2) v_1 (1, 0),$">|; 

$key = q/{displaymath}frac{partialU}{partialrho_1}=frac{partialU}{partialrho_2}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img2597.png"
 ALT="\begin{displaymath}
\frac{\partial U}{\partial \rho_1} = \frac{\partial U}{\partial \rho_2} = 0,
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{f}=-frac{G,m_1,m_2}{r^3},{{bf{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img929.png"
 ALT="\begin{displaymath}
{\bf f} = - \frac{G m_1 m_2}{r^3} {\bf r},
\end{displaymath}">|; 

$key = q/P_1(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2382.png"
 ALT="$P_1(0) = 0$">|; 

$key = q/displaystyleomega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2003.png"
 ALT="$\displaystyle \omega_1$">|; 

$key = q/{displaymath}K=frac{1}{2},M,dot{x}^{,2}+frac{1}{2},m,(dot{x}^{,2}+2,dot{x},dot{x}',costheta+dot{x}'^{,2}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1649.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} M \dot{x}^{ 2} + \frac{1}{2} m (\dot{x}^{ 2} + 2 \dot{x} \dot{x}' \cos\theta+ \dot{x}'^{ 2}),
\end{displaymath}">|; 

$key = q/dot{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1660.png"
 ALT="$\dot{x}$">|; 

$key = q/displaystylea',sin(n',t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2720.png"
 ALT="$\displaystyle a' \sin(n' t),$">|; 

$key = q/1630;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3015.png"
 ALT="$1630$">|; 

$key = q/{displaymath}K=frac{1}{2}sum_{k,l=1,{calF}}dot{eta}_k,dot{eta}_l,{{bf{x}_k^T,{{bf{M},{{bf{x}_l.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1976.png"
 ALT="\begin{displaymath}
K = \frac{1}{2}\sum_{k,l=1,{\cal F}} \dot{\eta}_k \dot{\eta}_l {\bf x}_k^T {\bf M} {\bf x}_l.
\end{displaymath}">|; 

$key = q/{{bf{-p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img946.png"
 ALT="${\bf -p}$">|; 

$key = q/displaystyle{{bf{A},{{bf{x}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1299.png"
 ALT="$\displaystyle {\bf A} {\bf x}_i$">|; 

$key = q/{displaymath}{{rm{Re}[{calL}(x)]equiv{calL}({{rm{Re}[x]),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img313.png"
 ALT="\begin{displaymath}
{\rm Re}[{\cal L}(x)]\equiv {\cal L}({\rm Re}[x]),
\end{displaymath}">|; 

$key = q/{{bf{a}=d{{bf{v}slashdtequiv(dv_xslashdt,,dv_yslashdt,,dv_zslashdt);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="${\bf a} = d{\bf v}/dt \equiv (dv_x/dt, dv_y/dt, dv_z/dt)$">|; 

$key = q/{displaymath}m,frac{dv}{dt}=m,g-c,v^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img253.png"
 ALT="\begin{displaymath}
m \frac{dv}{dt} = m  g - c v^2,
\end{displaymath}">|; 

$key = q/{displaymath}Phi(r)=-frac{G,M}{a}sum_{n=0,infty}[P_n(0)]^2left(frac{r}{a}right)^{n},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img2388.png"
 ALT="\begin{displaymath}
\Phi(r) = - \frac{G M}{a}\sum_{n=0,\infty} [P_n(0)]^2\left(\frac{r}{a}\right)^{n},
\end{displaymath}">|; 

$key = q/z'simeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1184.png"
 ALT="$z'\simeq 0$">|; 

$key = q/displaystylex_0+sum_{j>0}x_j,cos(omega_j,T-alpha_j),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2836.png"
 ALT="$\displaystyle x_0 + \sum_{j&gt;0}x_j \cos(\omega_j T-\alpha_j),$">|; 

$key = q/displaystyle{{bf{A},{{bf{x}_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1301.png"
 ALT="$\displaystyle {\bf A} {\bf x}_j$">|; 

$key = q/displaystylem,ddot{r}-m,r,dot{theta}^{,2}-m,g,costheta-lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1835.png"
 ALT="$\displaystyle m \ddot{r} -m r \dot{\theta}^{ 2} - m g \cos\theta - \lambda$">|; 

$key = q/{displaymath}{mitDelta}K=-{mitDelta}U{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="\begin{displaymath}
{\mit\Delta} K = - {\mit\Delta} U
\end{displaymath}">|; 

$key = q/v_c=sqrt{G,Mslashr_a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img782.png"
 ALT="$v_c=\sqrt{G M/r_a}$">|; 

$key = q/displaystyleleft.+sum_{j>i}left(frac{M_j}{M_0}right)left(frac{R_i}{R_j}right)^3l2+frac{175}{64}left(frac{R_i}{R_j}right)^4+cdotsright]right}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="488" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2417.png"
 ALT="$\displaystyle \left.+ \sum_{j&gt;i}\left(\frac{M_j}{M_0}\right)
\left(\frac{R_i}{R...
...right)^2 + \frac{175}{64}\left(\frac{R_i}{R_j}
\right)^4+\cdots\right]\right\}.$">|; 

$key = q/displaystyle-omega_0^{,2},x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img529.png"
 ALT="$\displaystyle - \omega_0^{ 2} x,$">|; 

$key = q/displaystyleG,M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img767.png"
 ALT="$\displaystyle G M$">|; 

$key = q/{displaymath}frac{partialF}{partialy'}=frac{y'}{sqrt{1+y'^{,2}}}=c,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img1746.png"
 ALT="\begin{displaymath}
\frac{\partial F}{\partial y'} = \frac{y'}{\sqrt{1+y'^{ 2}}} = c,
\end{displaymath}">|; 

$key = q/h=-kslashrho,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1768.png"
 ALT="$h=-k/\rho g$">|; 

$key = q/R=1.50times10^{11},{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2247.png"
 ALT="$R=1.50\times 10^{11} {\rm m}$">|; 

$key = q/{displaymath}frac{d}{dx}!left(frac{partialF}{partialy'}right)-frac{partialF}{partialy}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1735.png"
 ALT="\begin{displaymath}
\frac{d}{dx}\!\left(\frac{\partial F}{\partial y'}\right)-\frac{\partial F}{\partial y} = 0.
\end{displaymath}">|; 

$key = q/{displaymath}m,{{bf{a}'={{bf{f}-m,mbox{{boldmath{{Omega{}times(mbox{{boldmath{{Omega{}times{{bf{r}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1123.png"
 ALT="\begin{displaymath}
m {\bf a}' = {\bf f} - m \mbox{\boldmath$\Omega$}\times(\mbox{\boldmath$\Omega$}\times{\bf r}).
\end{displaymath}">|; 

$key = q/{displaymath}L=frac{1}{2},M,dot{x}^{,2}+frac{1}{2},m,(dot{x}^{,2}+2,dot{x},dot{xtheta+dot{x}'^{,2})+m,g,x',sintheta+{{rm{const}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="537" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1651.png"
 ALT="\begin{displaymath}
L = \frac{1}{2} M \dot{x}^{ 2} + \frac{1}{2} m (\dot{x}...
...os\theta+ \dot{x}'^{ 2})+ m g x' \sin\theta + {\rm const}.
\end{displaymath}">|; 

$key = q/Q_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3099.png"
 ALT="$Q_n$">|; 

$key = q/displaystylex_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img722.png"
 ALT="$\displaystyle x_c$">|; 

$key = q/ngeq3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img183.png"
 ALT="$n\geq 3$">|; 

$key = q/V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img507.png"
 ALT="$V$">|; 

$key = q/{displaymath}{{bf{r}_msimeqa_m,left(coslambda_m,,sinlambda_m,,iota_m,sin(lambda_m-varpi_n)right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="366" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2309.png"
 ALT="\begin{displaymath}
{\bf r}_m \simeq a_m \left(\cos\lambda_m, \sin\lambda_m, \iota_m \sin(\lambda_m-\varpi_n)\right),
\end{displaymath}">|; 

$key = q/{displaymath}frac{m_1,m_2}{m_1+m_2},frac{d^2{{bf{r}}{dt^2}=-frac{G,m_1,m_2}{r^3},{{bf{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img931.png"
 ALT="\begin{displaymath}
\frac{m_1 m_2}{m_1+m_2}  \frac{d^2{\bf r}}{dt^2} =-
\frac{G m_1 m_2}{r^3}  {\bf r},
\end{displaymath}">|; 

$key = q/sqrt{a^2+b^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1092.png"
 ALT="$\sqrt{a^2+b^2}$">|; 

$key = q/displaystyleleft[frac{(I_{z'z'}-I_{x'x'}),omega_{x'}}{I_{y'y'}}right]mu,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1539.png"
 ALT="$\displaystyle \left[\frac{(I_{z'z'}-I_{x'x'}) \omega_{x'}}{I_{y'y'}}\right]\mu,$">|; 

$key = q/displaystylefrac{d}{dt}!left(I_perp,sin^2theta,dot{phi}+I_{parallel},costheta,omegaright)+epsilon,beta_m,sin(2,theta),sin(varpi_n-phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="483" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2339.png"
 ALT="$\displaystyle \frac{d}{dt}\!\left(I_\perp \sin^2\theta \dot{\phi} + I_{\paral...
...theta \omega\right) + \epsilon \beta_m \sin(2 \theta) \sin(\varpi_n-\phi),$">|; 

$key = q/{displaymath}m,frac{d^2x}{dt^2}=f(x).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img288.png"
 ALT="\begin{displaymath}
m \frac{d^2 x}{d t^2} = f(x).
\end{displaymath}">|; 

$key = q/psi_{{rm{max}=m_2slashm_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1099.png"
 ALT="$\psi_{\rm max}=m_2/m_1$">|; 

$key = q/{displaymath}tilde{{bf{I},hat{mbox{{boldmath{{omega{}}_i=lambda_i,hat{mbox{{boldmath{{omega{}}_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1318.png"
 ALT="\begin{displaymath}
\tilde{\bf I}  \hat{\mbox{\boldmath$\omega$}}_i = \lambda_i \hat{\mbox{\boldmath$\omega$}}_i ,
\end{displaymath}">|; 

$key = q/y''';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1427.png"
 ALT="$y'''$">|; 

$key = q/{displaymath}Phi(r)=-frac{G,M}{a}left[1+frac{1}{4}left(frac{r}{a}right)^2+frac{9}{64}left(frac{r}{a}right)^4+cdotsright].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="384" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2389.png"
 ALT="\begin{displaymath}
\Phi(r) = - \frac{G M}{a}\left[1 + \frac{1}{4}\left(\frac{r...
...ght)^2 + \frac{9}{64}\left(\frac{r}{a}\right)^4+\cdots\right].
\end{displaymath}">|; 

$key = q/d=l,costheta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1691.png"
 ALT="$d=l \cos\theta_0$">|; 

$key = q/displaystyley_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2843.png"
 ALT="$\displaystyle y_j$">|; 

$key = q/omega_{y'}=omega,sinalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1456.png"
 ALT="$\omega_{y'}=\omega \sin\alpha$">|; 

$key = q/Phi(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2076.png"
 ALT="$\Phi(r,\theta)$">|; 

$key = q/sin(lambda_m-phi),sin(lambda_m-varpi_m);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2323.png"
 ALT="$\sin(\lambda_m-\phi) \sin(\lambda_m-\varpi_m)$">|; 

$key = q/r_2>r_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img783.png"
 ALT="$r_2&gt; r_1$">|; 

$key = q/stackrel{displaystylerightarrow}{AC};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3506.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{AC}$">|; 

$key = q/Omega=7.27times10^{-5},{{rm{rad.slashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2180.png"
 ALT="$\Omega = 7.27\times 10^{-5} {\rm rad./s}$">|; 

$key = q/l=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3178.png"
 ALT="$l=0$">|; 

$key = q/displaystyledeltaY,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2773.png"
 ALT="$\displaystyle \delta Y,$">|; 

$key = q/K+U;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img172.png"
 ALT="$K+U$">|; 

$key = q/displaystyle2,{mitOmega},sinlambda,dot{y}',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1149.png"
 ALT="$\displaystyle 2 {\mit\Omega} \sin\lambda \dot{y}',$">|; 

$key = q/v(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img246.png"
 ALT="$v(t)$">|; 

$key = q/displaystyle-frac{3}{2},m^2,sin[2,(1-m),T]-frac{3}{2},m^2,sin[2,(1-m),T],deltaX;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="417" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2791.png"
 ALT="$\displaystyle -\frac{3}{2} m^2 \sin[2 (1-m) T]-\frac{3}{2} m^2 \sin[2 (1-m) T] \delta X$">|; 

$key = q/{displaymath}I=int_a^bF(y_1,y_2,cdots,y_{calF},y_1',y_2',cdots,y_{calF}',x),dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1781.png"
 ALT="\begin{displaymath}
I = \int_a^b F(y_1,y_2,\cdots,y_{\cal F},y_1',y_2',\cdots,y_{\cal F}',x) dx.
\end{displaymath}">|; 

$key = q/B=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img569.png"
 ALT="$B=1$">|; 

$key = q/x=x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img219.png"
 ALT="$x=x_1$">|; 

$key = q/x_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1562.png"
 ALT="$x_j$">|; 

$key = q/r=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2140.png"
 ALT="$r=a$">|; 

$key = q/n=0,infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2104.png"
 ALT="$n=0,\infty$">|; 

$key = q/{{bf{e}_{x'''};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1429.png"
 ALT="${\bf e}_{x'''}$">|; 

$key = q/{displaymath}frac{v_t}{v_c}=sqrt{1+e}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img778.png"
 ALT="\begin{displaymath}
\frac{v_t}{v_c} = \sqrt{1+e}.
\end{displaymath}">|; 

$key = q/displaystylefrac{r}{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2965.png"
 ALT="$\displaystyle \frac{r}{a}$">|; 

$key = q/{displaymath}{{bf{L}=sum_{i=1,N}m_i,{{bf{r}_itimesfrac{d{{bf{r}_i}{dt}=sum_{i=1,}_icdotmbox{{boldmath{{omega{}),{{bf{r}_iright],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="597" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1241.png"
 ALT="\begin{displaymath}
{\bf L} = \sum_{i=1,N} m_i {\bf r}_i\times\frac{d{\bf r}_i}...
...}- ({\bf r}_i\cdot\mbox{\boldmath$\omega$}) {\bf r}_i\right],
\end{displaymath}">|; 

$key = q/{{bf{a}'={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="${\bf a}'= {\bf0}$">|; 

$key = q/displaystylemleft(array{{cc}1+alpha<col_mark>alpha<row_mark>alpha<col_mark>1+alphaarray{right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2036.png"
 ALT="$\displaystyle m\left(
\begin{array}{cc}
1+\alpha&amp; \alpha\\\\
\alpha&amp; 1+\alpha\end{array}\right),$">|; 

$key = q/y_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2724.png"
 ALT="$y_1$">|; 

$key = q/{{bf{x}_i^T{{bf{x}_j=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1310.png"
 ALT="${\bf x}_i^T{\bf x}_j = 0$">|; 

$key = q/F=sin(omega,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img514.png"
 ALT="$F = \sin(\omega t)$">|; 

$key = q/{displaymath}ddot{{bf{r}+2,mbox{{boldmath{{omega{}timesdot{{bf{r}+mbox{{boldmathr}|^{,3}}-frac{{{bf{r}'}{|{{bf{r}'|^{,3}}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="554" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2740.png"
 ALT="\begin{displaymath}
\ddot{\bf r}+ 2 \mbox{\boldmath$\omega$}\times\dot{\bf r} ...
...rt^{ 3}}
-\frac{{\bf r}'}{\vert{\bf r}'\vert^{ 3}}\right],
\end{displaymath}">|; 

$key = q/displaystyledot{phi}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2348.png"
 ALT="$\displaystyle \dot{\phi}_1$">|; 

$key = q/{displaymath}U=-m_1,g,x-m_2,g,(l-x).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1625.png"
 ALT="\begin{displaymath}
U = -m_1 g x - m_2 g (l-x).
\end{displaymath}">|; 

$key = q/{displaymath}frac{a_+-a_-}{a}=-epsilon=frac{15}{4},frac{m'}{m}left(frac{a}{R}right)^3.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2239.png"
 ALT="\begin{displaymath}
\frac{a_+-a_-}{a} = -\epsilon = \frac{15}{4} \frac{m'}{m}\left(\frac{a}{R}\right)^3.
\end{displaymath}">|; 

$key = q/j=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2912.png"
 ALT="$j=3$">|; 

$key = q/displaystylex',sintheta+y',costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img552.png"
 ALT="$\displaystyle x' \sin\theta + y' \cos\theta.$">|; 

$key = q/2pislashomega=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1477.png"
 ALT="$2\pi/\omega = 1$">|; 

$key = q/m,g,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1829.png"
 ALT="$m g \sin\theta$">|; 

$key = q/f_i=-k_i,dot{x}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1713.png"
 ALT="$f_i = -k_i \dot{x}_i$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter13slashfig13.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="239" BORDER="0"
 SRC="|."$dir".q|img2623.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter13/fig13.04.eps}}
\end{figure}">|; 

$key = q/displaystyley_0+deltay,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2635.png"
 ALT="$\displaystyle y_0 + \delta y,$">|; 

$key = q/{displaymath}y'simeq-g,{mitOmega},coslambda,frac{t^{,3}}{3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1160.png"
 ALT="\begin{displaymath}
y' \simeq - g {\mit\Omega} \cos\lambda \frac{t^{ 3}}{3}.
\end{displaymath}">|; 

$key = q/p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img895.png"
 ALT="$p$">|; 

$key = q/displaystyleT_{y'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1364.png"
 ALT="$\displaystyle T_{y'}$">|; 

$key = q/U(x_1,x_2,cdots,x_{calF},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1579.png"
 ALT="$U(x_1,x_2,\cdots,x_{\cal F},t)$">|; 

$key = q/{displaymath}{{bf{g}'=frac{2,G,m',a}{R^3},{{bf{e}_z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2256.png"
 ALT="\begin{displaymath}
{\bf g}' = \frac{2 G m' a}{R^3} {\bf e}_z.
\end{displaymath}">|; 

$key = q/gamma_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2305.png"
 ALT="$\gamma_m$">|; 

$key = q/k';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1984.png"
 ALT="$k'$">|; 

$key = q/{displaymath}x(t)={{rm{Re}left[a_+,{{rm{e}^{-{{rm{i},omega_+,t}+a_-,{{rm{e}^{-{{rm{i},omega_-,t}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img320.png"
 ALT="\begin{displaymath}
x(t) = {\rm Re}\left[a_+ {\rm e}^{-{\rm i} \omega_+ t}
+ a_- {\rm e}^{-{\rm i} \omega_- t}\right],
\end{displaymath}">|; 

$key = q/m'=7.35times10^{22},{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2243.png"
 ALT="$m'=7.35\times 10^{22} {\rm kg}$">|; 

$key = q/{displaymath}frac{dtanpsi}{dtheta}=frac{1+(m_1slashm_2),costheta}{(costheta+m_1slashm_2)^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1057.png"
 ALT="\begin{displaymath}
\frac{d\tan\psi}{d\theta} = \frac{1+(m_1/m_2) \cos\theta}{(\cos\theta+m_1/m_2)^2}.
\end{displaymath}">|; 

$key = q/a'=149,598,261,{{rm{km};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2699.png"
 ALT="$a'=149,598,261 {\rm km}$">|; 

$key = q/displaystylesum_{i=1,N}(y_i^{,2}+z_i^{,2}),m_i=int(y^2+z^2),dm,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1244.png"
 ALT="$\displaystyle \sum_{i=1,N}(y_i^{ 2}+z_i^{ 2})  m_i= \int(y^2+ z^2) dm,$">|; 

$key = q/{displaymath}frac{d}{dt}!left(frac{partialK}{partialdot{q}_i}right)-frac{partialK}{partialq_i}+frac{partialU}{partialq_i}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1892.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial K}{\partial \dot{q}_i}\ri...
...partial K}{\partial q_i}+ \frac{\partial U}{\partial q_i} = 0,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{Chapter15slashfig15.14.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="539" HEIGHT="402" BORDER="0"
 SRC="|."$dir".q|img3081.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{Chapter15/fig15.14.eps}}
\end{figure}">|; 

$key = q/{displaymath}frac{dsigma(psi)}{d{mitOmega}'}=frac{sintheta}{sinpsi},frac{dtheta}{dpsi},frac{dsigma(theta)}{d{mitOmega}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1082.png"
 ALT="\begin{displaymath}
\frac{d\sigma(\psi)}{d{\mit\Omega}'} = \frac{\sin\theta}{\si...
...,\frac{d\theta}{d\psi} \frac{d\sigma(\theta)}{d{\mit\Omega}}.
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{2},m_1,dot{x}^{,2}+frac{1}{2},m_2,(-dot{x}+dot{x}')^2+frac{1}{2},m_3,(-dot{x}-dot{x}')^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1635.png"
 ALT="$\displaystyle \frac{1}{2} m_1 \dot{x}^{ 2} + \frac{1}{2} m_2 (-\dot{x}+ \dot{x}')^2 + \frac{1}{2} m_3 (-\dot{x} - \dot{x}')^2$">|; 

$key = q/{mitOmega}_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1547.png"
 ALT="${\mit\Omega}_{x'}$">|; 

$key = q/varpi_m=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2360.png"
 ALT="$\varpi_m=0$">|; 

$key = q/{{bf{M};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1916.png"
 ALT="${\bf M}$">|; 

$key = q/displaystylea_z';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="$\displaystyle a_z'$">|; 

$key = q/{{bf{a},%,{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3247.png"
 ALT="${\bf a} \% {\bf b}$">|; 

$key = q/{displaymath}frac{alpha}{u}+frac{beta}{v}+frac{gamma}{w}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3535.png"
 ALT="\begin{displaymath}
\frac{\alpha}{u} +\frac{\beta}{v} + \frac{\gamma}{w} = 0.
\end{displaymath}">|; 

$key = q/Q=4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$Q=4$">|; 

$key = q/{displaymath}{mitOmega}=omega,cosalphaleft(frac{I_parallel}{I_perp}-1right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1397.png"
 ALT="\begin{displaymath}
{\mit\Omega} =\omega \cos\alpha \left(\frac{I_\parallel}{I_\perp}-1\right).
\end{displaymath}">|; 

$key = q/-frac{3}{4},m^2,x_5+frac{3}{4},m^2,y_5+frac{3}{2},y_4,x_5-frac{3}{2},y_5,x_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="317" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2890.png"
 ALT="$-\frac{3}{4} m^2 x_5
+\frac{3}{4} m^2 y_5+\frac{3}{2} y_4 x_5-\frac{3}{2} y_5 x_4$">|; 

$key = q/displaystylerho_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2586.png"
 ALT="$\displaystyle \rho_2$">|; 

$key = q/stackrel{displaystylerightarrow}{BA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2222.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{BA}$">|; 

$key = q/deltay=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1732.png"
 ALT="$\delta y=0$">|; 

$key = q/trightarrowt+tau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3091.png"
 ALT="$t\rightarrow t+\tau$">|; 

$key = q/{mitDelta}thetaslash2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="${\mit \Delta }\theta /2\pi $">|; 

$key = q/m^2,deltaX^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2786.png"
 ALT="$m^2 \delta X^{ 2}$">|; 

$key = q/{displaymath}{{bf{a}={{bf{a}'+mbox{{boldmath{{Omega{}times(mbox{{boldmath{{Omega{{bf{r})+2,mbox{{boldmath{{Omega{}times{{bf{v}'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1115.png"
 ALT="\begin{displaymath}
{\bf a} = {\bf a}' + \mbox{\boldmath$\Omega$}\times(\mbox{\b...
...a$}\times{\bf r}) + 2 \mbox{\boldmath$\Omega$}\times{\bf v}'.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{AppendixAslashfigA.25.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="305" BORDER="0"
 SRC="|."$dir".q|img3487.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{AppendixA/figA.25.eps}}
\end{figure}">|; 

$key = q/(dsigma(psi)slashd{mitOmega}'),d{mitOmega}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1076.png"
 ALT="$(d\sigma(\psi)/d{\mit\Omega}') d{\mit\Omega}'$">|; 

$key = q/{{bf{h};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img671.png"
 ALT="${\bf h}$">|; 

$key = q/{{bf{c}slashgamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3546.png"
 ALT="${\bf c}/\gamma$">|; 

$key = q/21.4,{{rm{km};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2185.png"
 ALT="$21.4 {\rm km}$">|; 

$key = q/{displaymath}ddot{lambda}+left[frac{(I_{y'y'}-I_{x'x'}),(I_{z'z'}-I_{x'x'})}{I_{y'y'},I_{z'z'}}right]omega_{x'}^{,2},lambda=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1543.png"
 ALT="\begin{displaymath}
\ddot{\lambda} + \left[\frac{(I_{y'y'}-I_{x'x'}) (I_{z'z'}-...
...})}{I_{y'y'} I_{z'z'}}\right]\omega_{x'}^{ 2}  \lambda = 0.
\end{displaymath}">|; 

$key = q/displaystylefrac{765}{128},m^3,e,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2943.png"
 ALT="$\displaystyle \frac{765}{128} m^3 e,$">|; 

$key = q/{displaymath}E=frac{1}{2},m,omega_0^{,2},a^2,cos^2(omega_0,t-phi_0)+frac{1}{2},m(omega_0,t-phi_0)=frac{1}{2},m,omega_0^{,2},a^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="587" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img305.png"
 ALT="\begin{displaymath}
E = \frac{1}{2} m \omega_0^{ 2} a^2 \cos^2(\omega_0 t-...
...n^2(\omega_0 t-\phi_0)
= \frac{1}{2} m \omega_0^{ 2} a^2,
\end{displaymath}">|; 

$key = q/{{bf{A}({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3406.png"
 ALT="${\bf A}({\bf r})$">|; 

$key = q/rho_1=rho_2=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2680.png"
 ALT="$\rho_1=\rho_2=1$">|; 

$key = q/{displaymath}1+costheta=2,cos^2(thetaslash2)=frac{2,(1-e),cos^2(Eslash2)}{1-e,cosE},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img813.png"
 ALT="\begin{displaymath}
1+\cos\theta = 2 \cos^2(\theta/2) = \frac{2 (1-e) \cos^2( E/2)}{1-e \cos E},
\end{displaymath}">|; 

$key = q/V=C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$V=C$">|; 

$key = q/displaystylefrac{d}{dt}!left(frac{partialL}{partialdot{r}}right)-frac{partialL}{partialr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1613.png"
 ALT="$\displaystyle \frac{d}{dt}\!\left(\frac{\partial L}{\partial\dot{r}}\right)- \frac{\partial L}{\partial r}$">|; 

$key = q/-{{bf{p}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img948.png"
 ALT="$-{\bf p}'$">|; 

$key = q/{displaymath}K=frac{1}{2}left(I_{xx},omega_x^{,2}+I_{yy},omega_y^{,2}+I_{zz},omega_z^{,2}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1327.png"
 ALT="\begin{displaymath}
K = \frac{1}{2}\left(I_{xx} \omega_x^{ 2} + I_{yy} \omega_y^{ 2}
+ I_{zz} \omega_z^{ 2}\right).
\end{displaymath}">|; 

$key = q/O;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1483.png"
 ALT="$O$">|; 

$key = q/mu_1=G,m_1simeqG,(m_1+m_2)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2523.png"
 ALT="$\mu_1=G m_1\simeq G (m_1+m_2)=1$">|; 

$key = q/m'slashm=(rho'slashrho),(a'slasha)^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2262.png"
 ALT="$m'/m = (\rho'/\rho) (a'/a)^3$">|; 

$key = q/displaystyle2,e,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2948.png"
 ALT="$\displaystyle 2 e,$">|; 

$key = q/U_x=U_y=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2643.png"
 ALT="$U_x=U_y=0$">|; 

$key = q/{displaymath}({{bf{G}-gamma^2,{{bf{M}),delta{bf{{q}={{bf{0}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1913.png"
 ALT="\begin{displaymath}
({\bf G} - \gamma^2 {\bf M}) \delta\bf {q} = {\bf0}.
\end{displaymath}">|; 

$key = q/v,{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3533.png"
 ALT="$v {\bf b}$">|; 

$key = q/displaystyleI_perp,(2,costheta,dot{theta},dot{phi}+sintheta,ddot{phi})-L_psi,dot{theta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1488.png"
 ALT="$\displaystyle I_\perp (2 \cos\theta \dot{\theta} \dot{\phi} +\sin\theta \ddot{\phi}) - L_\psi \dot{\theta},$">|; 

$key = q/displaystyle{{rm{RHS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3360.png"
 ALT="$\displaystyle {\rm RHS}$">|; 

$key = q/{displaymath}f(r,theta)=r-a=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1831.png"
 ALT="\begin{displaymath}
f(r,\theta) = r - a = 0.
\end{displaymath}">|; 

$key = q/{displaymath}deltaH=sum_{i=1,{calF}}left(dot{q}_i,deltap_i-dot{p}_i,deltaq_iright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1850.png"
 ALT="\begin{displaymath}
\delta H = \sum_{i=1,{\cal F}} \left(\dot{q}_i \delta p_i - \dot{p}_i \delta q_i\right).
\end{displaymath}">|; 

$key = q/{displaymath}T=2pi,sqrt{frac{a}{g}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2143.png"
 ALT="\begin{displaymath}
T = 2\pi \sqrt{\frac{a}{g}},
\end{displaymath}">|; 

$key = q/l_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1705.png"
 ALT="$l_0$">|; 

$key = q/Q=1.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$Q=1.5$">|; 

$key = q/{displaymath}{{bf{a}cdot{{bf{b}=|{{bf{a}|,|{{bf{b}|,costhetaleq|{{bf{a}|,|{{bf{b}|{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3511.png"
 ALT="\begin{displaymath}
{\bf a}\cdot{\bf b}= \vert{\bf a}\vert \vert{\bf b}\vert \cos\theta\leq \vert{\bf a}\vert \vert{\bf b}\vert
\end{displaymath}">|; 

$key = q/{displaymath}frac{q_1}{q_2}=frac{k}{k+k'+lambda,m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img2007.png"
 ALT="\begin{displaymath}
\frac{q_1}{q_2}= \frac{k}{k+k'+\lambda m}.
\end{displaymath}">|; 

$key = q/{displaymath}f(q_1,q_2,t)=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1795.png"
 ALT="\begin{displaymath}
f(q_1,q_2,t) = 0,
\end{displaymath}">|; 

$key = q/displaystyle-V_0,sintheta-2,{mitOmega},V_0,sinlambda,costheta,t.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1173.png"
 ALT="$\displaystyle - V_0 \sin\theta - 2 {\mit\Omega} V_0 \sin\lambda \cos\theta t.$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter03slashfig3.05.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="575" HEIGHT="249" BORDER="0"
 SRC="|."$dir".q|img375.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter03/fig3.05.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{(omega_j^{,2}+3+3,m^2slash2),b_j-2,omega_j,a_j}{omega_j^{,2},(1-3,m^2slash2-omega_j^{,2})},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2844.png"
 ALT="$\displaystyle \frac{(\omega_j^{ 2}+ 3+3 m^2/2) b_j - 2 \omega_j a_j}{\omega_j^{ 2} (1-3 m^2/2-\omega_j^{ 2})},$">|; 

$key = q/displaystylefrac{9}{16},m^3,iota.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2945.png"
 ALT="$\displaystyle \frac{9}{16} m^3 \iota.$">|; 

$key = q/{displaymath}n_s=frac{dlambda_s}{dt}=left(frac{G,M_s}{a_s^{,3}}right)^{1slash2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img2285.png"
 ALT="\begin{displaymath}
n_s =\frac{d\lambda_s}{dt}= \left(\frac{G M_s}{a_s^{ 3}}\right)^{1/2}
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{x}_k^T,{{bf{x}_k=m^{-1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2008.png"
 ALT="\begin{displaymath}
{\bf x}_k^T {\bf x}_k = m^{-1},
\end{displaymath}">|; 

$key = q/L_parallel;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img202.png"
 ALT="$L_\parallel$">|; 

$key = q/v^2=dot{x}^2+dot{y}^2+dot{z}^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2557.png"
 ALT="$v^2=\dot{x}^2+\dot{y}^2+\dot{z}^2$">|; 

$key = q/{displaymath}dot{{bf{e}_theta=dot{theta},(-costheta,,-sintheta)=-dot{theta},{{bf{e}_r.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img690.png"
 ALT="\begin{displaymath}
\dot{\bf e}_\theta = \dot{\theta} (-\cos\theta, -\sin\theta) = - \dot{\theta} {\bf e}_r.
\end{displaymath}">|; 

$key = q/{displaymath}L=frac{1}{2},m,dot{x}^{,2}+frac{1}{2},I,dot{phi}^{,2}+m,g,x,sintheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1819.png"
 ALT="\begin{displaymath}
L = \frac{1}{2} m \dot{x}^{ 2} + \frac{1}{2} I \dot{\phi}^{ 2} + m g x \sin\theta,
\end{displaymath}">|; 

$key = q/displaystyleleft(frac{G,M}{a^3}right)^{1slash2}t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img832.png"
 ALT="$\displaystyle \left(\frac{G M}{a^3}\right)^{1/2} t,$">|; 

$key = q/0.41;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2472.png"
 ALT="$0.41$">|; 

$key = q/z=r,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2213.png"
 ALT="$z=r \cos\theta$">|; 

$key = q/displaystyle{{bf{r}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img921.png"
 ALT="$\displaystyle {\bf r}_1$">|; 

$key = q/displaystyle-2,pi,G,r^nint_r^inftyint_0^pir'^{,1-n},rho(r',theta'),P_n(costheta'),sintheta',dr',dtheta'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="448" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2123.png"
 ALT="$\displaystyle -2 \pi G r^n\int_r^\infty \int_0^\pi r'^{ 1-n} 
\rho(r',\theta') P_n(\cos\theta') \sin\theta' dr' d\theta'.$">|; 

$key = q/{displaymath}m,g,l=-I_perp,costheta,dot{phi}^{,2}+L_psi,dot{phi},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1520.png"
 ALT="\begin{displaymath}
m g l = -I_\perp \cos\theta \dot{\phi}^{ 2}+ L_\psi \dot{\phi},
\end{displaymath}">|; 

$key = q/U(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img950.png"
 ALT="$U(r)$">|; 

$key = q/{displaymath}frac{dA}{dt}=lim_{deltatrightarrow0}frac{r^2,deltatheta}{2,delta{t}}=frac{r^2}{2},frac{dtheta}{dt}=frac{h}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img742.png"
 ALT="\begin{displaymath}
\frac{dA}{dt} = \lim_{\delta t\rightarrow 0}\frac{r^2 \delt...
... \delta{t}}= \frac{r^2}{2} \frac{d\theta}{dt} = \frac{h}{2}.
\end{displaymath}">|; 

$key = q/displaystyleI_{yz}=I_{zy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1251.png"
 ALT="$\displaystyle I_{yz}=I_{zy}$">|; 

$key = q/displaystyle{{bf{g};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1138.png"
 ALT="$\displaystyle {\bf g}$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{AppendixAslashfigA.13.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="247" BORDER="0"
 SRC="|."$dir".q|img3417.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{AppendixA/figA.13.eps}}
\end{figure}">|; 

$key = q/displaystyleleft.int_0^1x,y^2,dxright|_{y=0}+left.int_0^1x,y^2,dyright|_{x=1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3381.png"
 ALT="$\displaystyle \left.\int_0^1 x y^2 dx\right\vert _{y=0}+\left.\int_0^1 x y^2  dy
\right\vert _{x=1}$">|; 

$key = q/displaystyleJ_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2159.png"
 ALT="$\displaystyle J_2$">|; 

$key = q/E_1=(1slash2),m_1,v_1^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1037.png"
 ALT="$E_1=(1/2) m_1 v_1^{ 2}$">|; 

$key = q/displaystylev_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1013.png"
 ALT="$\displaystyle v_1'$">|; 

$key = q/x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img370.png"
 ALT="$x_1$">|; 

$key = q/displaystylefrac{d}{dt}!left(frac{partialL}{partialdot{q}_1}right)-frac{partialL}{partialq_1}-lambda(t),frac{partialf}{partialq_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1806.png"
 ALT="$\displaystyle \frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot{q}_1}\right)-\frac{\partial L}{\partial q_1} - \lambda(t) \frac{\partial f}{\partial q_1}$">|; 

$key = q/psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img869.png"
 ALT="$\psi$">|; 

$key = q/x''';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1426.png"
 ALT="$x'''$">|; 

$key = q/displaystyle-(I_{x'x'}-I_{y'y'}),omega_{x'},omega_{y'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1372.png"
 ALT="$\displaystyle - (I_{x'x'}-I_{y'y'}) \omega_{x'} \omega_{y'}.$">|; 

$key = q/yz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1260.png"
 ALT="$yz$">|; 

$key = q/|deltaX|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2776.png"
 ALT="$\vert\delta X\vert$">|; 

$key = q/displaystyleI_{xx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1243.png"
 ALT="$\displaystyle I_{xx}$">|; 

$key = q/{displaymath}deltadot{psi}simeqfrac{0.0383}{R,T}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2470.png"
 ALT="\begin{displaymath}
\delta\dot{\psi} \simeq \frac{0.0383}{R T}
\end{displaymath}">|; 

$key = q/dot{theta}=ddot{theta}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1519.png"
 ALT="$\dot{\theta}=\ddot{\theta}=0$">|; 

$key = q/Omega_3simeq2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2918.png"
 ALT="$\Omega_3\simeq 2$">|; 

$key = q/E_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img810.png"
 ALT="$E_n$">|; 

$key = q/{displaymath}dot{phi}simeq-frac{4,epsilon,alpha_s,costheta}{I_{parallel},omega}=Omega_phi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2295.png"
 ALT="\begin{displaymath}
\dot{\phi} \simeq
-\frac{4 \epsilon \alpha_s \cos\theta}{I_{\parallel} \omega} = \Omega_\phi,
\end{displaymath}">|; 

$key = q/v=dxslashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img231.png"
 ALT="$v=dx/dt$">|; 

$key = q/F=4.69;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3101.png"
 ALT="$F=4.69$">|; 

$key = q/{displaymath}I_perp,ddot{theta}-frac{partial{calL}}{partialtheta}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2290.png"
 ALT="\begin{displaymath}
I_\perp \ddot{\theta} - \frac{\partial {\cal L}}{\partial \theta} = 0.
\end{displaymath}">|; 

$key = q/{displaymath}E=K+U={{rm{constant}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="\begin{displaymath}
E = K + U = {\rm constant}.
\end{displaymath}">|; 

$key = q/eta_k(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1971.png"
 ALT="$\eta_k(t)$">|; 

$key = q/0.615;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2424.png"
 ALT="$0.615$">|; 

$key = q/displaystyleT_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1362.png"
 ALT="$\displaystyle T_{x'}$">|; 

$key = q/0leqtleqT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img429.png"
 ALT="$0\leq t\leq T$">|; 

$key = q/displaystylefrac{partialK}{partialdot{theta}}=m,r^2,dot{theta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1871.png"
 ALT="$\displaystyle \frac{\partial K}{\partial \dot{\theta}} = m r^2 \dot{\theta},$">|; 

$key = q/PQRS;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3194.png"
 ALT="$PQRS$">|; 

$key = q/{{bf{r}'=(x',,y',,z');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2054.png"
 ALT="${\bf r}' = (x', y', z')$">|; 

$key = q/phi_1simeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img381.png"
 ALT="$\phi_1\simeq 0$">|; 

$key = q/delta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3335.png"
 ALT="$\delta$">|; 

$key = q/rho({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2071.png"
 ALT="$\rho({\bf r})$">|; 

$key = q/displaystylevartheta_0left[omega_0^{,2}-omega^2-(1slash8),omega_0^{,2},vartheta_0^{,2}right]cos(omega,t)+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img474.png"
 ALT="$\displaystyle \vartheta_0\left[\omega_0^{ 2}-\omega^2 - (1/8) \omega_0^{ 2} \vartheta_0^{ 2} \right] \cos(\omega t) +$">|; 

$key = q/displaystylephi(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2342.png"
 ALT="$\displaystyle \phi(t)$">|; 

$key = q/m=n'slashn;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2980.png"
 ALT="$m=n'/n$">|; 

$key = q/{displaymath}frac{d}{dt}!left(frac{partial{calL}}{partialdot{theta}}right)-frac{partial{calL}}{partialtheta}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2289.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial{\cal L}}{\partial \dot{\theta}}\right)- \frac{\partial {\cal L}}{\partial\theta} = 0,
\end{displaymath}">|; 

$key = q/{displaymath}psi=pileft[3+r_c,frac{2,k,r_c^{-3}+4,epsilon,r_c^{-5}}{-k,r_c^{-2}-)}{1+epsilonslash(k,r_c^{,2})}right]^{-1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="546" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img880.png"
 ALT="\begin{displaymath}
\psi = \pi\left[3 + r_c \frac{2 k r_c^{-3} + 4 \epsilon\...
...ilon/(k r_c^{ 2})}{1+\epsilon/(k r_c^{ 2})}\right]^{-1/2}.
\end{displaymath}">|; 

$key = q/{displaymath}gamma=gamma_0-gamma_1,n',t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2990.png"
 ALT="\begin{displaymath}
\gamma = \gamma_0 - \gamma_1 n' t,
\end{displaymath}">|; 

$key = q/{displaymath}theta=pi-2int_0^{1slasha}frac{b,du}{sqrt{1-b^2,u^2}}=pi-2,sin^{-1}(bslasha).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img985.png"
 ALT="\begin{displaymath}
\theta = \pi - 2\int_0^{1/a}\frac{b du}{\sqrt{1-b^2 u^2}} = \pi - 2 \sin^{-1}(b/a).
\end{displaymath}">|; 

$key = q/displaystylex-u,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$\displaystyle x - u t,$">|; 

$key = q/29.46;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2436.png"
 ALT="$29.46$">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}=omega,(0,,sintheta,,costheta),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2276.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$}= \omega (0, \sin\theta, \cos\theta),
\end{displaymath}">|; 

$key = q/displaystyle-(I_{z'z'}-I_{x'x'}),omega_{z'},omega_{x'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1371.png"
 ALT="$\displaystyle - (I_{z'z'}-I_{x'x'}) \omega_{z'} \omega_{x'},$">|; 

$key = q/displaystyledot{p}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1855.png"
 ALT="$\displaystyle \dot{p}_i$">|; 

$key = q/r<r';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2117.png"
 ALT="$r &lt; r'$">|; 

$key = q/6,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1473.png"
 ALT="$6 {\rm m}$">|; 

$key = q/a_e=a,(1+epsilonslash3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2176.png"
 ALT="$a_e = a (1+\epsilon/3)$">|; 

$key = q/displaystyleY;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2755.png"
 ALT="$\displaystyle Y$">|; 

$key = q/{displaymath}U(r)=frac{q^2}{4pi,epsilon_0,r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1003.png"
 ALT="\begin{displaymath}
U(r) = \frac{q^2}{4\pi \epsilon_0 r}.
\end{displaymath}">|; 

$key = q/y(pmaslash2)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1774.png"
 ALT="$y(\pm a/2)= 0$">|; 

$key = q/f_r=-m,partialPhislashpartialr=-(G,m,Mslasha^3),r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2142.png"
 ALT="$f_r = -m \partial\Phi/\partial r = -(G m M/a^3) r$">|; 

$key = q/{displaymath}V(u)=-h^2left(2,r_0,u^3+frac{u^2}{2}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img843.png"
 ALT="\begin{displaymath}
V(u) = -h^2\left(2 r_0 u^3 + \frac{u^2}{2}\right),
\end{displaymath}">|; 

$key = q/displaystyledot{x}_jleft(sum_{k=1,{calF}}frac{partial^2x_j}{partialq_i,partialq_k},dot{q}_k+frac{partial^2x_j}{partialq_i,partialt}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1596.png"
 ALT="$\displaystyle \dot{x}_j\left(\sum_{k=1,{\cal F}}\frac{\partial^2 x_j}{\partial ...
...rtial q_k} \dot{q}_k +
\frac{\partial^2 x_j}{\partial q_i \partial t}\right)$">|; 

$key = q/{displaymath}frac{mu_2}{mu_1}=frac{(1-rho_1^{,3}),(1+rho_1)^2}{rho_1^{,3},(rho_1^{,2}+3,rho_1+3)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2590.png"
 ALT="\begin{displaymath}
\frac{\mu_2}{\mu_1} = \frac{(1-\rho_1^{ 3}) (1+\rho_1)^2}{\rho_1^{ 3} (\rho_1^{ 2}+3 \rho_1+3)}.
\end{displaymath}">|; 

$key = q/{displaymath}R=frac{v_0,v_t,costheta}{g}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img604.png"
 ALT="\begin{displaymath}
R= \frac{v_0 v_t \cos\theta}{g}
\end{displaymath}">|; 

$key = q/{displaymath}dot{x}_j=sum_{i=1,{calF}}frac{partialx_j}{partialq_i},dot{q}_i+frac{partialx_j}{partialt},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1586.png"
 ALT="\begin{displaymath}
\dot{x}_j= \sum_{i=1,{\cal F}} \frac{\partial x_j}{\partial q_i} \dot{q}_i
+ \frac{\partial x_j}{\partial t},
\end{displaymath}">|; 

$key = q/displaystyleh,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1619.png"
 ALT="$\displaystyle h,$">|; 

$key = q/-frac{3}{4},iota^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2895.png"
 ALT="$-\frac{3}{4} \iota^2$">|; 

$key = q/{{bf{e}_xtimes{{bf{e}_y={{bf{e}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3307.png"
 ALT="${\bf e}_x\times
{\bf e}_y={\bf e}_z$">|; 

$key = q/s=4,A,sinphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img660.png"
 ALT="$s=4 A \sin\phi$">|; 

$key = q/rho_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2577.png"
 ALT="$\rho_2$">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}_psi=dot{psi},{{bf{e}_{z'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1440.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$}_\psi= \dot{\psi} {\bf e}_{z'}.
\end{displaymath}">|; 

$key = q/{displaymath}g_{{rm{eff}=frac{G,m}{a^2}left(1-2,frac{m'}{m},frac{a^3}{R^3}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2258.png"
 ALT="\begin{displaymath}
g_{\rm eff} = \frac{G m}{a^2}\left(1- 2 \frac{m'}{m} \frac{a^3}{R^3}\right).
\end{displaymath}">|; 

$key = q/2.45times10^{-6};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2423.png"
 ALT="$2.45\times 10^{-6}$">|; 

$key = q/{displaymath}ddot{{bf{r}+2,mbox{{boldmath{{omega{}timesdot{{bf{r}=-mu_1,frac{({{ega{}times(mbox{{boldmath{{omega{}times{{bf{r}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="465" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2536.png"
 ALT="\begin{displaymath}
\ddot{\bf r} + 2 \mbox{\boldmath$\omega$}\times \dot{\bf r}...
...ldmath$\omega$}\times(\mbox{\boldmath$\omega$}\times {\bf r}),
\end{displaymath}">|; 

$key = q/x=a_n,exp(-{{rm{i},n,omega,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img417.png"
 ALT="$x = a_n \exp(-{\rm i} n \omega t)$">|; 

$key = q/h(x,,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3432.png"
 ALT="$h(x, y)$">|; 

$key = q/T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img393.png"
 ALT="$T$">|; 

$key = q/{displaymath}left(m_1+m_2+frac{I}{a^2}right)ddot{x}-g,(m_1-m_2)=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1628.png"
 ALT="\begin{displaymath}
\left(m_1+ m_2+\frac{I}{a^2}\right)\ddot{x} - g (m_1-m_2) = 0,
\end{displaymath}">|; 

$key = q/oint{{bf{A}cdotd{{bf{r}neq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3475.png"
 ALT="$\oint {\bf A}\cdot d{\bf r} \neq 0$">|; 

$key = q/displaystylealpha_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1966.png"
 ALT="$\displaystyle \alpha_k$">|; 

$key = q/{displaymath}{{bf{r}_s=a_s,(coslambda_s,,sinlambda_s,,0).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2277.png"
 ALT="\begin{displaymath}
{\bf r}_s = a_s (\cos\lambda_s, \sin\lambda_s, 0).
\end{displaymath}">|; 

$key = q/{displaymath}gamma_1=frac{3}{4},m-frac{9}{32},m^2+{calO}(m^3).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2991.png"
 ALT="\begin{displaymath}
\gamma_1 = \frac{3}{4} m - \frac{9}{32} m^2 + {\cal O}(m^3).
\end{displaymath}">|; 

$key = q/{{bf{v}=d{{bf{r}slashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="${\bf v} = d{\bf r}/dt$">|; 

$key = q/{displaymath}frac{d}{dt}!left(frac{partialL}{partialdot{theta}}right)-frac{partialL}{partialtheta}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1681.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot{\theta}}\right) - \frac{\partial L}{\partial\theta} = 0,
\end{displaymath}">|; 

$key = q/displaystylea_{y'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3242.png"
 ALT="$\displaystyle a_{y'}$">|; 

$key = q/b>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img698.png"
 ALT="$b&gt;0$">|; 

$key = q/{displaymath}theta=pi-2,{mitTheta}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img960.png"
 ALT="\begin{displaymath}
\theta = \pi - 2 {\mit\Theta}.
\end{displaymath}">|; 

$key = q/rho_a,A,v^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img278.png"
 ALT="$\rho_a A v^2$">|; 

$key = q/displaystylex_0+deltax,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2634.png"
 ALT="$\displaystyle x_0 + \delta x,$">|; 

$key = q/ddot{x}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1642.png"
 ALT="$\ddot{x}'$">|; 

$key = q/x_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img724.png"
 ALT="$x_c$">|; 

$key = q/6.55;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2454.png"
 ALT="$6.55$">|; 

$key = q/eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2486.png"
 ALT="$\eta$">|; 

$key = q/I';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2529.png"
 ALT="$I'$">|; 

$key = q/2+(c+g),m^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2861.png"
 ALT="$2+(c+g) m^2$">|; 

$key = q/displaystylec_x,costheta+c_y,sintheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3299.png"
 ALT="$\displaystyle c_x \cos\theta
+c_y \sin\theta.$">|; 

$key = q/{displaymath}chi+Phi'=-frac{G,m'}{R^3},(z^2-x^2slash2-y^2slash2)+{{rm{const}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2253.png"
 ALT="\begin{displaymath}
\chi+\Phi' = - \frac{G m'}{R^3} (z^2-x^2/2-y^2/2) + {\rm const}.
\end{displaymath}">|; 

$key = q/displaystyleI_perp,ddot{theta}_1+I_{parallel},sintheta_0,omega,dot{phi}_1+2,alpha,sin(2,theta_0)+2,beta_m,cos(2,theta_0),cos(Omega_n,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="534" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2345.png"
 ALT="$\displaystyle I_\perp \ddot{\theta}_1 + I_{\parallel} \sin\theta_0 \omega \...
... \alpha \sin(2 \theta_0) + 2 \beta_m \cos(2 \theta_0) \cos(\Omega_n t),$">|; 

$key = q/{displaymath}frac{dsigma}{dOmega'}simeqfrac{q_1^{,2},q_2^{,2}}{4,(4piepsilon_0)^ft[1-(psislashpsi_{{rm{max})^2right]^{1slash2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="512" HEIGHT="62" BORDER="0"
 SRC="|."$dir".q|img1098.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d\Omega'}\simeq \frac{q_1^{ 2} q_2^{ 2}}{4...
...rm max})^2}]^2 \left[1-(\psi/\psi_{\rm max})^2\right]^{1/2}},
\end{displaymath}">|; 

$key = q/displaystyleddot{z}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1152.png"
 ALT="$\displaystyle \ddot{z}'$">|; 

$key = q/v_perp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img637.png"
 ALT="$v_\perp$">|; 

$key = q/displaystyle(x+mu_2)^2+y^2+z^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2537.png"
 ALT="$\displaystyle (x+\mu_2)^2+y^2 + z^2,$">|; 

$key = q/omega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img300.png"
 ALT="$\omega_0$">|; 

$key = q/displaystyledeltaddot{y}+2,deltadot{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2647.png"
 ALT="$\displaystyle \delta\ddot{y} + 2 \delta\dot{x}$">|; 

$key = q/{{bf{q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3536.png"
 ALT="${\bf q}$">|; 

$key = q/displaystylefrac{partialU}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2568.png"
 ALT="$\displaystyle \frac{\partial U}{\partial y}$">|; 

$key = q/{displaymath}K=frac{1}{2}left(I_perp,dot{theta}^2+I_perp,sin^2theta,dot{phi}^2+I_parallel,omega^2right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2286.png"
 ALT="\begin{displaymath}
K = \frac{1}{2}\left(I_\perp \dot{\theta}^2 + I_\perp \sin^2\theta \dot{\phi}^2 + I_\parallel \omega^2\right),
\end{displaymath}">|; 

$key = q/{displaymath}G=6.67300times10^{-11},{{rm{m^3,kg^{-1},s^{-2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img664.png"
 ALT="\begin{displaymath}
G = 6.67300\times 10^{-11} {\rm m^3 kg^{-1} s^{-2}}.
\end{displaymath}">|; 

$key = q/dFslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1742.png"
 ALT="$dF/dx$">|; 

$key = q/-infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img828.png"
 ALT="$-\infty$">|; 

$key = q/m_1slashm_2=0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img941.png"
 ALT="$m_1/m_2=0.5$">|; 

$key = q/displaystyle{{bf{V}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1031.png"
 ALT="$\displaystyle {\bf V}_1$">|; 

$key = q/displaystylev_parallel,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img641.png"
 ALT="$\displaystyle v_\parallel t,$">|; 

$key = q/{displaymath}-frac{h^2}{r_c^{,3}}=f(r_c),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img848.png"
 ALT="\begin{displaymath}
-\frac{h^2}{r_c^{ 3}} = f(r_c),
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}cdot{{bf{b}times{{bf{c}=-{{bf{a}cdot{{bf{c}times{{bf{b}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3341.png"
 ALT="\begin{displaymath}
{\bf a} \cdot {\bf b}\times{\bf c} = - {\bf a} \cdot {\bf c}\times{\bf b}.
\end{displaymath}">|; 

$key = q/{displaymath}phi_1=tan^{-1}left(frac{2,nu,omega}{omega_0^{,2}-omega^2}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img374.png"
 ALT="\begin{displaymath}
\phi_1 = \tan^{-1}\left(\frac{2 \nu \omega}{\omega_0^{ 2}-\omega^2}\right).
\end{displaymath}">|; 

$key = q/{displaymath}frac{1}{3}left(frac{2,R}{g}right)^{1slash2},left[left(1+frac{h}{R}right)^{3slash2}-1right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img489.png"
 ALT="\begin{displaymath}
\frac{1}{3}\left(\frac{2 R}{g}\right)^{1/2} \left[\left(1+\frac{h}{R}\right)^{3/2}-1\right].
\end{displaymath}">|; 

$key = q/{displaymath}frac{d{{bf{L}}{dt}={{bf{T},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img170.png"
 ALT="\begin{displaymath}
\frac{d{\bf L}}{dt} = {\bf T},
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=1.in{centerline{epsffile{Chapter12slashfig12.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="456" HEIGHT="114" BORDER="0"
 SRC="|."$dir".q|img2204.png"
 ALT="\begin{figure}
\epsfysize =1.in
\centerline{\epsffile{Chapter12/fig12.03.eps}}
\end{figure}">|; 

$key = q/{displaymath}-c,r_c^{,n}-frac{c,n}{3},r_c^{,n}<0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img859.png"
 ALT="\begin{displaymath}
-c r_c^{ n} -\frac{c n}{3} r_c^{ n} &lt; 0,
\end{displaymath}">|; 

$key = q/m=n'slashn=0.07480;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2769.png"
 ALT="$m=n'/n=0.07480$">|; 

$key = q/{displaymath}{{bf{T}=sum_{i=1,N}{{bf{r}_itimes{{bf{F}_i{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img171.png"
 ALT="\begin{displaymath}
{\bf T} = \sum_{i=1,N} {\bf r}_i\times {\bf F}_i
\end{displaymath}">|; 

$key = q/Q=2.13;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$Q=2.13$">|; 

$key = q/+pislash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img298.png"
 ALT="$+\pi/4$">|; 

$key = q/displaystyle-frac{partialH}{partialq_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1856.png"
 ALT="$\displaystyle -\frac{\partial H}{\partial q_i},$">|; 

$key = q/{displaymath}H=K+U=frac{1}{2},frac{p^{,2}}{m}+frac{1}{2},k,x^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1861.png"
 ALT="\begin{displaymath}
H = K + U = \frac{1}{2} \frac{p^{ 2}}{m} + \frac{1}{2} k x^2.
\end{displaymath}">|; 

$key = q/{displaymath}m,ddot{{bf{r}'=m,{{bf{g}+{{bf{T}-2,m,mbox{{boldmath{{Omega{}timesdot{{bf{r}'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1180.png"
 ALT="\begin{displaymath}
m \ddot{\bf r}' = m {\bf g} + {\bf T} - 2 m \mbox{\boldmath$\Omega$}\times \dot{\bf r}'.
\end{displaymath}">|; 

$key = q/displaystyleI_perp,(ddot{theta}-costheta,sintheta,dot{phi}^{,2})+L_psi,sintheta,dot{phi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1487.png"
 ALT="$\displaystyle I_\perp (\ddot{\theta}- \cos\theta \sin\theta \dot{\phi}^{ 2}) + L_\psi \sin\theta \dot{\phi},$">|; 

$key = q/displaystyle-frac{v}{Q}-theta+A,cos(omega,t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3043.png"
 ALT="$\displaystyle -\frac{v}{Q} -\theta + A \cos(\omega  t).$">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}=omega_{x'},{{bf{e}_{x'}+lambda,{{bf{e}_{y'}+mu,{{bf{e}_{z'},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1532.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$} = \omega_{x'} {\bf e}_{x'} + \lambda {\bf e}_{y'} + \mu {\bf e}_{z'},
\end{displaymath}">|; 

$key = q/{displaymath}frac{1}{2},I_perp,sin(2,theta),dot{phi}^{,2}-I_parallel,sintheta,omega,dot{phi}-2,epsilon,alpha_s,sin(2,theta)=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="424" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2293.png"
 ALT="\begin{displaymath}
\frac{1}{2} I_\perp \sin(2 \theta) \dot{\phi}^{ 2} - I_...
...mega \dot{\phi} - 2 \epsilon \alpha_s \sin(2 \theta) = 0,
\end{displaymath}">|; 

$key = q/{displaymath}M=m_1+m_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img933.png"
 ALT="\begin{displaymath}
M = m_1+ m_2.
\end{displaymath}">|; 

$key = q/P_n(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2102.png"
 ALT="$P_n(x)$">|; 

$key = q/theta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1515.png"
 ALT="$\theta_1$">|; 

$key = q/r_1=mu_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2496.png"
 ALT="$r_1 = \mu_2$">|; 

$key = q/d^3{{bf{r}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2067.png"
 ALT="$d^3{\bf r}'$">|; 

$key = q/displaystyle-dot{y},frac{partialU}{partialy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2552.png"
 ALT="$\displaystyle -\dot{y} \frac{\partial U}{\partial y},$">|; 

$key = q/dot{x}_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1590.png"
 ALT="$\dot{x}_j$">|; 

$key = q/x=-mu_2+rho_1=mu_1+rho_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2581.png"
 ALT="$x = -\mu_2+\rho_1=\mu_1+\rho_2$">|; 

$key = q/displaystyle+frac{1}{omega_ast}left{deltav_0+frac{deltatheta_0}{2Q}right}sin(omega_ast,{mitDelta}t),,{{rm{e}^{-{mitDelta}tslash2Q},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3141.png"
 ALT="$\displaystyle + \frac{1}{\omega_\ast}\left\{\delta v_0 + \frac{\delta\theta_0}{2Q}\right\}
\sin(\omega_\ast  {\mit\Delta} t)  {\rm e}^{-{\mit\Delta} t/2Q},$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{AppendixAslashfigA.02b.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="190" BORDER="0"
 SRC="|."$dir".q|img3216.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.02b.eps}}
\end{figure}">|; 

$key = q/1slashe;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img509.png"
 ALT="$1/e$">|; 

$key = q/displaystyleomega_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2005.png"
 ALT="$\displaystyle \omega_2$">|; 

$key = q/displaystylea_x,b_x+a_y,b_y+a_z,b_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3259.png"
 ALT="$\displaystyle a_x  b_x + a_y  b_y + a_z  b_z$">|; 

$key = q/{displaymath}p=frac{partialK}{partialdot{x}}=frac{partialL}{partialdot{x}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1658.png"
 ALT="\begin{displaymath}
p = \frac{\partial K}{\partial \dot{x}}= \frac{\partial L}{\partial\dot{x}},
\end{displaymath}">|; 

$key = q/{displaymath}deltaW=sum_{j=1,{calF}}f_j,deltax_j{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1569.png"
 ALT="\begin{displaymath}
\delta W = \sum_{j=1,{\cal F}} f_j \delta x_j
\end{displaymath}">|; 

$key = q/{displaymath}alpha=alpha_0+alpha_1,n',t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img2979.png"
 ALT="\begin{displaymath}
\alpha = \alpha_0 + \alpha_1 n' t,
\end{displaymath}">|; 

$key = q/{{bf{v}_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1009.png"
 ALT="${\bf v}_1'$">|; 

$key = q/{displaymath}frac{d}{dx}!left(y',frac{partialF}{partialy'}right)=frac{dF}{dx},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1743.png"
 ALT="\begin{displaymath}
\frac{d}{dx}\!\left(y' \frac{\partial F}{\partial y'}\right) = \frac{dF}{dx},
\end{displaymath}">|; 

$key = q/{{bf{e}_{z'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1436.png"
 ALT="${\bf e}_{z'}$">|; 

$key = q/{displaymath}frac{dh}{dr}=left(frac{dh}{dr}right)_{{rm{max},costheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3436.png"
 ALT="\begin{displaymath}
\frac{dh}{dr}= \left(\frac{dh}{dr}\right)_{\rm max}  \cos\theta,
\end{displaymath}">|; 

$key = q/pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img875.png"
 ALT="$\pi/2$">|; 

$key = q/f(x)=-k,x^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img491.png"
 ALT="$f(x) = - k x^{-2}$">|; 

$key = q/cosbeta=sinalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1336.png"
 ALT="$\cos\beta=\sin\alpha$">|; 

$key = q/theta>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3074.png"
 ALT="$\theta&gt;0$">|; 

$key = q/Q=1.36;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="$Q=1.36$">|; 

$key = q/displaystylefrac{mu_2}{mu_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2592.png"
 ALT="$\displaystyle \frac{\mu_2}{\mu_1}$">|; 

$key = q/displaystyle{{bf{x}_i^T,{{bf{A},{{bf{x}_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1306.png"
 ALT="$\displaystyle {\bf x}_i^T {\bf A} {\bf x}_j$">|; 

$key = q/M_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2706.png"
 ALT="$M_M$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.28.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="274" BORDER="0"
 SRC="|."$dir".q|img3146.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.28.eps}}
\end{figure}">|; 

$key = q/{displaymath}U=m,g,z=-m,g,l,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1675.png"
 ALT="\begin{displaymath}
U = m g z= - m g l \cos\theta.
\end{displaymath}">|; 

$key = q/v(t=0)=v_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img245.png"
 ALT="$v(t=0)=v_0$">|; 

$key = q/a';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2251.png"
 ALT="$a'$">|; 

$key = q/Q=3.9;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$Q=3.9$">|; 

$key = q/displaystyleX_0+deltaX,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2772.png"
 ALT="$\displaystyle X_0 + \delta X,$">|; 

$key = q/a_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3221.png"
 ALT="$a_z$">|; 

$key = q/{displaymath}frac{d{{bf{a}}{dt}=lim_{deltatrightarrow0}left[frac{{{bf{a}(t+deltat)-{{bf{a}(t)}{deltat}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3364.png"
 ALT="\begin{displaymath}
\frac{d {\bf a}}{dt} = \lim_{\delta t\rightarrow 0} \left[\frac{{\bf a}(t+\delta t) - {\bf a}(t)}
{\delta t}\right].
\end{displaymath}">|; 

$key = q/displaystylex',costheta-y',sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img551.png"
 ALT="$\displaystyle x' \cos\theta - y' \sin \theta,$">|; 

$key = q/(m,l^2,omegaslash12),sinalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1557.png"
 ALT="$(m l^2 \omega/12) \sin\alpha$">|; 

$key = q/displaystylem_1,ddot{x}+m_2,(ddot{x}-ddot{x}')+m_3,(ddot{x}+ddot{x}')-g,(m_1-m_2-m_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1639.png"
 ALT="$\displaystyle m_1 \ddot{x} + m_2 (\ddot{x}- \ddot{x}') + m_3 (\ddot{x}+\ddot{x}')
- g (m_1-m_2-m_3)$">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}=lim_{deltatrightarrow0}frac{deltambox{{boldmath{{theta{}}{deltat},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3336.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$} = \lim_{\delta t\rightarrow 0} \frac{\delta
\mbox{\boldmath$\theta$} }{\delta t},
\end{displaymath}">|; 

$key = q/L_psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1494.png"
 ALT="$L_\psi$">|; 

$key = q/nu_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img333.png"
 ALT="$\nu_+$">|; 

$key = q/r_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img709.png"
 ALT="$r_1$">|; 

$key = q/{displaymath}costheta=-frac{m_2}{m_1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img1059.png"
 ALT="\begin{displaymath}
\cos\theta = -\frac{m_2}{m_1}.
\end{displaymath}">|; 

$key = q/{displaymath}x(t)=sum_{n=0}^inftyX_n,a_n,{{rm{e}^{-{{rm{i},n,omega,t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="61" BORDER="0"
 SRC="|."$dir".q|img414.png"
 ALT="\begin{displaymath}
x(t) = \sum_{n=0}^\infty X_n a_n {\rm e}^{-{\rm i} n \omega t},
\end{displaymath}">|; 

$key = q/K=I+lambda,J;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1750.png"
 ALT="$K = I + \lambda J$">|; 

$key = q/{displaymath}costheta=frac{cosE-e}{1-e,cosE}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img812.png"
 ALT="\begin{displaymath}
\cos\theta = \frac{\cos E - e}{1-e \cos E}.
\end{displaymath}">|; 

$key = q/{{bf{a}'=d^2{{bf{r}slashdt'^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1117.png"
 ALT="${\bf a}'= d^2{\bf r}/dt'^2$">|; 

$key = q/{displaymath}K=sum_{i,j=1,{calF}}m_{ij},dot{q}_i,dot{q}_j,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1841.png"
 ALT="\begin{displaymath}
K = \sum_{i,j = 1,{\cal F}} m_{ij} \dot{q}_i \dot{q}_j,
\end{displaymath}">|; 

$key = q/q_islashm_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img198.png"
 ALT="$q_i/m_i$">|; 

$key = q/Q_{n}-Q_{n-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3107.png"
 ALT="$Q_{n} - Q_{n-1}$">|; 

$key = q/{displaymath}frac{d}{dt}!left(frac{partialL}{partialdot{q}_i}right)-frac{partialtialq_i}-lambda(t),frac{partialf}{partialq_i}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1815.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot{q}_i}\ri...
...artial q_i} - \lambda(t) \frac{\partial f}{\partial q_i} = 0,
\end{displaymath}">|; 

$key = q/displaystylesqrt{frac{m}{2}},(q_1-q_2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2047.png"
 ALT="$\displaystyle \sqrt{\frac{m}{2}} (q_1-q_2),$">|; 

$key = q/{displaymath}psi=frac{T}{2},dot{theta}=pileft[3+r_c,frac{f'(r_c)}{f(r_c)}right]^{-1slash2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img870.png"
 ALT="\begin{displaymath}
\psi = \frac{T}{2}  \dot{\theta} = \pi \left[3+r_c \frac{f'(r_c)}{f(r_c)}\right]^{-1/2}
\end{displaymath}">|; 

$key = q/displaystylefrac{r}{a}-1+frac{m^2}{6};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2798.png"
 ALT="$\displaystyle \frac{r}{a}-1+\frac{m^2}{6}$">|; 

$key = q/{{bf{a}times{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3300.png"
 ALT="${\bf a}\times{\bf b}$">|; 

$key = q/displaystyleddot{r}-r,dot{theta}^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1617.png"
 ALT="$\displaystyle \ddot{r} - r \dot{\theta}^{ 2}$">|; 

$key = q/n>N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img432.png"
 ALT="$n&gt;N$">|; 

$key = q/displaystyley_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2908.png"
 ALT="$\displaystyle y_2$">|; 

$key = q/{{bf{V}_2'={{bf{v}_2'-{{bf{v}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1020.png"
 ALT="${\bf V}_2' = {\bf v}_2'-{\bf v}_2$">|; 

$key = q/displaystylelambda_j,{{bf{x}_i^T{{bf{x}_j.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1308.png"
 ALT="$\displaystyle \lambda_j {\bf x}_i^T{\bf x}_j.$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter02slashfig2.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="407" HEIGHT="254" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter02/fig2.01.eps}}
\end{figure}">|; 

$key = q/{displaymath}Q_{i,0}=-frac{partialU(q_{1,0},q_{2,0},cdots,q_{{calF},0})}{partialq_i}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1895.png"
 ALT="\begin{displaymath}
Q_{i 0} = - \frac{\partial U(q_{1 0}, q_{2 0},\cdots,q_{{\cal F} 0})}{\partial q_i} = 0,
\end{displaymath}">|; 

$key = q/Phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2061.png"
 ALT="$\Phi$">|; 

$key = q/{displaymath}{{bf{grad},(phi,psi)=phi,,{{bf{grad},psi+psi,,{{bf{grad},phi{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3478.png"
 ALT="\begin{displaymath}
{\bf grad} (\phi  \psi) = \phi   {\bf grad} \psi +\psi   {\bf grad} \phi
\end{displaymath}">|; 

$key = q/theta(0)slashA=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$\theta (0)/A=0$">|; 

$key = q/{displaymath}l=int_A^B[dx^2+dy^2]^{1slash2}=int_a^b[1+y'^{,2}(x)]^{1slash2},dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1717.png"
 ALT="\begin{displaymath}
l = \int_A^B [dx^2 + dy^2]^{1/2} = \int_a^b [1 + y'^{ 2}(x)]^{1/2} dx,
\end{displaymath}">|; 

$key = q/displaystyle(m_1slashm_2),v_1,(1-costheta,,-sintheta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1036.png"
 ALT="$\displaystyle (m_1/m_2) v_1 (1-\cos\theta, -\sin\theta).$">|; 

$key = q/{displaymath}alphasimeq-frac{omega_0^{,2}}{192}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img480.png"
 ALT="\begin{displaymath}
\alpha\simeq -\frac{\omega_0^{ 2}}{192}.
\end{displaymath}">|; 

$key = q/displaystylefrac{d}{dt}!left(frac{partialL}{partialdot{q}_2}right)-frac{partialL}{partialq_2}-lambda(t),frac{partialf}{partialq_2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1807.png"
 ALT="$\displaystyle \frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot{q}_2}\right)-\frac{\partial L}{\partial q_2} - \lambda(t) \frac{\partial f}{\partial q_2}$">|; 

$key = q/Phi({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2070.png"
 ALT="$\Phi({\bf r})$">|; 

$key = q/mbox{{boldmath{{omega{}'=n',{{bf{e}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2716.png"
 ALT="$\mbox{\boldmath$\omega$}'=n' {\bf e}_z$">|; 

$key = q/{displaymath}Q_infty=Q_1+(Q_2-Q_1)sum_{j=0}^{infty}frac{1}{F^j}equivQ_1+(Q_2-Q_1),frac{F}{F-1}=1.3758.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="526" HEIGHT="63" BORDER="0"
 SRC="|."$dir".q|img3105.png"
 ALT="\begin{displaymath}
Q_\infty = Q_1 + (Q_2-Q_1)\sum_{j=0}^{\infty}\frac{1}{F^j} \equiv Q_1 + (Q_2-Q_1) \frac{F}{F-1}=
1.3758.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter04slashfig4.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="433" HEIGHT="359" BORDER="0"
 SRC="|."$dir".q|img567.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter04/fig4.01.eps}}
\end{figure}">|; 

$key = q/{displaymath}{{bf{L}=left(I_{xx},omega_x,,I_{yy},omega_y,I_{zz},omega_zright),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1326.png"
 ALT="\begin{displaymath}
{\bf L} = \left(I_{xx} \omega_x, I_{yy} \omega_y,I_{zz} \omega_z\right),
\end{displaymath}">|; 

$key = q/n,n'>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img407.png"
 ALT="$n, n'&gt;0$">|; 

$key = q/{{bf{a}cdot{{bf{b}times{{bf{c}neq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3349.png"
 ALT="$ {\bf a}\cdot{\bf b} \times{\bf c} \neq 0$">|; 

$key = q/displaystylev(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3047.png"
 ALT="$\displaystyle v(t)$">|; 

$key = q/theta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img464.png"
 ALT="$\theta_0$">|; 

$key = q/a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img177.png"
 ALT="$a$">|; 

$key = q/delta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3157.png"
 ALT="$\delta_0$">|; 

$key = q/theta_0=23.44^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2344.png"
 ALT="$\theta_0=23.44^\circ$">|; 

$key = q/F(P);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3395.png"
 ALT="$F(P)$">|; 

$key = q/u=costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1501.png"
 ALT="$u=\cos\theta$">|; 

$key = q/|omega|=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3061.png"
 ALT="$\vert\omega\vert=1$">|; 

$key = q/{displaymath}E=K+U=frac{1}{2},mleft(frac{dx}{dt}right)^2+frac{1}{2},m,omega_0^{,2},x^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img304.png"
 ALT="\begin{displaymath}
E = K + U = \frac{1}{2} m\left(\frac{dx}{dt}\right)^2+ \frac{1}{2} m \omega_0^{ 2} x^2,
\end{displaymath}">|; 

$key = q/displaystyleddot{z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2543.png"
 ALT="$\displaystyle \ddot{z}$">|; 

$key = q/{displaymath}f(x)simeq-m,omega_0^{,2},x,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img290.png"
 ALT="\begin{displaymath}
f(x) \simeq - m \omega_0^{ 2} x,
\end{displaymath}">|; 

$key = q/{displaymath}|{{bf{r}'-{{bf{r}|^{-1}=(r^{2}-2,{{bf{r}cdot{{bf{r}'+r'^{,2})^{-1slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2083.png"
 ALT="\begin{displaymath}
\vert{\bf r}'-{\bf r}\vert^{-1} = (r^{2}-2 {\bf r}\cdot{\bf r}' + r'^{ 2})^{-1/2},
\end{displaymath}">|; 

$key = q/{{bf{r}_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2275.png"
 ALT="${\bf r}_s$">|; 

$key = q/{{bf{a}'neq{{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="${\bf a}'\neq{\bf0}$">|; 

$key = q/dx=dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3390.png"
 ALT="$dx=dy$">|; 

$key = q/{displaymath}M,frac{d^2{{bf{r}_{cm}}{dt^2}={{bf{0},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="\begin{displaymath}
M \frac{d^2 {\bf r}_{cm}}{dt^2}= {\bf0},
\end{displaymath}">|; 

$key = q/{{bf{T}=mbox{{boldmath{{omega{}times{{bf{L};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1374.png"
 ALT="${\bf T}= \mbox{\boldmath$\omega$}\times{\bf L}$">|; 

$key = q/{displaymath}frac{dp_i}{dt}=frac{partialL}{partialq_i},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1663.png"
 ALT="\begin{displaymath}
\frac{d p_i}{dt} = \frac{\partial L}{\partial q_i},
\end{displaymath}">|; 

$key = q/Gamma=gamma^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2675.png"
 ALT="$\Gamma=\gamma^2$">|; 

$key = q/displaystyleI_{xy}=I_{yx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1249.png"
 ALT="$\displaystyle I_{xy}=I_{yx}$">|; 

$key = q/~^ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1282.png"
 ALT="$&nbsp;^\ast$">|; 

$key = q/{{bf{r}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="${\bf r}_i$">|; 

$key = q/M,(a+b,v^2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img483.png"
 ALT="$M (a+b v^2)$">|; 

$key = q/{displaymath}K=frac{m}{2}left[(1+alpha),dot{q}_1^{,2}+2,alpha,dot{q}_1,dot{q}_2+(1+alpha),dot{q}_2^{,2}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img2033.png"
 ALT="\begin{displaymath}
K = \frac{m}{2}\left[(1+\alpha) \dot{q}_1^{ 2} + 2 \alpha \dot{q}_1 \dot{q}_2 + (1+\alpha) \dot{q}_2^{ 2}\right],
\end{displaymath}">|; 

$key = q/{displaymath}1,,2,,2times2,,6,,5,,3,,2times3,,5,,6,,4,,6,,5,,6.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3170.png"
 ALT="\begin{displaymath}
1,  2,  2\times 2,  6,  5,  3,  2\times 3,  5,  6,  4,  6,  5,  6.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter05slashfig5.05.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="251" BORDER="0"
 SRC="|."$dir".q|img715.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter05/fig5.05.eps}}
\end{figure}">|; 

$key = q/displaystyleomega,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1469.png"
 ALT="$\displaystyle \omega,$">|; 

$key = q/alpha=1.79;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2479.png"
 ALT="$\alpha=1.79$">|; 

$key = q/d{{bf{L}slashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1454.png"
 ALT="$d{\bf L}/dt$">|; 

$key = q/{displaymath}Q_i=sum_{j=1,{calF}}f_j,frac{partialx_j}{partialq_i}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1575.png"
 ALT="\begin{displaymath}
Q_i = \sum_{j=1,{\cal F}} f_j \frac{\partial x_j}{\partial q_i}.
\end{displaymath}">|; 

$key = q/{displaymath}dot{theta}simeqfrac{h}{r_c^{,2}}=left[-frac{f(r_c)}{r_c}right]^{1slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img868.png"
 ALT="\begin{displaymath}
\dot{\theta} \simeq \frac{h}{r_c^{ 2}} = \left[-\frac{f(r_c)}{r_c}\right]^{1/2},
\end{displaymath}">|; 

$key = q/displaystylesinpsi,sintheta,dot{phi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1450.png"
 ALT="$\displaystyle \sin\psi \sin\theta \dot{\phi},$">|; 

$key = q/-m,mbox{{boldmath{{Omega{}times(mbox{{boldmath{{Omega{}times{{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1146.png"
 ALT="$-m \mbox{\boldmath$\Omega$}\times(\mbox{\boldmath$\Omega$}\times{\bf r})$">|; 

$key = q/displaystyleR_X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2830.png"
 ALT="$\displaystyle R_X$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter05slashfig5.07.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img788.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter05/fig5.07.eps}}
\end{figure}">|; 

$key = q/{displaymath}-ln!left[1-left(frac{v}{v_t}right)^2right]=frac{x}{x_t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img273.png"
 ALT="\begin{displaymath}
-\ln\!\left[1-\left(\frac{v}{v_t}\right)^2\right] = \frac{x}{x_t},
\end{displaymath}">|; 

$key = q/P_3(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2384.png"
 ALT="$P_3(0)=0$">|; 

$key = q/-alpha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2866.png"
 ALT="$-\alpha_0$">|; 

$key = q/I_{xy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1257.png"
 ALT="$I_{xy}$">|; 

$key = q/displaystylev_z';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="$\displaystyle v_z'$">|; 

$key = q/displaystyledot{v}_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1166.png"
 ALT="$\displaystyle \dot{v}_{x'}$">|; 

$key = q/{displaymath}int_P^QnablaUcdotd{{bf{r}equiv{mitDelta}U=U(Q)-U(P),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="\begin{displaymath}
\int_P^Q \nabla U\cdot d{\bf r} \equiv {\mit\Delta} U = U(Q) - U(P),
\end{displaymath}">|; 

$key = q/rho_2^{,2}=(x-mu_1)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2668.png"
 ALT="$\rho_2^{ 2} = (x-\mu_1)^2$">|; 

$key = q/{{bf{r}=(x,,y,,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2053.png"
 ALT="${\bf r}=(x,  y,  z)$">|; 

$key = q/n=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img877.png"
 ALT="$n=3$">|; 

$key = q/displaystylep_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1870.png"
 ALT="$\displaystyle p_\theta$">|; 

$key = q/stackrel{displaystylerightarrow}{DB};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3507.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{DB}$">|; 

$key = q/{displaymath}Q_i=-sum_{j=1,{calF}}frac{partialU}{partialx_j},frac{partialx_j}{partialq_i}=-frac{partialU}{partialq_i},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1580.png"
 ALT="\begin{displaymath}
Q_i = - \sum_{j=1,{\cal F}} \frac{\partial U}{\partial x_j}\...
...artial x_j}{\partial q_i} = - \frac{\partial U}{\partial q_i},
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2{{bf{r}}{dt^2}=-frac{G,M}{r^3},{{bf{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img932.png"
 ALT="\begin{displaymath}
\frac{d^2{\bf r}}{dt^2} =-
\frac{G M}{r^3}  {\bf r},
\end{displaymath}">|; 

$key = q/S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2711.png"
 ALT="$S$">|; 

$key = q/{displaymath}nablaV=frac{partialV}{partialr},{{bf{e}_r+frac{1}{r}frac{partialV}{intheta},frac{partialV}{partialphi},{{bf{e}_phi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img3502.png"
 ALT="\begin{displaymath}
\nabla V = \frac{\partial V}{\partial r} {\bf e}_r
+ \frac{...
... \sin\theta} \frac{\partial V}{\partial \phi} {\bf e}_\phi.
\end{displaymath}">|; 

$key = q/MslashM_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2418.png"
 ALT="$M/M_0$">|; 

$key = q/displaystyle+2,epsilon,beta_m,cos(2,theta),cos(varpi_n-phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2338.png"
 ALT="$\displaystyle + 2 \epsilon \beta_m \cos(2 \theta) \cos(\varpi_n-\phi),$">|; 

$key = q/{displaymath}{{bf{f}=q,left({{bf{E}+{{bf{v}times{{bf{B}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img622.png"
 ALT="\begin{displaymath}
{\bf f} = q \left({\bf E} + {\bf v}\times{\bf B}\right),
\end{displaymath}">|; 

$key = q/displaystylelambda_j,{{bf{x}_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1302.png"
 ALT="$\displaystyle \lambda_j {\bf x}_j,$">|; 

$key = q/3.04times10^{-6};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2426.png"
 ALT="$3.04\times 10^{-6}$">|; 

$key = q/displaystyle+y'^{,2}left[frac{sin^2theta}{A^2}+frac{2,costheta,sintheta,cos{mitDelta}}{A,B}+frac{cos^2theta}{B^2}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img554.png"
 ALT="$\displaystyle + y'^{ 2}\left[\frac{\sin^2\theta}{A^2} +\frac{2 \cos\theta \sin\theta \cos{\mit\Delta}}{A B} + \frac{\cos^2\theta}{B^2}\right]$">|; 

$key = q/bar{theta}=alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2978.png"
 ALT="$\bar{\theta}=\alpha$">|; 

$key = q/r_2=r,costheta+d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img708.png"
 ALT="$r_2 = r \cos\theta+d$">|; 

$key = q/m_1<m_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1065.png"
 ALT="$m_1&lt;m_2$">|; 

$key = q/{{bf{a}slashalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3544.png"
 ALT="${\bf a}/\alpha$">|; 

$key = q/{displaymath}dot{theta}^{,2}=frac{2,g}{a},costheta+c,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1837.png"
 ALT="\begin{displaymath}
\dot{\theta}^{ 2} = \frac{2 g}{a} \cos\theta + c,
\end{displaymath}">|; 

$key = q/{displaymath}omega,t=phi+k,2pi{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img3068.png"
 ALT="\begin{displaymath}
\omega t = \phi + k 2\pi
\end{displaymath}">|; 

$key = q/v_1(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3135.png"
 ALT="$v_1(t)$">|; 

$key = q/delta(t)simdelta_0,exp(lambda,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3158.png"
 ALT="$\delta(t)\sim \delta_0 \exp(\lambda t)$">|; 

$key = q/E_2=(1slash2),m_2,v_2^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1038.png"
 ALT="$E_2=(1/2) m_2 v_2^{ 2}$">|; 

$key = q/{displaymath}v_x=v_0,costheta,{{rm{e}^{-g,tslashv_t}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img585.png"
 ALT="\begin{displaymath}
v_x = v_0 \cos\theta {\rm e}^{-g t/v_t}.
\end{displaymath}">|; 

$key = q/{displaymath}theta(t)=theta_0,cos(omega_0,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img463.png"
 ALT="\begin{displaymath}
\theta(t) = \theta_0 \cos(\omega_0 t).
\end{displaymath}">|; 

$key = q/displaystyle{{bf{v}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1025.png"
 ALT="$\displaystyle {\bf v}_2$">|; 

$key = q/34.36^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2985.png"
 ALT="$34.36^\circ$">|; 

$key = q/{displaymath}{{bf{x}_k^T,{{bf{M},{{bf{x}_l=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1945.png"
 ALT="\begin{displaymath}
{\bf x}_k^T {\bf M} {\bf x}_l = 0,
\end{displaymath}">|; 

$key = q/5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2864.png"
 ALT="$5$">|; 

$key = q/{displaymath}frac{d{{bf{L}}{dt}=frac{d{{bf{L}}{dt'}+mbox{{boldmath{{omega{}times{{bf{L}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1354.png"
 ALT="\begin{displaymath}
\frac{d{\bf L}}{dt} = \frac{d{\bf L}}{dt'} + \mbox{\boldmath$\omega$}\times{\bf L}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{partialdot{x}_j}{partialdot{q}_i}=frac{partialx_j}{partialq_i},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1588.png"
 ALT="\begin{displaymath}
\frac{\partial \dot{x}_j}{\partial\dot{q}_i} = \frac{\partial x_j}{\partial q_i},
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter03slashfig3.08.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="284" BORDER="0"
 SRC="|."$dir".q|img448.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter03/fig3.08.eps}}
\end{figure}">|; 

$key = q/{{bf{0}equiv(0,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3226.png"
 ALT="${\bf0} \equiv (0, 0, 0)$">|; 

$key = q/w,{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3534.png"
 ALT="$w {\bf c}$">|; 

$key = q/alll;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1621.png"
 ALT="$a\ll l$">|; 

$key = q/stackrel{displaystylerightarrow}{BC};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3508.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{BC}$">|; 

$key = q/lambda_i=lambda_j=lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1312.png"
 ALT="$\lambda_i=\lambda_j=\lambda$">|; 

$key = q/v_0slashv_t=4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img618.png"
 ALT="$v_0/v_t=4$">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1504.png"
 ALT="$u$">|; 

$key = q/{{bf{V}_1'={{bf{v}_1'-{{bf{v}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1019.png"
 ALT="${\bf V}_1' = {\bf v}_1'-{\bf v}_2$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter10slashfig10.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="281" BORDER="0"
 SRC="|."$dir".q|img1817.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter10/fig10.02.eps}}
\end{figure}">|; 

$key = q/v(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$v(0)=0$">|; 

$key = q/{displaymath}overline{z}=left.int!int!intz,dVrightslashint!int!intdV.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3423.png"
 ALT="\begin{displaymath}
\overline{z} = \left. \int\!\int\!\int z dV\right/ \int\!\int\!\int dV.
\end{displaymath}">|; 

$key = q/x_a(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img412.png"
 ALT="$x_a(t)$">|; 

$key = q/m=0.07480;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2984.png"
 ALT="$m=0.07480$">|; 

$key = q/{displaymath}f_{ext}(t)=m,omega_0^{,2},X(t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img394.png"
 ALT="\begin{displaymath}
f_{ext}(t) = m \omega_0^{ 2} X(t),
\end{displaymath}">|; 

$key = q/q_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img197.png"
 ALT="$q_i$">|; 

$key = q/{displaymath}ddot{x}-frac{h^2}{r_c^{,3}}left(1-3,frac{x}{r_c}right)=f(r_c)+f'(r_c),x,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img854.png"
 ALT="\begin{displaymath}
\ddot{x} - \frac{h^2}{r_c^{ 3}}\left(1-3 \frac{x}{r_c}\right)=
f(r_c) + f'(r_c) x,
\end{displaymath}">|; 

$key = q/APB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1701.png"
 ALT="$APB$">|; 

$key = q/{displaymath}frac{partialF}{partialy'}={{rm{const}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1738.png"
 ALT="\begin{displaymath}
\frac{\partial F}{\partial y'} = {\rm const}.
\end{displaymath}">|; 

$key = q/{{bf{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img620.png"
 ALT="${\bf E}$">|; 

$key = q/x(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img442.png"
 ALT="$x(0)=0$">|; 

$key = q/displaystyle-2,{mitOmega},V_0,sinlambda,sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1167.png"
 ALT="$\displaystyle -2 {\mit\Omega} V_0 \sin\lambda \sin\theta,$">|; 

$key = q/displaystyle{{bf{e}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1441.png"
 ALT="$\displaystyle {\bf e}_z$">|; 

$key = q/rho_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img505.png"
 ALT="$\rho_0$">|; 

$key = q/C<C_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2628.png"
 ALT="$C &lt; C_4$">|; 

$key = q/displaystylen,t+2,e,sin(n,t-alpha_0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2817.png"
 ALT="$\displaystyle n t+2 e \sin(n t-\alpha_0),$">|; 

$key = q/k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img498.png"
 ALT="$k$">|; 

$key = q/displaystylemu_1left(frac{1-rho_1^{,3}}{rho_1^{,2}}right)left(frac{y}{rho_1}righfrac{1-rho_2^{,3}}{rho_2^{,2}}right)left(frac{y}{rho_2}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2571.png"
 ALT="$\displaystyle \mu_1\left(\frac{1-\rho_1^{ 3}}{\rho_1^{ 2}}\right)\left(\frac{...
..._2\left(\frac{1-\rho_2^{ 3}}{\rho_2^{ 2}}\right)\left(\frac{y}{\rho_2}\right)$">|; 

$key = q/dot{phi}=h=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1685.png"
 ALT="$\dot{\phi}=h=0$">|; 

$key = q/{{bf{r}'equiv(x',,y',,z');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="${\bf r}'\equiv (x', y', z')$">|; 

$key = q/{displaymath}{calE}=frac{u_c^{,2},h^2}{2},(e^2-1)=frac{G,M}{2,r_p},(e-1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img775.png"
 ALT="\begin{displaymath}
{\cal E} = \frac{u_c^{ 2} h^2}{2}  (e^2 -1) = \frac{G M}{2 r_p} (e-1).
\end{displaymath}">|; 

$key = q/Q=5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$Q=5$">|; 

$key = q/{displaymath}b(theta)=a,cos(thetaslash2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img986.png"
 ALT="\begin{displaymath}
b(\theta) = a \cos(\theta/2).
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter07slashfig7.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="438" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img1122.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter07/fig7.01.eps}}
\end{figure}">|; 

$key = q/{displaymath}H=sum_{i=1,{calF}}dot{q}_i,p_i-L=sum_{i=1,{calF}}dot{q}_i,p_i-K+U.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1845.png"
 ALT="\begin{displaymath}
H = \sum_{i=1,{\cal F}} \dot{q}_i p_i - L = \sum_{i=1,{\cal F}} \dot{q}_i p_i -K + U.
\end{displaymath}">|; 

$key = q/x=x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img220.png"
 ALT="$x=x_2$">|; 

$key = q/displaystyle-frac{v}{Q}-sintheta+A,cos(omega,t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3041.png"
 ALT="$\displaystyle -\frac{v}{Q} -\sin\theta + A \cos(\omega  t).$">|; 

$key = q/{displaymath}frac{x^2}{A^2}-2,frac{x,y}{A,B},cos{mitDelta}+frac{y^2}{B^2}=sin^2{mitDelta}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img547.png"
 ALT="\begin{displaymath}
\frac{x^2}{A^2} - 2 \frac{x y}{A B}  \cos{\mit\Delta} + \frac{y^2}{B^2}
= \sin^2{\mit\Delta}.
\end{displaymath}">|; 

$key = q/{displaymath}sigma=intfrac{dsigma}{d{mitOmega}},d{mitOmega},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img979.png"
 ALT="\begin{displaymath}
\sigma = \int \frac{d\sigma}{d{\mit\Omega}} d{\mit\Omega},
\end{displaymath}">|; 

$key = q/{displaymath}frac{dtheta}{dt}simeq2,{mitOmega},sinlambda.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1176.png"
 ALT="\begin{displaymath}
\frac{d\theta}{dt} \simeq 2 {\mit\Omega} \sin\lambda.
\end{displaymath}">|; 

$key = q/displaystylefrac{3}{2},frac{epsilon,iota_m,mu_m,T_n({{rm{yr})}{T_s({{rm{day}),[T_m({{rm{yr})]^2},frac{cos(2,theta_0)}{sintheta_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2370.png"
 ALT="$\displaystyle \frac{3}{2} \frac{\epsilon \iota_m \mu_m T_n({\rm yr})}{T_s({\rm day}) [T_m({\rm yr})]^2} \frac{\cos(2 \theta_0)}{\sin\theta_0},$">|; 

$key = q/{displaymath}{{bf{g}=G,m',frac{({{bf{r}'-{{bf{r})}{|{{bf{r}'-{{bf{r}|^{,3}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2051.png"
 ALT="\begin{displaymath}
{\bf g} = G m' \frac{({\bf r}'-{\bf r})}{\vert{\bf r}'-{\bf r}\vert^{ 3}}.
\end{displaymath}">|; 

$key = q/displaystyletan(thetaslash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img819.png"
 ALT="$\displaystyle \tan(\theta/2)$">|; 

$key = q/mu_2slashmu_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2596.png"
 ALT="$\mu_2/\mu_1$">|; 

$key = q/{displaymath}L_phi=I_perp,sin^2theta,dot{phi}+L_psi,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1493.png"
 ALT="\begin{displaymath}
L_\phi = I_\perp \sin^2\theta \dot{\phi} + L_\psi \cos\theta.
\end{displaymath}">|; 

$key = q/{displaymath}p_theta=frac{partialL}{partialdot{theta}}=m,r^2,dot{theta}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1667.png"
 ALT="\begin{displaymath}
p_\theta = \frac{\partial L}{\partial\dot{\theta}} = m r^2 \dot{\theta}
\end{displaymath}">|; 

$key = q/M=sum_{i=1,N}m_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1233.png"
 ALT="$M=\sum_{i=1,N} m_i$">|; 

$key = q/{displaymath}K=frac{1}{2},mbox{{boldmath{{omega{}^T,tilde{{bf{I},mbox{{boldmath{{omega{}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1272.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} \mbox{\boldmath$\omega$}^T \tilde{\bf I} \mbox{\boldmath$\omega$}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{dsigma(psi)}{d{mitOmega}'}=4,cospsi,,frac{dsigma(theta=2,psi)}{d{mitOmega}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1084.png"
 ALT="\begin{displaymath}
\frac{d\sigma(\psi)}{d{\mit\Omega}'} = 4 \cos\psi  \frac{d\sigma(\theta=2 \psi)}{d{\mit\Omega}}.
\end{displaymath}">|; 

$key = q/displaystyle(a_x^{~2}+a_y^{~2}+a_z^{~2}),(b_x^{~2}+b_y^{~2}+b_z^{~2})-(a_x,b_x+a_y,b_y+a_z,b_z)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="494" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3311.png"
 ALT="$\displaystyle (a_x^{&nbsp;2}+a_y^{&nbsp;2}+a_z^{&nbsp;2}) (b_x^{&nbsp;2}+b_y^{&nbsp;2}+b_z^{&nbsp;2}) -
(a_x  b_x + a_y  b_y + a_z  b_z)^2$">|; 

$key = q/{displaymath}I=int_a^bF(y,y',x),dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1726.png"
 ALT="\begin{displaymath}
I = \int_a^b F(y, y',x) dx,
\end{displaymath}">|; 

$key = q/displaystyleK;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2023.png"
 ALT="$\displaystyle K$">|; 

$key = q/nuslashomega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img376.png"
 ALT="$\nu/\omega_0$">|; 

$key = q/{displaymath}Phi(a_theta,theta)+chi(a_theta,theta)=c,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2172.png"
 ALT="\begin{displaymath}
\Phi(a_\theta,\theta)+\chi(a_\theta,\theta) = c,
\end{displaymath}">|; 

$key = q/displaystylesqrt{-lambda_2}=sqrt{frac{k,(1+2,alpha)}{m}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2042.png"
 ALT="$\displaystyle \sqrt{-\lambda_2} = \sqrt{\frac{k (1+2 \alpha)}{m}}.$">|; 

$key = q/16.28;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2452.png"
 ALT="$16.28$">|; 

$key = q/delta_k({mitDelta}t)simeqdelta_k(0),exp(lambda_k,{mitDelta}t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3156.png"
 ALT="$\delta_k({\mit\Delta}t)\simeq \delta_k(0) \exp(\lambda_k {\mit\Delta}t)$">|; 

$key = q/omega_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2016.png"
 ALT="$\omega_2$">|; 

$key = q/M=1.90times10^{27},{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2188.png"
 ALT="$M=1.90\times 10^{27} {\rm kg}$">|; 

$key = q/|{{bf{r}-({{bf{r}cdot{{bf{n}),{{bf{n}|=d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3529.png"
 ALT="$\vert{\bf r} -({\bf r}\cdot{\bf n}) {\bf n}\vert = d$">|; 

$key = q/{displaymath}frac{d{{bf{r}_{cm}}{dt}=frac{sum_{i=1}^Nm_i,d{{bf{r}_islashdt}{sum_{i=1}^Nm_i},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img148.png"
 ALT="\begin{displaymath}
\frac{d{\bf r}_{cm} }{dt}= \frac{\sum_{i=1}^N m_i  d{\bf r}_i/dt}{\sum_{i=1}^N m_i},
\end{displaymath}">|; 

$key = q/displaystyle-frac{k}{m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2039.png"
 ALT="$\displaystyle - \frac{k}{m},$">|; 

$key = q/(c-g),m^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2856.png"
 ALT="$(c-g) m^2$">|; 

$key = q/{displaymath}frac{d{{bf{r}_i}{dt}=mbox{{boldmath{{omega{}times{{bf{r}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1238.png"
 ALT="\begin{displaymath}
\frac{d{\bf r}_i}{dt} = \mbox{\boldmath$\omega$}\times {\bf r}_i.
\end{displaymath}">|; 

$key = q/displaystyle{{bf{M};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1991.png"
 ALT="$\displaystyle {\bf M}$">|; 

$key = q/{displaymath}m,(q_1+q_2)+M,q_3=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2031.png"
 ALT="\begin{displaymath}
m (q_1+q_2) + M q_3 = 0.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d}{dx}!left(y',frac{partialF}{partialy'}right)=y',frac{d}{dx}!{partialy'}right)+y'',frac{partialF}{partialy'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1740.png"
 ALT="\begin{displaymath}
\frac{d}{dx}\!\left(y' \frac{\partial F}{\partial y'}\right...
...l F}{\partial y'}\right)+ y'' \frac{\partial F}{\partial y'}.
\end{displaymath}">|; 

$key = q/langle|{{bf{r}'-{{bf{r}|^{-1}rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2115.png"
 ALT="$\langle\vert{\bf r}'-{\bf r}\vert^{-1}\rangle$">|; 

$key = q/z(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img655.png"
 ALT="$z(x)$">|; 

$key = q/(x,,y,,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2532.png"
 ALT="$(x, y, z)$">|; 

$key = q/{{bf{e}_{y''};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1421.png"
 ALT="${\bf e}_{y''}$">|; 

$key = q/p_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1670.png"
 ALT="$p_x$">|; 

$key = q/{displaymath}{{bf{A}=frac{x,{{bf{e}_x+y,{{bf{e}_y}{sqrt{x^2+y^2}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img3550.png"
 ALT="\begin{displaymath}
{\bf A} = \frac{x {\bf e}_x + y {\bf e}_y}{\sqrt{x^2 + y^2}}
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2x}{dt^2}+2,nu,frac{dx}{dt}+omega_0^{,2},x=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img309.png"
 ALT="\begin{displaymath}
\frac{d^2 x}{dt^2} + 2 \nu \frac{dx}{dt} + \omega_0^{ 2} x
= 0.
\end{displaymath}">|; 

$key = q/v_0>2,EslashB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img651.png"
 ALT="$v_0&gt; 2 E/B$">|; 

$key = q/frac{9}{4},m^2,e-3,e,x_4-frac{3}{2},e,y_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2900.png"
 ALT="$\frac{9}{4} m^2 e-3 e x_4 - \frac{3}{2} e y_4$">|; 

$key = q/costheta,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img360.png"
 ALT="$\cos\theta \sin\theta$">|; 

$key = q/v_0,v_tslashg;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img607.png"
 ALT="$v_0 v_t/g$">|; 

$key = q/T_{x'}=T_{y'}=T_{z'}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1413.png"
 ALT="$T_{x'}=T_{y'}=T_{z'}=0$">|; 

$key = q/{displaymath}H=frac{1}{2,m}left(p_r^{,2}+frac{p_theta^{,2}}{r^2}right)+U(r).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1873.png"
 ALT="\begin{displaymath}
H = \frac{1}{2 m}\left(p_r^{ 2} + \frac{p_\theta^{ 2}}{r^2}\right)
+ U(r).
\end{displaymath}">|; 

$key = q/{calF}=3,N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1561.png"
 ALT="${\cal F}= 3 N$">|; 

$key = q/D=45^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3000.png"
 ALT="$D=45^\circ$">|; 

$key = q/7.7;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1101.png"
 ALT="$7.7$">|; 

$key = q/displaystylemu_1,(cosomegat,,sinomegat,,0).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2501.png"
 ALT="$\displaystyle \mu_1 (\cos \omega t, \sin\omega t, 0).$">|; 

$key = q/F=sin^2(omega,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img515.png"
 ALT="$F= \sin^2(\omega t)$">|; 

$key = q/displaystylec,[1-cos(omega,t)],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img647.png"
 ALT="$\displaystyle c [1-\cos(\omega t)],$">|; 

$key = q/displaystylebar{theta}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2975.png"
 ALT="$\displaystyle \bar{\theta}'$">|; 

$key = q/displaystyleE_1'=left(frac{m_2}{m_1+m_2}right)E,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1047.png"
 ALT="$\displaystyle E_1'= \left(\frac{m_2}{m_1+m_2}\right) E,$">|; 

$key = q/v_tslashg;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img586.png"
 ALT="$v_t/g$">|; 

$key = q/{displaymath}beta=iota,sin(theta-gamma_0).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2821.png"
 ALT="\begin{displaymath}
\beta= \iota \sin(\theta-\gamma_0).
\end{displaymath}">|; 

$key = q/1.4215;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3163.png"
 ALT="$1.4215$">|; 

$key = q/{displaymath}U=-m,g,x',sintheta+{{rm{const}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1650.png"
 ALT="\begin{displaymath}
U = - m g x' \sin\theta + {\rm const}.
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{r}left[1+left(frac{r'}{r}right)costheta,costheta'right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2100.png"
 ALT="$\displaystyle \frac{1}{r}\left[
1 + \left(\frac{r'}{r}\right)\cos\theta \cos\theta' \right.$">|; 

$key = q/5.32;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2464.png"
 ALT="$5.32$">|; 

$key = q/{displaymath}K=frac{1}{2},m,dot{x}^{,2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1656.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} m \dot{x}^{ 2},
\end{displaymath}">|; 

$key = q/displaystylefrac{7}{12}left(frac{mu_2}{mu_1}right)-frac{7}{12}left(frac{mu_2}{mufrac{mu_2}{mu_1}right)^3+{calO}left(frac{mu_2}{mu_1}right)^4,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2595.png"
 ALT="$\displaystyle \frac{7}{12}\left(\frac{\mu_2}{\mu_1}\right) - \frac{7}{12}\left(...
...left(\frac{\mu_2}{\mu_1}\right)^3 + {\cal O}\left(\frac{\mu_2}{\mu_1}\right)^4,$">|; 

$key = q/{displaymath}x=frac{v_0,v_t,costheta}{g}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img593.png"
 ALT="\begin{displaymath}
x = \frac{v_0 v_t \cos\theta}{g}.
\end{displaymath}">|; 

$key = q/I_{y'y'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1360.png"
 ALT="$I_{y'y'}$">|; 

$key = q/displaystyle-l,costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1674.png"
 ALT="$\displaystyle -l \cos\theta.$">|; 

$key = q/{{bf{x}_k^{ast,T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1929.png"
 ALT="${\bf x}_k^{\ast T}$">|; 

$key = q/e=0.01673;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img893.png"
 ALT="$e=0.01673$">|; 

$key = q/8.85;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2693.png"
 ALT="$8.85$">|; 

$key = q/I_{z'z'}>I_{y'y'}>I_{x'x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1530.png"
 ALT="$I_{z'z'} &gt; I_{y'y'} &gt; I_{x'x'}$">|; 

$key = q/~^T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1277.png"
 ALT="$&nbsp;^T$">|; 

$key = q/5pislash2<omega,t<7pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img522.png"
 ALT="$5\pi/2&lt;\omega t&lt;7\pi/2$">|; 

$key = q/{figure}{epsfysize{=1.25in{centerline{epsffile{Chapter11slashfig11.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="444" HEIGHT="140" BORDER="0"
 SRC="|."$dir".q|img2028.png"
 ALT="\begin{figure}
\epsfysize =1.25in
\centerline{\epsffile{Chapter11/fig11.02.eps}}
\end{figure}">|; 

$key = q/{displaymath}x(t)=-left(frac{v_0+nu_-,x_0}{nu_+-nu_-}right){{rm{e}^{-nu_+,t}+lefv_0+nu_+,x_0}{nu_+-nu_-}right){{rm{e}^{-nu_-,t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="426" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img331.png"
 ALT="\begin{displaymath}
x(t) = -\left(\frac{v_0 + \nu_- x_0}{\nu_+-\nu_-}\right){\r...
...(\frac{v_0+\nu_+ x_0}{\nu_+-\nu_-}\right){\rm e}^{-\nu_- t},
\end{displaymath}">|; 

$key = q/infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3415.png"
 ALT="$\infty$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{AppendixAslashfigA.12.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="200" BORDER="0"
 SRC="|."$dir".q|img3393.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.12.eps}}
\end{figure}">|; 

$key = q/{displaymath}f(x)simeqf(0)+frac{df(0)}{dx},x+{calO}(x^2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img289.png"
 ALT="\begin{displaymath}
f(x) \simeq f(0) + \frac{df(0)}{dx} x + {\cal O}(x^2).
\end{displaymath}">|; 

$key = q/{displaymath}dot{phi}=omegaleft[1+left(frac{I_parallel^{,2}}{I_perp^{,2}}-1right)cos^2alpharight]^{1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="61" BORDER="0"
 SRC="|."$dir".q|img1459.png"
 ALT="\begin{displaymath}
\dot{\phi} = \omega\left[1 + \left(\frac{I_\parallel^{ 2}}{I_\perp^{ 2}}-1\right)\cos^2\alpha\right]^{1/2}.
\end{displaymath}">|; 

$key = q/{displaymath}M,frac{d^2{{bf{r}_{cm}}{dt^2}={{bf{F}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1232.png"
 ALT="\begin{displaymath}
M \frac{d^2{\bf r}_{cm}}{dt^2} = {\bf F}.
\end{displaymath}">|; 

$key = q/{displaymath}m_i,frac{d^2{{bf{r}_i}{dt^2}=sum_{j=1,N}^{jneqi}!{{bf{f}_{ij}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="67" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="\begin{displaymath}
m_i \frac{d^2 {\bf r}_i}{dt^2} = \sum_{j=1,N}^{j\neq i}\! {\bf f}_{ij}.
\end{displaymath}">|; 

$key = q/p_i=partialLslashpartialdot{q}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1848.png"
 ALT="$p_i=
\partial L/\partial \dot{q}_i$">|; 

$key = q/j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="$j$">|; 

$key = q/{displaymath}int_0^vfrac{v',dv'}{1-(v'slashv_t)^2}=g,x.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img270.png"
 ALT="\begin{displaymath}
\int_0^v\frac{v' dv'}{1-(v'/v_t)^2} = g x.
\end{displaymath}">|; 

$key = q/2times3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3173.png"
 ALT="$2\times 3$">|; 

$key = q/omega_{z'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1381.png"
 ALT="$\omega_{z'}$">|; 

$key = q/displaystylex,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2108.png"
 ALT="$\displaystyle x,$">|; 

$key = q/2.2,{{rm{cm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1163.png"
 ALT="$2.2 {\rm cm}$">|; 

$key = q/P=(x_0,,y_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3444.png"
 ALT="$P=(x_0,  y_0)$">|; 

$key = q/0^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1165.png"
 ALT="$0^\circ$">|; 

$key = q/dslashdtneqdslashdt';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1108.png"
 ALT="$d/dt\neq d/dt'$">|; 

$key = q/{{bf{a},%,{{bf{b}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3250.png"
 ALT="${\bf a} \% {\bf b}= 1$">|; 

$key = q/r_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3122.png"
 ALT="$r_n$">|; 

$key = q/nu_pm=nupmsqrt{nu^2-omega_0^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img332.png"
 ALT="$\nu_\pm = \nu\pm \sqrt{\nu^2-\omega_0^{ 2}}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.10.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="274" BORDER="0"
 SRC="|."$dir".q|img3076.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.10.eps}}
\end{figure}">|; 

$key = q/{displaymath}{{bf{g}=-frac{G,m}{a^2},{{bf{e}_z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2257.png"
 ALT="\begin{displaymath}
{\bf g} = - \frac{G m}{a^2} {\bf e}_z.
\end{displaymath}">|; 

$key = q/a=r_pslash(1-e);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img807.png"
 ALT="$a = r_p/(1-e)$">|; 

$key = q/R<R_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2259.png"
 ALT="$R&lt; R_c$">|; 

$key = q/{displaymath}K=frac{1}{2},,mbox{{boldmath{{omega{}cdot{{bf{L}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1271.png"
 ALT="\begin{displaymath}
K = \frac{1}{2}   \mbox{\boldmath$\omega$} \cdot {\bf L}.
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{a^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img561.png"
 ALT="$\displaystyle \frac{1}{a^2}$">|; 

$key = q/{displaymath}{mitTheta}=int_0^{u_{max}}frac{b,du}{sqrt{1-b^2,u^2-k,uslashE}}=int_0^{x_{max}}frac{dx}{sqrt{1-x^2-alpha,x}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img991.png"
 ALT="\begin{displaymath}
{\mit\Theta} = \int_0^{u_{max}}\frac{b du}{\sqrt{1-b^2 u^2
- k u/E}} = \int_0^{x_{max}}\frac{dx}{\sqrt{1-x^2- \alpha x}},
\end{displaymath}">|; 

$key = q/{displaymath}int_P^Qleft[y,dx+x^3,dyright]=int_{0}^1left[y,dy+(y+1)^3,dyright]=frac{17}{4}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img3391.png"
 ALT="\begin{displaymath}
\int_P^Q \left[ y dx + x^3 dy\right]= \int_{0}^1\left[y dy + (y+1)^3 dy\right] = \frac{17}{4}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{x}_k^{T},{{bf{M},delta{{bf{q}^{(0)}=alpha_k+beta_k,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1964.png"
 ALT="\begin{displaymath}
{\bf x}_k^{T} {\bf M} \delta {\bf q}^{(0)} = \alpha_k + \beta_k,
\end{displaymath}">|; 

$key = q/{displaymath}oint{{bf{A}cdotd{{bf{r}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img3473.png"
 ALT="\begin{displaymath}
\oint {\bf A}\cdot d{\bf r} = 0,
\end{displaymath}">|; 

$key = q/a_x,,a_y,,a_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3245.png"
 ALT="$a_x,  a_y,  a_z$">|; 

$key = q/Vgeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2607.png"
 ALT="$V\geq 0$">|; 

$key = q/displaystyleB,cos(omega_0,t-phi_2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img536.png"
 ALT="$\displaystyle B \cos(\omega_0 t-\phi_2).$">|; 

$key = q/a=sqrt{2,Eslashk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img238.png"
 ALT="$a = \sqrt{2 E/k}$">|; 

$key = q/displaystyleI_{x'x'},dot{omega}_{x'}-(I_{y'y'}-I_{z'z'}),omega_{y'},omega_{z'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1363.png"
 ALT="$\displaystyle I_{x'x'} \dot{\omega}_{x'} - (I_{y'y'}-I_{z'z'}) \omega_{y'} \omega_{z'},$">|; 

$key = q/gamma_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2274.png"
 ALT="$\gamma_s$">|; 

$key = q/u_c=r_c^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img773.png"
 ALT="$u_c = r_c^{-1}$">|; 

$key = q/z=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2255.png"
 ALT="$z=a$">|; 

$key = q/displaystylehat{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3029.png"
 ALT="$\displaystyle \hat{t}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter03slashfig3.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="272" BORDER="0"
 SRC="|."$dir".q|img302.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter03/fig3.03.eps}}
\end{figure}">|; 

$key = q/{displaymath}frac{dsigma(psi)}{d{mitOmega}'}=frac{d{mitOmega}}{d{mitOmega}'},frac{dsigma(theta)}{d{mitOmega}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1081.png"
 ALT="\begin{displaymath}
\frac{d\sigma(\psi)}{d{\mit\Omega}'} = \frac{d{\mit\Omega}}{d{\mit\Omega}'} \frac{d\sigma(\theta)}{d{\mit\Omega}},
\end{displaymath}">|; 

$key = q/mu_2=G,m_2=1-mu_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2495.png"
 ALT="$\mu_2=G m_2=1-\mu_1$">|; 

$key = q/{displaymath}{{bf{a}+{{bf{b}equiv(a_x+b_x,,a_y+b_y,,a_z+b_z).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img3231.png"
 ALT="\begin{displaymath}
{\bf a} + {\bf b} \equiv (a_x+b_x, a_y+b_y, a_z+b_z).
\end{displaymath}">|; 

$key = q/omegaequiv(omega_{x'}^{,2}+omega_{y'}^{,2}+omega_{z'}^{,2})^{1slash2}=dot{phi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1453.png"
 ALT="$ \omega\equiv (\omega_{x'}^{ 2}+\omega_{y'}^{ 2}+\omega_{z'}^{ 2})^{1/2}=\dot{\phi}$">|; 

$key = q/theta_0(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3127.png"
 ALT="$\theta_0(t)$">|; 

$key = q/y=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2572.png"
 ALT="$y=0$">|; 

$key = q/{displaymath}epsilon=frac{5}{4},frac{{mitOmega}^2,a^3}{G,M}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2174.png"
 ALT="\begin{displaymath}
\epsilon = \frac{5}{4} \frac{{\mit\Omega}^2 a^3}{G M}.
\end{displaymath}">|; 

$key = q/I_0=(2slash5),M,a^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2193.png"
 ALT="$I_0 = (2/5) M a^2$">|; 

$key = q/displaystyle{{bf{x}_j';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1315.png"
 ALT="$\displaystyle {\bf x}_j'$">|; 

$key = q/m=A,d,rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img282.png"
 ALT="$m = A d \rho$">|; 

$key = q/ddot{~}equivd^2slashdT^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2770.png"
 ALT="$\ddot{&nbsp;}\equiv
d^2/dT^2$">|; 

$key = q/displaystyle2,e,sin(T-alpha_0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2810.png"
 ALT="$\displaystyle 2 e \sin(T-\alpha_0),$">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}=omega_{x'},{{bf{e}_{x'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1531.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$} = \omega_{x'} {\bf e}_{x'}.
\end{displaymath}">|; 

$key = q/G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img663.png"
 ALT="$G$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{AppendixAslashfigA.14.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="252" BORDER="0"
 SRC="|."$dir".q|img3440.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{AppendixA/figA.14.eps}}
\end{figure}">|; 

$key = q/hat{mbox{{boldmath{{omega{}}=(cosalpha,,cos,beta,,cosgamma);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1330.png"
 ALT="$\hat{\mbox{\boldmath$\omega$}} = (\cos\alpha, \cos \beta, \cos\gamma)$">|; 

$key = q/{displaymath}f(t,x_1,t,x_2,cdots,t,x_n)=t^a,f(x_1,x_2,cdots,x_n){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="334" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img174.png"
 ALT="\begin{displaymath}
f(t x_1, t x_2,\cdots,t x_n) = t^a f(x_1,x_2,\cdots,x_n)
\end{displaymath}">|; 

$key = q/{displaymath}frac{d}{dx}!left(frac{partialF}{partialy'}right)-frac{partialF}{paright)-frac{partial[lambda,G]}{partialy}right]=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1756.png"
 ALT="\begin{displaymath}
\frac{d}{dx}\!\left(\frac{\partial F}{\partial y'}\right)-\f...
...y'}\right)-\frac{\partial [\lambda G]}{\partial y}\right]= 0.
\end{displaymath}">|; 

$key = q/n=1,l=+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3180.png"
 ALT="$n=1, l=+1$">|; 

$key = q/y'siml,theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1182.png"
 ALT="$y'\sim l \theta$">|; 

$key = q/I,sigma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img980.png"
 ALT="$I \sigma$">|; 

$key = q/{figure}{epsfysize{=1.25in{centerline{epsffile{Chapter12slashfig12.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="454" HEIGHT="137" BORDER="0"
 SRC="|."$dir".q|img2209.png"
 ALT="\begin{figure}
\epsfysize =1.25in
\centerline{\epsffile{Chapter12/fig12.04.eps}}
\end{figure}">|; 

$key = q/d^3{{bf{r}'rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2068.png"
 ALT="$d^3{\bf r}'\rightarrow 0$">|; 

$key = q/1-2,m-g,m^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2867.png"
 ALT="$1-2 m-g m^2$">|; 

$key = q/{displaymath}-frac{sin(2,theta)}{A^2}-frac{2,cos(2,theta),cos{mitDelta}}{A,B}+frac{sin(2,theta)}{B^2}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img558.png"
 ALT="\begin{displaymath}
-\frac{\sin (2 \theta)}{A^2} - \frac{2 \cos (2 \theta) \cos{\mit\Delta}}{A B} + \frac{\sin (2 \theta)}{B^2}= 0,
\end{displaymath}">|; 

$key = q/n=-2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img873.png"
 ALT="$n=-2$">|; 

$key = q/U=m,Phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2062.png"
 ALT="$U=m \Phi$">|; 

$key = q/x_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img176.png"
 ALT="$x_i$">|; 

$key = q/{displaymath}m,v,frac{dv}{dx}=f(v),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img247.png"
 ALT="\begin{displaymath}
m v \frac{dv}{dx} = f(v),
\end{displaymath}">|; 

$key = q/1slashr^5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1094.png"
 ALT="$1/r^5$">|; 

$key = q/displaystylefrac{d^2x}{dt^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img528.png"
 ALT="$\displaystyle \frac{d^2 x}{d t^2}$">|; 

$key = q/1.34870;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3110.png"
 ALT="$1.34870$">|; 

$key = q/B=C=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2669.png"
 ALT="$B=C=0$">|; 

$key = q/{{bf{x}_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1304.png"
 ALT="${\bf x}_j$">|; 

$key = q/{{bf{f}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="${\bf f}_1$">|; 

$key = q/dot{x}_j=dot{x}_j(dot{q}_1,dot{q}_2,cdots,dot{q}_{calF},q_1,q_2,cdots,q_{calF},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1587.png"
 ALT="$\dot{x}_j = \dot{x}_j(\dot{q}_1,\dot{q}_2,\cdots,
\dot{q}_{\cal F},q_1,q_2,\cdots,q_{\cal F},t)$">|; 

$key = q/x(t_f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img602.png"
 ALT="$x(t_f)$">|; 

$key = q/displaystyle{{bf{a}cdot({{bf{b}+{{bf{c});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3261.png"
 ALT="$\displaystyle {\bf a}\cdot({\bf b}+{\bf c})$">|; 

$key = q/partialfslashpartialx=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1820.png"
 ALT="$\partial f/\partial x = 1$">|; 

$key = q/{{bf{r}_{cm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img146.png"
 ALT="${\bf r}_{cm}$">|; 

$key = q/displaystylecostheta,dot{phi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1452.png"
 ALT="$\displaystyle \cos\theta \dot{\phi},$">|; 

$key = q/{displaymath}{{bf{r}cdot{{bf{b}times{{bf{c}=alpha,{{bf{a}cdot{{bf{b}times{{bf{c},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3347.png"
 ALT="\begin{displaymath}
{\bf r} \cdot {\bf b} \times{\bf c} = \alpha  {\bf a}\cdot{\bf b} \times{\bf c},
\end{displaymath}">|; 

$key = q/displaystyle{{bf{e}_{x''};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1443.png"
 ALT="$\displaystyle {\bf e}_{x''}$">|; 

$key = q/m_4,m_5,m_6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1583.png"
 ALT="$m_4, m_5, m_6$">|; 

$key = q/{{bf{b}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3367.png"
 ALT="${\bf b}(t)$">|; 

$key = q/displaystylex_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2841.png"
 ALT="$\displaystyle x_j$">|; 

$key = q/langlecdotsrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2082.png"
 ALT="$\langle\cdots\rangle$">|; 

$key = q/displaystyledot{mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1540.png"
 ALT="$\displaystyle \dot{\mu}$">|; 

$key = q/aslashR;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2235.png"
 ALT="$a/R$">|; 

$key = q/z=c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1885.png"
 ALT="$z=c$">|; 

$key = q/displaystyle-frac{3}{2},m^2,cos[2,(1-m),T],deltaY+3,deltaX,deltaY,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2792.png"
 ALT="$\displaystyle -\frac{3}{2} m^2 \cos[2 (1-m) T] \delta Y +3 \delta X \delta Y,$">|; 

$key = q/z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img548.png"
 ALT="$z$">|; 

$key = q/N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="$N$">|; 

$key = q/4.69pm0.01;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3113.png"
 ALT="$4.69\pm 0.01$">|; 

$key = q/{{bf{r}_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2306.png"
 ALT="${\bf r}_m$">|; 

$key = q/{displaymath}frac{d{{bf{r}}{dt}={{bf{v}'+mbox{{boldmath{{Omega{}times{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1106.png"
 ALT="\begin{displaymath}
\frac{d{\bf r}}{dt} = {\bf v}'+ \mbox{\boldmath$\Omega$}\times{\bf r}.
\end{displaymath}">|; 

$key = q/d{{bf{r}equiv(dx,,dy);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3433.png"
 ALT="$d{\bf r}\equiv(dx, dy)$">|; 

$key = q/displaystylelambda_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2001.png"
 ALT="$\displaystyle \lambda_2$">|; 

$key = q/V({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3493.png"
 ALT="$V({\bf r})$">|; 

$key = q/vartheta_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img471.png"
 ALT="$\vartheta_0=0$">|; 

$key = q/v^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img210.png"
 ALT="$v^2$">|; 

$key = q/{displaymath}f_{drag}=C,rho_a,A,v^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img279.png"
 ALT="\begin{displaymath}
f_{drag} = C \rho_a A v^2
\end{displaymath}">|; 

$key = q/{displaymath}ddot{theta}+frac{g}{l},sintheta=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1686.png"
 ALT="\begin{displaymath}
\ddot{\theta} + \frac{g}{l} \sin\theta= 0.
\end{displaymath}">|; 

$key = q/{displaymath}K=frac{1}{2}sum_{i=1,N}m_ileft(frac{d{{bf{r}_i}{dt}right)^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="63" BORDER="0"
 SRC="|."$dir".q|img1269.png"
 ALT="\begin{displaymath}
K = \frac{1}{2}\sum_{i=1,N} m_i\left(\frac{d{\bf r}_i}{dt}\right)^2.
\end{displaymath}">|; 

$key = q/displaystyle(0,,0,,R),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1137.png"
 ALT="$\displaystyle (0, 0, R),$">|; 

$key = q/displaystyledeltav_0,cos(omega_ast,{mitDelta}t),,{{rm{e}^{-{mitDelta}tslash2Q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3142.png"
 ALT="$\displaystyle \delta v_0 \cos(\omega_\ast  {\mit\Delta} t)  {\rm e}^{-{\mit\Delta} t/2Q}$">|; 

$key = q/{mitDelta}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img570.png"
 ALT="${\mit\Delta}=0$">|; 

$key = q/F(t)=F_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img517.png"
 ALT="$F(t) = F_0$">|; 

$key = q/displaystylerho_1,dot{rho}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2515.png"
 ALT="$\displaystyle \rho_1 \dot{\rho}_1$">|; 

$key = q/{displaymath}X(t)=sum_{n=0}^{infty}X_{n},cos(n,omega,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="61" BORDER="0"
 SRC="|."$dir".q|img397.png"
 ALT="\begin{displaymath}
X(t) = \sum_{n=0}^{\infty} X_{n} \cos(n \omega t),
\end{displaymath}">|; 

$key = q/cos2D=+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3011.png"
 ALT="$\cos 2D = +1$">|; 

$key = q/|{{bf{a}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3187.png"
 ALT="$\vert{\bf a}\vert$">|; 

$key = q/{displaymath}L=frac{1}{2}left(m_1+m_2+frac{I}{a^2}right)dot{x}^{,2}+g,(m_1-m_2),x+{{rm{const}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="429" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1626.png"
 ALT="\begin{displaymath}
L=\frac{1}{2}\left(m_1+ m_2+\frac{I}{a^2}\right)\dot{x}^{ 2} + g (m_1-m_2) x + {\rm const}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{x_1(omega=omega_0)}{x_1(omega=0)}=frac{omega_0}{2,nu}=Q.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img384.png"
 ALT="\begin{displaymath}
\frac{x_1(\omega=\omega_0)}{x_1(\omega = 0)} = \frac{\omega_0}{2 \nu}
=Q.
\end{displaymath}">|; 

$key = q/displaystyle{{bf{x}_k^T,{{bf{G},{{bf{x}_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1941.png"
 ALT="$\displaystyle {\bf x}_k^T {\bf G} {\bf x}_l$">|; 

$key = q/displaystyle+x',y'left[-frac{2,sintheta,costheta}{A^2}+frac{2,(sin^2theta-cos^2teta),cos{mitDelta}}{A,B}+frac{2,costheta,sintheta}{B^2}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="543" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img555.png"
 ALT="$\displaystyle +x'  y'\left[-\frac{2 \sin\theta \cos\theta}{A^2} + \frac{2 (...
...\theta) \cos{\mit\Delta}}{A B} + \frac{2 \cos\theta \sin\theta}{B^2}\right]$">|; 

$key = q/displaystyleA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2653.png"
 ALT="$\displaystyle A$">|; 

$key = q/displaystyle-frac{(2,k+k')}{m}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2002.png"
 ALT="$\displaystyle - \frac{(2 k+k')}{m}.$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter03slashfig3.06.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="565" HEIGHT="249" BORDER="0"
 SRC="|."$dir".q|img434.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter03/fig3.06.eps}}
\end{figure}">|; 

$key = q/{bf{b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3275.png"
 ALT="$\bf b$">|; 

$key = q/displaystyle-frac{G,M_0}{R_i^{,2}}-frac{G}{R_i^{,2}}sum_{j<i}M_jleft[1+frac{3}{4}right)^2+frac{45}{64}left(frac{R_j}{R_i}right)^4+cdotsright];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="452" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2399.png"
 ALT="$\displaystyle -\frac{G M_0}{R_i^{ 2}}-\frac{G}{R_i^{ 2}} \sum_{j&lt; i}M_j\left...
...R_j}{R_i}\right)^2
+ \frac{45}{64}\left(\frac{R_j}{R_i}\right)^4 +\cdots\right]$">|; 

$key = q/lambda_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1934.png"
 ALT="$\lambda_l$">|; 

$key = q/displaystylez_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2845.png"
 ALT="$\displaystyle z_j$">|; 

$key = q/d^3{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3420.png"
 ALT="$d^3{\bf r}$">|; 

$key = q/displaystylehat{A};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3035.png"
 ALT="$\displaystyle \hat{A}$">|; 

$key = q/{displaymath}leftlangle|{{bf{r}'-{{bf{r}|^{-1}rightrangle=frac{1}{r'}sum_{n=0,infrac{r}{r'}right)^nP_n(costheta),P_n(costheta').{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img2119.png"
 ALT="\begin{displaymath}
\left\langle\vert{\bf r}'-{\bf r}\vert^{-1}\right\rangle = \...
...\left(\frac{r}{r'}\right)^n P_n(\cos\theta) P_n(\cos\theta').
\end{displaymath}">|; 

$key = q/displaystyle-frac{a_0}{3,(1+m^2slash2)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2840.png"
 ALT="$\displaystyle -\frac{a_0}{3 (1+m^2/2)},$">|; 

$key = q/{mitrho};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img643.png"
 ALT="${\mit\rho}$">|; 

$key = q/0.36;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2460.png"
 ALT="$0.36$">|; 

$key = q/Q=1.3757;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$Q=1.3757$">|; 

$key = q/U(x,y,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2638.png"
 ALT="$U(x,y,0)$">|; 

$key = q/(dTslashdr)_{{rm{max};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3454.png"
 ALT="$(dT/dr)_{\rm max}$">|; 

$key = q/omega_i^{,2}=G,M_0slashR_i^{,3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2414.png"
 ALT="$\omega_i^{ 2} = G M_0/R_i^{ 3}$">|; 

$key = q/{displaymath}Phi(r,theta)=sum_{n=0,infty}Phi_n(r),P_n(costheta),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2120.png"
 ALT="\begin{displaymath}
\Phi(r,\theta) = \sum_{n=0,\infty} \Phi_n(r) P_n(\cos\theta),
\end{displaymath}">|; 

$key = q/displaystyleU_{xx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2661.png"
 ALT="$\displaystyle U_{xx}$">|; 

$key = q/c_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2884.png"
 ALT="$c_j$">|; 

$key = q/displaystyleddot{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1225.png"
 ALT="$\displaystyle \ddot{y}$">|; 

$key = q/-mbox{{boldmath{{Omega{}times(mbox{{boldmath{{Omega{}times{{bf{R});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1130.png"
 ALT="$-\mbox{\boldmath$\Omega$}\times(\mbox{\boldmath$\Omega$}\times{\bf R})$">|; 

$key = q/displaystyle-dot{z},frac{partialU}{partialz}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2554.png"
 ALT="$\displaystyle -\dot{z} \frac{\partial U}{\partial z}.$">|; 

$key = q/{displaymath}f_j=-frac{partialU}{partialx_j},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1578.png"
 ALT="\begin{displaymath}
f_j = -\frac{\partial U}{\partial x_j},
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{Chapter15slashfig15.25.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="517" HEIGHT="219" BORDER="0"
 SRC="|."$dir".q|img3120.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{Chapter15/fig15.25.eps}}
\end{figure}">|; 

$key = q/AB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1698.png"
 ALT="$AB$">|; 

$key = q/theta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1516.png"
 ALT="$\theta_2$">|; 

$key = q/X(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img396.png"
 ALT="$X(t)$">|; 

$key = q/{displaymath}Phi(r,theta)=-frac{G,M}{r}+frac{2}{5}frac{G,M,a^2}{r^3},epsilon,P_2(costheta)+{calO}(epsilon^2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="404" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2161.png"
 ALT="\begin{displaymath}
\Phi(r,\theta) = - \frac{G M}{r} +\frac{2}{5}\frac{G M a^2}{r^3} \epsilon P_2(\cos\theta) + {\cal O}(\epsilon^2).
\end{displaymath}">|; 

$key = q/2,(1+g,m^2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2859.png"
 ALT="$2 (1+g m^2)$">|; 

$key = q/{displaymath}mbox{{boldmath{{Omega{}={mitOmega},(coslambda,,0,,sinlambda),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1135.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\Omega$}= {\mit\Omega} (\cos\lambda, 0, \sin\lambda),
\end{displaymath}">|; 

$key = q/displaystylefrac{c_j}{1+3,m^2slash2-Omega_j^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2846.png"
 ALT="$\displaystyle \frac{c_j}{1+3 m^2/2-\Omega_j^{ 2}},$">|; 

$key = q/v_0(t_0)+deltav_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3133.png"
 ALT="$v_0(t_0)
+\delta v_0$">|; 

$key = q/v_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img484.png"
 ALT="$v_0 $">|; 

$key = q/(x',,y',,z');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3234.png"
 ALT="$(x',  y',  z')$">|; 

$key = q/1>27,mu_2,(1-mu_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2689.png"
 ALT="$1 &gt; 27 \mu_2 (1-\mu_2)$">|; 

$key = q/{displaymath}frac{d^2theta}{dt^2}-omega^{,2},sintheta,costheta=-frac{g}{l},sintheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1702.png"
 ALT="\begin{displaymath}
\frac{d^2\theta}{dt^2} -\omega^{ 2} \sin\theta \cos\theta = -\frac{g}{l} \sin\theta,
\end{displaymath}">|; 

$key = q/m=5.97times10^{24},{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2242.png"
 ALT="$m=5.97\times 10^{24} {\rm kg}$">|; 

$key = q/{displaymath}alpha_s=frac{3}{8},I_parallel,n_s^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2283.png"
 ALT="\begin{displaymath}
\alpha_s = \frac{3}{8} I_\parallel n_s^{ 2}.
\end{displaymath}">|; 

$key = q/{displaymath}tanzeta=frac{sintheta}{1-costheta}=tanleft(frac{pi}{2}-frac{theta}{2}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1055.png"
 ALT="\begin{displaymath}
\tan\zeta = \frac{\sin\theta}{1-\cos\theta} = \tan\left(\frac{\pi}{2}-\frac{\theta}{2}\right).
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{2},(dot{eta}_1^{,2}+dot{eta}_2^{,2}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2024.png"
 ALT="$\displaystyle \frac{1}{2} (\dot{\eta}_1^{ 2} + \dot{\eta}_2^{ 2}),$">|; 

$key = q/rho_1=1-beta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2591.png"
 ALT="$\rho_1=1-\beta$">|; 

$key = q/{{bf{a}ast{{bf{b}={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3291.png"
 ALT="${\bf a} \ast {\bf b}= {\bf0}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter03slashfig3.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="272" BORDER="0"
 SRC="|."$dir".q|img335.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter03/fig3.04.eps}}
\end{figure}">|; 

$key = q/deltatheta_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3332.png"
 ALT="$\delta\theta_y$">|; 

$key = q/{displaymath}T_phi({{rm{yr})=frac{n_s}{Omega_phi}=frac{2,T_s({{rm{day})}{epsilon,(1+mu_mslash[T_m({{rm{yr})]^2),costheta_0},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2361.png"
 ALT="\begin{displaymath}
T_\phi({\rm yr}) = \frac{n_s}{\Omega_\phi} = \frac{2 T_s({\rm day})}{\epsilon (1+\mu_m/[T_m({\rm yr})]^2) \cos\theta_0},
\end{displaymath}">|; 

$key = q/{displaymath}U=-frac{1}{2}sum_{k=1,{calF}}lambda_k,eta_k^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1980.png"
 ALT="\begin{displaymath}
U = -\frac{1}{2}\sum_{k=1,{\cal F}}\lambda_k  \eta_k^{ 2}.
\end{displaymath}">|; 

$key = q/I_parallel<I_perp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1462.png"
 ALT="$I_\parallel&lt;I_\perp$">|; 

$key = q/M=5.97times10^{24},{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2181.png"
 ALT="$M = 5.97\times 10^{24} {\rm kg}$">|; 

$key = q/displaystyle(xi-xi_1)^2+(eta-eta_1)^2+zeta^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2510.png"
 ALT="$\displaystyle (\xi-\xi_1)^2+(\eta-\eta_1)^2 + \zeta^2,$">|; 

$key = q/X(t)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img428.png"
 ALT="$X(t)=0$">|; 

$key = q/h=1250;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1217.png"
 ALT="$h=1250$">|; 

$key = q/omega_r=sqrt{omega_0^{,2}-nu^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img324.png"
 ALT="$\omega_r = \sqrt{\omega_0^{ 2}-\nu^2}$">|; 

$key = q/displaystyle+frac{3}{4}sum_{j>i}left(frac{M_j}{M_0}right)left(frac{R_i}{R_j}righight)^2+frac{175}{64}left(frac{R_i}{R_j}right)^4+cdotsright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="484" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2406.png"
 ALT="$\displaystyle + \frac{3}{4}\sum_{j&gt;i}\left(\frac{M_j}{M_0}\right)
\left(\frac{R...
...i}{R_j}\right)^2 + \frac{175}{64}\left(\frac{R_i}{R_j}
\right)^4+\cdots\right].$">|; 

$key = q/T_s({{rm{day})=omegaslashn_s=365.24;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2300.png"
 ALT="$T_s({\rm day}) = \omega/n_s = 365.24$">|; 

$key = q/{displaymath}z_5simeqfrac{3}{8}m,iota.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2939.png"
 ALT="\begin{displaymath}
z_5\simeq \frac{3}{8} m \iota.
\end{displaymath}">|; 

$key = q/5.50;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2447.png"
 ALT="$5.50$">|; 

$key = q/P=(infty,0,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3411.png"
 ALT="$P=(\infty, 0, 0)$">|; 

$key = q/{displaymath}{{bf{rho}=frac{v_perp}{{mitOmega}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img642.png"
 ALT="\begin{displaymath}
{\bf\rho} = \frac{v_\perp}{{\mit\Omega}}
\end{displaymath}">|; 

$key = q/{displaymath}v^2=dot{x}^{,2}+dot{y}^{,2}+dot{z}^{,2}=l^2,(dot{theta}^{,2}+sin^2theta,dot{phi}^{,2}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1676.png"
 ALT="\begin{displaymath}
v^2 = \dot{x}^{ 2} + \dot{y}^{ 2} + \dot{z}^{ 2} = l^2 (\dot{\theta}^{ 2}+ \sin^2\theta \dot{\phi}^{ 2}).
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{grad},hequivleft(frac{partialh}{partialx},,frac{partialh}{partialy}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3443.png"
 ALT="\begin{displaymath}
{\bf grad} h \equiv \left(\frac{\partial h}{\partial x},  \frac{\partial h}{\partial y}
\right).
\end{displaymath}">|; 

$key = q/{displaymath}k=left[frac{(I_{y'y'}-I_{x'x'}),(I_{z'z'}-I_{y'y'})}{I_{x'x'},I_{z'z'}}right]^{1slash2}omega_{y'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1552.png"
 ALT="\begin{displaymath}
k= \left[\frac{(I_{y'y'}-I_{x'x'}) (I_{z'z'}-I_{y'y'})}{I_{x'x'} I_{z'z'}}\right]^{1/2}\omega_{y'}.
\end{displaymath}">|; 

$key = q/0.387;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2422.png"
 ALT="$0.387$">|; 

$key = q/{displaymath}m,v,frac{dv}{dx}=m,g-c,v^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img269.png"
 ALT="\begin{displaymath}
m v \frac{dv}{dx} = m g - c v^2.
\end{displaymath}">|; 

$key = q/{mitDelta}phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1697.png"
 ALT="${\mit\Delta}\phi$">|; 

$key = q/displaystyleomega_perp,sin({mitOmega},t'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1388.png"
 ALT="$\displaystyle \omega_\perp \sin({\mit\Omega} t'),$">|; 

$key = q/x=y+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3389.png"
 ALT="$x=y+1$">|; 

$key = q/{displaymath}frac{dz}{dx}=-h^{-1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1770.png"
 ALT="\begin{displaymath}
\frac{dz}{dx} = -h^{-1}.
\end{displaymath}">|; 

$key = q/displaystyle-kleft(array{{cc}1+2,alpha+2,alpha^2<col_mark>2,alpha,(1+alpha)<row_k>2,alpha,(1+alpha)<col_mark>1+2,alpha+2,alpha^2array{right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2037.png"
 ALT="$\displaystyle -k\left(\begin{array}{cc}
1+2 \alpha+2 \alpha^2&amp; 2 \alpha (1+\alpha)\\\\
2 \alpha (1+\alpha)&amp;1+2 \alpha+2 \alpha^2 \end{array}\right).$">|; 

$key = q/displaystylefrac{T}{m}-g-2,{mitOmega},coslambda,dot{y}'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1191.png"
 ALT="$\displaystyle \frac{T}{m} - g - 2 {\mit\Omega} \cos\lambda \dot{y}'.$">|; 

$key = q/{{bf{x}_icdot{{bf{x}_j=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1311.png"
 ALT="${\bf x}_i \cdot{\bf x}_j=0$">|; 

$key = q/dot{}equivdslashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1154.png"
 ALT="$\dot{}\equiv d/dt$">|; 

$key = q/displaystylec_5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2930.png"
 ALT="$\displaystyle c_5$">|; 

$key = q/-aslash2leqxleq+aslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1759.png"
 ALT="$-a/2\leq x \leq +a/2$">|; 

$key = q/displaystyler^2,dot{theta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img765.png"
 ALT="$\displaystyle r^2 \dot{\theta}$">|; 

$key = q/{displaymath}omega^2+{{rm{i},2,nu,omega-omega_0^{,2}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img318.png"
 ALT="\begin{displaymath}
\omega^2 + {\rm i} 2 \nu \omega - \omega_0^{ 2} = 0.
\end{displaymath}">|; 

$key = q/{{bf{h}={{bf{r}timesdot{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2521.png"
 ALT="${\bf h} = {\bf r}\times \dot{\bf r}$">|; 

$key = q/{displaymath}alpha=rho_2+frac{rho_2^{,2}}{3}+frac{rho_2^{,3}}{3}+frac{51,rho_2^{,4}}{81}+{calO}(rho_2^{,5}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2578.png"
 ALT="\begin{displaymath}
\alpha = \rho_2 + \frac{\rho_2^{ 2}}{3} + \frac{\rho_2^{ 3}}{3} + \frac{51 \rho_2^{ 4}}{81}+ {\cal O}(\rho_2^{ 5}).
\end{displaymath}">|; 

$key = q/epsilon=0.00335;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2184.png"
 ALT="$\epsilon=0.00335$">|; 

$key = q/{displaymath}f(q_1,q_2,cdots,q_{calF},t)=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1814.png"
 ALT="\begin{displaymath}
f(q_1,q_2,\cdots,q_{\cal F},t) = 0.
\end{displaymath}">|; 

$key = q/U_0'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2332.png"
 ALT="$U_0''$">|; 

$key = q/P_4(0)=3slash8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2385.png"
 ALT="$P_4(0)=3/8$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter15slashfig15.32.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="345" BORDER="0"
 SRC="|."$dir".q|img3164.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter15/fig15.32.eps}}
\end{figure}">|; 

$key = q/tsimeqsqrt{2,hslashg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1161.png"
 ALT="$t\simeq \sqrt{2 h/g}$">|; 

$key = q/a_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3219.png"
 ALT="$a_x$">|; 

$key = q/displaystyleddot{x}_1-2,n,dot{y}_1-left(n^2+n'^{,2}slash2right)x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2743.png"
 ALT="$\displaystyle \ddot{x}_1 -2 n \dot{y}_1-\left(n^2+n'^{ 2}/2\right)x_1$">|; 

$key = q/(X_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2779.png"
 ALT="$(X_0$">|; 

$key = q/nu=0.8,omega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img436.png"
 ALT="$\nu= 0.8 \omega_0$">|; 

$key = q/displaystylerho_2,dot{rho}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2517.png"
 ALT="$\displaystyle \rho_2 \dot{\rho}_2$">|; 

$key = q/{{bf{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3530.png"
 ALT="${\bf n}$">|; 

$key = q/psi+dpsi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1077.png"
 ALT="$\psi+d\psi$">|; 

$key = q/mbox{{boldmath{{omega{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1222.png"
 ALT="$\mbox{\boldmath$\omega$}$">|; 

$key = q/displaystylefrac{partialL}{partialdot{r}}=m,dot{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1608.png"
 ALT="$\displaystyle \frac{\partial L}{\partial\dot{r}} = m \dot{r},$">|; 

$key = q/{displaymath}dot{r}^{,2}=(e-1),frac{G,M}{r_p}-(e+1),frac{r_p,G,M}{r^2}+frac{2,G,M}{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img794.png"
 ALT="\begin{displaymath}
\dot{r}^{ 2} = (e-1) \frac{G M}{r_p} - (e+1) \frac{r_p G M}{r^2}
+ \frac{2 G M}{r}.
\end{displaymath}">|; 

$key = q/{displaymath}tanpsi=frac{sintheta}{costheta+m_1slashm_2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1054.png"
 ALT="\begin{displaymath}
\tan\psi = \frac{\sin\theta}{\cos\theta+m_1/m_2},
\end{displaymath}">|; 

$key = q/displaystylecospsi,sintheta,dot{phi}-sinpsi,dot{theta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1448.png"
 ALT="$\displaystyle \cos\psi \sin\theta \dot{\phi}-\sin\psi \dot{\theta},$">|; 

$key = q/{displaymath}R'^{,-1}=frac{1}{R}sum_{n=0,infty}left(frac{r}{R}right)^nP_n(costheta).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img2225.png"
 ALT="\begin{displaymath}
R'^{ -1} = \frac{1}{R}\sum_{n=0,\infty} \left(\frac{r}{R}\right)^n P_n(\cos\theta).
\end{displaymath}">|; 

$key = q/displaystyle-frac{15}{8},m,e,cos[(1-2,m-c,m^2),T+alpha_0],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2958.png"
 ALT="$\displaystyle -\frac{15}{8} m e \cos[(1-2 m-c m^2) T+\alpha_0],$">|; 

$key = q/{{bf{e}_thetaequiv{{bf{e}_ztimes{{bf{e}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img675.png"
 ALT="${\bf e}_\theta\equiv {\bf e}_z\times {\bf e}_r$">|; 

$key = q/I_perp>I_parallel;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1409.png"
 ALT="$I_\perp&gt;I_\parallel$">|; 

$key = q/displaystylecostheta,dot{phi}+dot{psi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1449.png"
 ALT="$\displaystyle \cos\theta \dot{\phi} +\dot{\psi}.$">|; 

$key = q/displaystylesqrt{frac{m}{2}},(q_1-q_2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2022.png"
 ALT="$\displaystyle \sqrt{\frac{m}{2}} (q_1-q_2).$">|; 

$key = q/{displaymath}mu,frac{d^2{{bf{r}}{dt^2}={{bf{f},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img926.png"
 ALT="\begin{displaymath}
\mu \frac{d^2{\bf r}}{d t^2} = {\bf f},
\end{displaymath}">|; 

$key = q/tllv_tslashg;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img260.png"
 ALT="$t\ll v_t/g$">|; 

$key = q/20^{''};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2377.png"
 ALT="$20^{''}$">|; 

$key = q/u=r^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img772.png"
 ALT="$u=r^{-1}$">|; 

$key = q/{displaymath}v^2=v_x^{,2}+v_y^{,2}=dot{x}^{,2}+2,dot{x},dot{x}',costheta+dot{x}'^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img1648.png"
 ALT="\begin{displaymath}
v^2 = v_x^{ 2} + v_y^{ 2} = \dot{x}^{ 2} + 2 \dot{x} \dot{x}' \cos\theta+ \dot{x}'^{ 2}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="276" BORDER="0"
 SRC="|."$dir".q|img3054.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.02.eps}}
\end{figure}">|; 

$key = q/{{bf{a},%,{{bf{b}=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3253.png"
 ALT="${\bf a} \% {\bf b} = 1/2$">|; 

$key = q/c=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1776.png"
 ALT="$c=0$">|; 

$key = q/displaystyle3+mu_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2618.png"
 ALT="$\displaystyle 3 + \mu_2,$">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{Chapter15slashfig15.24.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="517" HEIGHT="219" BORDER="0"
 SRC="|."$dir".q|img3119.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{Chapter15/fig15.24.eps}}
\end{figure}">|; 

$key = q/pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img230.png"
 ALT="$\pm$">|; 

$key = q/omega_5=omega_4-omega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2873.png"
 ALT="$\omega_5=\omega_4-\omega_1$">|; 

$key = q/dot{phi}=dot{phi}_0={{rm{const}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1689.png"
 ALT="$\dot{\phi} = \dot{\phi}_0 = {\rm const}.$">|; 

$key = q/displaystyleleft{v(0)-frac{A,omega^2slashQ}{[(1-omega^2)^2+omega^2slashQ^2]}right}{{rm{e}^{-tslash2Q},cosomega_astt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3048.png"
 ALT="$\displaystyle \left\{v(0) - \frac{A \omega^2/Q}{[(1-\omega^2)^2+\omega^2/Q^2]}\right\}
{\rm e}^{-t/2Q} \cos\omega_\ast t$">|; 

$key = q/{calL}(a,x)=a,{calL}(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img315.png"
 ALT="${\cal L}(a x)=a {\cal L}(x)$">|; 

$key = q/rho({{bf{r}'),d^3{{bf{r}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2065.png"
 ALT="$\rho({\bf r}') d^3{\bf r}'$">|; 

$key = q/{displaymath}x=a,{{rm{e}^{-{{rm{i},omega,t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img310.png"
 ALT="\begin{displaymath}
x = a {\rm e}^{-{\rm i} \omega t},
\end{displaymath}">|; 

$key = q/{{bf{a}ast{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3288.png"
 ALT="${\bf a} \ast {\bf b}$">|; 

$key = q/r=r_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img789.png"
 ALT="$r=r_p$">|; 

$key = q/{displaymath}U=U_0+U_x,deltax+U_y,deltay+frac{1}{2},U_{xx},(deltax)^2+U_{xy},deltax,deltay+frac{1}{2},U_{yy},(deltay)^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="539" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2639.png"
 ALT="\begin{displaymath}
U = U_0 + U_x \delta x+ U_y \delta y + \frac{1}{2} U_{xx}...
..._{xy} \delta x \delta y
+ \frac{1}{2} U_{yy} (\delta y)^2,
\end{displaymath}">|; 

$key = q/{{bf{a}={{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3264.png"
 ALT="${\bf a} = {\bf b}$">|; 

$key = q/displaystyleomega_{z'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1395.png"
 ALT="$\displaystyle \omega_{z'}$">|; 

$key = q/displaystyleI,ddot{phi}+lambda,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1823.png"
 ALT="$\displaystyle I \ddot{\phi} + \lambda a$">|; 

$key = q/{displaymath}{{bf{v}_{EB}=frac{{{bf{E}times{{bf{B}}{B^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img626.png"
 ALT="\begin{displaymath}
{\bf v}_{EB} = \frac{{\bf E}\times{\bf B}}{B^2},
\end{displaymath}">|; 

$key = q/omega_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2848.png"
 ALT="$\omega_j$">|; 

$key = q/{displaymath}r(theta)=frac{r_c}{1-e,costheta},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img751.png"
 ALT="\begin{displaymath}
r(\theta) = \frac{r_c}{1 - e \cos\theta},
\end{displaymath}">|; 

$key = q/m';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2049.png"
 ALT="$m'$">|; 

$key = q/stackrel{displaystylerightarrow}{SR};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3197.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{SR}$">|; 

$key = q/{displaymath}frac{dsigma(psi)}{d{mitOmega}'},d{mitOmega}'=frac{dsigma(theta)}{d{mitOmega}},d{mitOmega},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1080.png"
 ALT="\begin{displaymath}
\frac{d\sigma(\psi)}{d{\mit\Omega}'}  d{\mit\Omega}' =
\frac{d\sigma(\theta)}{d{\mit\Omega}} d{\mit\Omega},
\end{displaymath}">|; 

$key = q/0.09008;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2826.png"
 ALT="$0.09008$">|; 

$key = q/T({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3405.png"
 ALT="$T({\bf r})$">|; 

$key = q/displaystylesum_{j>0}z_j,sin(Omega_j,T-gamma_j).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2838.png"
 ALT="$\displaystyle \sum_{j&gt;0}z_j \sin(\Omega_j T-\gamma_j).$">|; 

$key = q/displaystyle-U_{xy},deltax-U_{yy},deltay,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2648.png"
 ALT="$\displaystyle - U_{xy} \delta x -U_{yy} \delta y,$">|; 

$key = q/displaystyle-V_0,sin(theta+2,{mitOmega},sinlambda,t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1175.png"
 ALT="$\displaystyle -V_0 \sin(\theta+2 {\mit\Omega} \sin\lambda t).$">|; 

$key = q/displaystyle2,e,sinM+frac{5}{4},e^2,sin2,M-frac{1}{4},iota^2,sin2,F+frac{11}{8},m^2,sin2,D+frac{15}{4},m,e,sin(2,D-M);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="640" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2999.png"
 ALT="$\displaystyle 2 e \sin M+ \frac{5}{4} e^2 \sin 2 M
- \frac{1}{4} \iota^2 \sin 2 F+ \frac{11}{8} m^2 \sin 2 D+\frac{15}{4} m e \sin(2 D-M)$">|; 

$key = q/{displaymath}T=frac{2pi}{dot{phi}}=0.997,,{{rm{days}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1481.png"
 ALT="\begin{displaymath}
T = \frac{2\pi}{\dot{\phi}} = 0.997  {\rm days}.
\end{displaymath}">|; 

$key = q/4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img617.png"
 ALT="$4$">|; 

$key = q/+pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img799.png"
 ALT="$+\pi$">|; 

$key = q/displaystyle{{bf{G};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1993.png"
 ALT="$\displaystyle {\bf G}$">|; 

$key = q/{mitDelta}=pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img572.png"
 ALT="${\mit\Delta}=\pi/2$">|; 

$key = q/displaystylex_1slasha,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2754.png"
 ALT="$\displaystyle x_1/a,$">|; 

$key = q/R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img196.png"
 ALT="$R$">|; 

$key = q/v_0<EslashB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img649.png"
 ALT="$v_0&lt;E/B$">|; 

$key = q/2,gamma_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2860.png"
 ALT="$2 \gamma_0$">|; 

$key = q/{displaymath}V(Q+dx)=V(Q)+int_Q^{Q+dx}{{bf{A}cdotd{{bf{r}=V(Q)+A_x,dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img3469.png"
 ALT="\begin{displaymath}
V(Q+dx) = V(Q) + \int_Q^{Q+dx} {\bf A}\cdot d{\bf r} = V(Q) + A_x dx.
\end{displaymath}">|; 

$key = q/xy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1258.png"
 ALT="$xy$">|; 

$key = q/displaystylex'^{,2}left[frac{cos^2theta}{A^2}-frac{2,costheta,sintheta,cos{mitDelta}}{A,B}+frac{sin^2theta}{B^2}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img553.png"
 ALT="$\displaystyle x'^{ 2}\left[\frac{\cos^2\theta}{A^2} - \frac{2 \cos\theta \sin\theta \cos{\mit\Delta}}{A B} + \frac{\sin^2\theta}{B^2}\right]$">|; 

$key = q/{displaymath}omega=left(frac{|U_0|}{I_0}right)^{1slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img187.png"
 ALT="\begin{displaymath}
\omega = \left(\frac{\vert U_0\vert}{I_0}\right)^{1/2},
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}+({{bf{b}+{{bf{c})=({{bf{a}+{{bf{b})+{{bf{c}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3199.png"
 ALT="\begin{displaymath}
{\bf a} +
({\bf b} + {\bf c}) = ({\bf a} + {\bf b}) + {\bf c}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d(r^2,dot{theta})}{dt}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img735.png"
 ALT="\begin{displaymath}
\frac{d(r^2 \dot{\theta})}{dt} = 0,
\end{displaymath}">|; 

$key = q/1.2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3071.png"
 ALT="$1.2$">|; 

$key = q/int_P^Q{{bf{A}cdotd{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3465.png"
 ALT="$\int_P^Q {\bf A}\cdot d{\bf r}$">|; 

$key = q/W;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3278.png"
 ALT="$W$">|; 

$key = q/{mitOmega}^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1158.png"
 ALT="${\mit\Omega}^2$">|; 

$key = q/q_2-q_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1986.png"
 ALT="$q_2-q_1$">|; 

$key = q/omega_1=1+c,m^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2950.png"
 ALT="$\omega_1=1+c m^2$">|; 

$key = q/Q=1.24;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$Q=1.24$">|; 

$key = q/{displaymath}a=frac{omega_0^{,2},X_1}{omega_0^{,2}-omega^2-{{rm{i},2,nu,omega}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img368.png"
 ALT="\begin{displaymath}
a = \frac{\omega_0^{ 2} X_1}{\omega_0^{ 2}-\omega^2 - {\rm i} 2 \nu \omega}.
\end{displaymath}">|; 

$key = q/{displaymath}R_c=left(2,frac{m'}{m}right)^{1slash3}a,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img2260.png"
 ALT="\begin{displaymath}
R_c = \left(2 \frac{m'}{m}\right)^{1/3} a,
\end{displaymath}">|; 

$key = q/displaystylex_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2914.png"
 ALT="$\displaystyle x_3$">|; 

$key = q/displaystylehat{omega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3031.png"
 ALT="$\displaystyle \hat{\omega}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="272" BORDER="0"
 SRC="|."$dir".q|img3064.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.04.eps}}
\end{figure}">|; 

$key = q/{displaymath}phi_n=tan^{-1}left(frac{2,nu,n,omega}{omega_0^{,2}-n^2,omega^2}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img424.png"
 ALT="\begin{displaymath}
\phi_n = \tan^{-1}\left(\frac{2 \nu n \omega}{\omega_0^{ 2}-n^2 \omega^2}\right).
\end{displaymath}">|; 

$key = q/I_{xz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1333.png"
 ALT="$I_{xz}$">|; 

$key = q/theta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img450.png"
 ALT="$\theta=0$">|; 

$key = q/{displaymath}K=frac{1}{2},m_1,dot{x}^{,2}+frac{1}{2},m_2,(-dot{x}+dot{x}')^2+frac{1}{2},m_3,(-dot{x}-dot{x}')^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1632.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} m_1 \dot{x}^{ 2} + \frac{1}{2} m_2 (-\dot{x}+ \dot{x}')^2 + \frac{1}{2} m_3 (-\dot{x} - \dot{x}')^2,
\end{displaymath}">|; 

$key = q/displaystylefrac{A}{m,g},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3036.png"
 ALT="$\displaystyle \frac{A}{m g},$">|; 

$key = q/{displaymath}m,frac{dv}{dt}=f(v).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img243.png"
 ALT="\begin{displaymath}
m \frac{dv}{dt} = f(v).
\end{displaymath}">|; 

$key = q/{{bf{x}_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1935.png"
 ALT="${\bf x}_l$">|; 

$key = q/displaystylebeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2594.png"
 ALT="$\displaystyle \beta$">|; 

$key = q/{displaymath}T^2=frac{4pi^2,a^3}{G,M}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img756.png"
 ALT="\begin{displaymath}
T^2 = \frac{4\pi^2 a^3}{G M}.
\end{displaymath}">|; 

$key = q/i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="$i$">|; 

$key = q/{displaymath}int_0^vfrac{dv'}{1-(v'slashv_t)^2}=g,t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img255.png"
 ALT="\begin{displaymath}
\int_0^v \frac{dv'}{1-(v'/v_t)^2} = g t,
\end{displaymath}">|; 

$key = q/{displaymath}deltal=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1720.png"
 ALT="\begin{displaymath}
\delta l = 0.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2x}{dt^2}+2,nu,frac{dx}{dt}+omega_0^{,2},x=omega_0^{,2},{{rm{e}^{-{{rm{i},n,omega,t}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img416.png"
 ALT="\begin{displaymath}
\frac{d^2 x}{dt^2} + 2 \nu \frac{dx}{dt} + \omega_0^{ 2} x
= \omega_0^{ 2} {\rm e}^{-{\rm i} n \omega t}.
\end{displaymath}">|; 

$key = q/xi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2485.png"
 ALT="$\xi$">|; 

$key = q/{displaymath}frac{Deltaomega}{omega_0}=frac{2,nu}{omega_0}=frac{1}{Q}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img389.png"
 ALT="\begin{displaymath}
\frac{\Delta \omega}{\omega_0} = \frac{2 \nu}{\omega_0} = \frac{1}{Q}.
\end{displaymath}">|; 

$key = q/{displaymath}C=-2,U-v^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img2556.png"
 ALT="\begin{displaymath}
C = - 2 U - v^2
\end{displaymath}">|; 

$key = q/{{bf{f}_{ij}=-{{bf{f}_{ji};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1231.png"
 ALT="${\bf f}_{ij} = - {\bf f}_{ji}$">|; 

$key = q/displaystyledot{q}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1853.png"
 ALT="$\displaystyle \dot{q}_i$">|; 

$key = q/{displaymath}frac{mu_2}{3,mu_1}=frac{rho_2^{,3},(1-rho_2+rho_2^{,2}slash3)}{(1+rho_2+rho_2^{,2}),(1-rho_2)^3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2575.png"
 ALT="\begin{displaymath}
\frac{\mu_2}{3 \mu_1} = \frac{\rho_2^{ 3} (1-\rho_2+\rho_2^{ 2}/3)}{(1+\rho_2+\rho_2^{ 2}) (1-\rho_2)^3}.
\end{displaymath}">|; 

$key = q/displaystylecospsi,sintheta,dot{phi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1451.png"
 ALT="$\displaystyle \cos\psi \sin\theta \dot{\phi},$">|; 

$key = q/displaystylefrac{1}{sin^2{mitDelta}}left[frac{cos^2theta}{A^2}-frac{2,costheta,sintheta,cos{mitDelta}}{A,B}+frac{sin^2theta}{B^2}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img562.png"
 ALT="$\displaystyle \frac{1}{\sin^2{\mit\Delta}}\left[\frac{\cos^2\theta}{A^2} - \fra...
...s\theta \sin\theta \cos{\mit\Delta}}{A B} + \frac{\sin^2\theta}{B^2}\right],$">|; 

$key = q/{displaymath}T_phisimeq79,200,,{{rm{years}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2301.png"
 ALT="\begin{displaymath}
T_\phi\simeq 79,200  {\rm years}.
\end{displaymath}">|; 

$key = q/displaystyleC_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2617.png"
 ALT="$\displaystyle C_3$">|; 

$key = q/tau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img452.png"
 ALT="$\tau$">|; 

$key = q/{displaymath}z'=h-frac{g,t^{,2}}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1157.png"
 ALT="\begin{displaymath}
z' = h - \frac{g t^{ 2}}{2}.
\end{displaymath}">|; 

$key = q/{displaymath}K=frac{m}{2},(dot{q}_1^{,2}+dot{q}_2^{,2}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1988.png"
 ALT="\begin{displaymath}
K = \frac{m}{2}  (\dot{q}_1^{ 2} + \dot{q}_2^{ 2}),
\end{displaymath}">|; 

$key = q/thetarightarrowtheta+2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img823.png"
 ALT="$\theta\rightarrow
\theta+2\pi$">|; 

$key = q/gsimeq9.8,{{rm{mslashs^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1129.png"
 ALT="$g\simeq 9.8 {\rm m/s^2}$">|; 

$key = q/a<l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1757.png"
 ALT="$a &lt; l$">|; 

$key = q/displaystyledeltaY;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2809.png"
 ALT="$\displaystyle \delta Y$">|; 

$key = q/{displaymath}U=-frac{1}{2}sum_{k,l=1,{calF}}eta_k,eta_l,{{bf{x}_k^T,{{bf{G},{{bf{x}_l.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1979.png"
 ALT="\begin{displaymath}
U =- \frac{1}{2}\sum_{k,l=1,{\cal F}} \eta_k \eta_l {\bf x}_k^T {\bf G} {\bf x}_l.
\end{displaymath}">|; 

$key = q/M=(4pislash3),a^3,gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2137.png"
 ALT="$M=(4\pi/3) a^3 \gamma$">|; 

$key = q/{displaymath}{{bf{g}'={{bf{g}-mbox{{boldmath{{Omega{}times(mbox{{boldmath{{Omega{}times{{bf{R}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1126.png"
 ALT="\begin{displaymath}
{\bf g}' = {\bf g} - \mbox{\boldmath$\Omega$}\times(\mbox{\boldmath$\Omega$}\times{\bf R}),
\end{displaymath}">|; 

$key = q/{displaymath}y'^{,2},(-rho,g,y+lambda),[1+y'^{,2}]^{-1slash2}-(-rho,g,y+lambda),[1+y'^{,2}]^{1slash2}=k,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="494" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1765.png"
 ALT="\begin{displaymath}
y'^{ 2} (-\rho g y+\lambda) [1+y'^{ 2}]^{-1/2} - (-\rho g y+\lambda) [1+y'^{ 2}]^{1/2} = k,
\end{displaymath}">|; 

$key = q/alpha=kslash(E,b);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img992.png"
 ALT="$\alpha = k/(E b)$">|; 

$key = q/-{{bf{v}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1017.png"
 ALT="$-{\bf v}_2$">|; 

$key = q/dV=dx,dy,dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3419.png"
 ALT="$dV = dx  dy  dz$">|; 

$key = q/2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img616.png"
 ALT="$2$">|; 

$key = q/{{bf{v}_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1011.png"
 ALT="${\bf v}_2'$">|; 

$key = q/{displaymath}ddot{s}=-frac{g}{l},s-{{rm{i},2,{mitOmega},sinlambda,dot{s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1198.png"
 ALT="\begin{displaymath}
\ddot{s} = -\frac{g}{l} s - {\rm i} 2 {\mit\Omega} \sin\lambda \dot{s}.
\end{displaymath}">|; 

$key = q/{{bf{r}_1=(xi_1,,eta_1,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2498.png"
 ALT="${\bf r}_1=(\xi_1, \eta_1, 0)$">|; 

$key = q/thetasimv_tslashv_0ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img609.png"
 ALT="$\theta\sim v_t/v_0\ll 1$">|; 

$key = q/iota=5.161^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2723.png"
 ALT="$\iota=5.161^\circ$">|; 

$key = q/displaystyleepsilon,phi_1(t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2343.png"
 ALT="$\displaystyle \epsilon \phi_1(t),$">|; 

$key = q/theta=alpha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2820.png"
 ALT="$\theta=\alpha_0$">|; 

$key = q/2,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1553.png"
 ALT="$2 a$">|; 

$key = q/M';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3020.png"
 ALT="$M'$">|; 

$key = q/phi_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img371.png"
 ALT="$\phi_1$">|; 

$key = q/{displaymath}frac{d^2I}{dt^2}=-2,U+c,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img184.png"
 ALT="\begin{displaymath}
\frac{d^2 I}{dt^2} = -2 U + c,
\end{displaymath}">|; 

$key = q/{displaymath}sum_{i=1,n}x_i,frac{partialf}{partialx_i}=a,f{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img178.png"
 ALT="\begin{displaymath}
\sum_{i=1,n} x_i \frac{\partial f}{\partial x_i} = a f
\end{displaymath}">|; 

$key = q/s_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1206.png"
 ALT="$s_-$">|; 

$key = q/(1+c,m^2),n,t-alpha_0=j,2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2977.png"
 ALT="$(1+c m^2) n t -\alpha_0=j 2\pi$">|; 

$key = q/delta_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3153.png"
 ALT="$\delta_k$">|; 

$key = q/x=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img285.png"
 ALT="$x=0$">|; 

$key = q/{{bf{x}_i^{ast,T}{{bf{x}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1297.png"
 ALT="${\bf x}_i^{\ast T}{\bf x}_i$">|; 

$key = q/omega_3simeq2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2913.png"
 ALT="$\omega_3\simeq 2$">|; 

$key = q/{displaymath}stackrel{displaystylerightarrow}{PR},=,stackrel{displaystylerightararrow}{PS}+stackrel{displaystylerightarrow}{SR}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img3195.png"
 ALT="\begin{displaymath}
\stackrel{\displaystyle \rightarrow}{PR}  =   \stackrel{\...
...le \rightarrow}{PS}+\stackrel{\displaystyle \rightarrow}{SR}.
\end{displaymath}">|; 

$key = q/{{bf{v}=d{{bf{r}slashdtequiv(dxslashdt,,dyslashdt,,dzslashdt);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="${\bf v} = d{\bf r}/dt \equiv (dx/dt, dy/dt, dz/dt)$">|; 

$key = q/0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img359.png"
 ALT="$0$">|; 

$key = q/{displaymath}{{bf{r}=alpha,{{bf{a}+beta,{{bf{b}+gamma,{{bf{c}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img3346.png"
 ALT="\begin{displaymath}
{\bf r} = \alpha  {\bf a} + \beta {\bf b} + \gamma  {\bf c}.
\end{displaymath}">|; 

$key = q/{mitDelta}R=13,{{rm{mi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img910.png"
 ALT="${\mit\Delta}R = 13 {\rm mi}$">|; 

$key = q/{displaymath}{{bf{F}=sum_{i=1,N}{{bf{F}_i{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img168.png"
 ALT="\begin{displaymath}
{\bf F} = \sum_{i=1,N} {\bf F}_i
\end{displaymath}">|; 

$key = q/{displaymath}s=s_0,{{rm{e}^{-{{rm{i},omega,t}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1199.png"
 ALT="\begin{displaymath}
s = s_0 {\rm e}^{-{\rm i} \omega t}.
\end{displaymath}">|; 

$key = q/{displaymath}Phi(r)=-frac{G}{r}int_0^a4pi,r'^{,2},rho(r'),dr'=-frac{G,M}{r}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2141.png"
 ALT="\begin{displaymath}
\Phi(r) = - \frac{G}{r}\int_0^a 4\pi r'^{ 2} \rho(r') dr' = -\frac{G M}{r}
\end{displaymath}">|; 

$key = q/r,sintheta,Omega^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2167.png"
 ALT="$r \sin\theta \Omega^2$">|; 

$key = q/displaystylexcostheta+ysintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3235.png"
 ALT="$\displaystyle x\cos\theta + y\sin \theta,$">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}=lambda,{{bf{e}_{x'}+omega_{y'},{{bf{e}_{y'}+mu,{{bf{e}_{z'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1549.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$} = \lambda {\bf e}_{x'} + \omega_{y'} {\bf e}_{y'} + \mu {\bf e}_{z'}.
\end{displaymath}">|; 

$key = q/displaystyle(a_x,costheta+a_y,sintheta),(b_x,costheta+b_y,sintheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3257.png"
 ALT="$\displaystyle (a_x  \cos\theta+a_y \sin\theta) (b_x \cos\theta + b_y \sin\theta)$">|; 

$key = q/z=z(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img654.png"
 ALT="$z=z(x)$">|; 

$key = q/25,800,{{rm{years};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2366.png"
 ALT="$25,800 {\rm years}$">|; 

$key = q/dot{q}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1589.png"
 ALT="$\dot{q}_i$">|; 

$key = q/{displaymath}|{{bf{r}'-{{bf{r}|^{-1}=(r^{2}-2,r,r',F+r'^{,2})^{-1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2086.png"
 ALT="\begin{displaymath}
\vert{\bf r}'-{\bf r}\vert^{-1} = (r^{2}-2 r r' F + r'^{ 2})^{-1/2}.
\end{displaymath}">|; 

$key = q/{{bf{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="${\bf p}$">|; 

$key = q/displaystyletheta_1(t_0+{mitDelta}t)-theta_0(t_0+{mitDelta}t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3137.png"
 ALT="$\displaystyle \theta_1(t_0+{\mit\Delta} t)-\theta_0(t_0+{\mit\Delta} t),$">|; 

$key = q/displaystylen,m,{{bf{a}=m,(n,{{bf{a}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3209.png"
 ALT="$\displaystyle n m {\bf a} = m (n {\bf a}),$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{Chapter15slashfig15.15.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="539" HEIGHT="402" BORDER="0"
 SRC="|."$dir".q|img3082.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{Chapter15/fig15.15.eps}}
\end{figure}">|; 

$key = q/lambda(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1810.png"
 ALT="$\lambda(t)$">|; 

$key = q/displaystylefrac{Aleft[(1-omega^2),cos(omega,t)+(omegaslashQ),sin(omega,t)right]}{left[(1-omega^2)^2+omega^2slashQ^2right]},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3055.png"
 ALT="$\displaystyle \frac{A\left[(1-\omega^2) \cos(\omega  t)+ (\omega/Q) \sin(\omega  t)\right]}
{\left[(1-\omega^2)^2+\omega^2/Q^2\right]},$">|; 

$key = q/{displaymath}{{bf{a},%,{{bf{b}equiva_x,b_z+a_y,b_x+a_z,b_y={{rm{scalar~number},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="389" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img3246.png"
 ALT="\begin{displaymath}
{\bf a} \% {\bf b} \equiv a_x  b_z + a_y  b_x + a_z  b_y = {\rm scalar&nbsp;number},
\end{displaymath}">|; 

$key = q/displaystyleiota,sin[(1+g,m^2),T-gamma_0]+e,iota,sin[(c-g),m^2,T-alpha_0+gamma_0];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="465" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2970.png"
 ALT="$\displaystyle \iota \sin[(1+g m^2) T-\gamma_0] + e \iota \sin[(c-g) m^2 T-\alpha_0+\gamma_0]$">|; 

$key = q/rleqa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2131.png"
 ALT="$r\leq a$">|; 

$key = q/cos(omega,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img392.png"
 ALT="$\cos(\omega t)$">|; 

$key = q/mu_m=0.0123;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2364.png"
 ALT="$\mu_m=0.0123$">|; 

$key = q/Q=1.30;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$Q=1.30$">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{Chapter15slashfig15.33.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="517" HEIGHT="219" BORDER="0"
 SRC="|."$dir".q|img3167.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{Chapter15/fig15.33.eps}}
\end{figure}">|; 

$key = q/{displaymath}V,frac{b^2}{a^2}left[left(frac{a^4}{b^4}+frac{1}{4}right)^{1slash2}-frac{1}{2}right]^{1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="69" BORDER="0"
 SRC="|."$dir".q|img1095.png"
 ALT="\begin{displaymath}
V \frac{b^2}{a^2}\left[\left(\frac{a^4}{b^4}+\frac{1}{4}\right)^{1/2}-\frac{1}{2}\right]^{1/2}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{AppendixAslashfigA.09.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="164" BORDER="0"
 SRC="|."$dir".q|img3345.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{AppendixA/figA.09.eps}}
\end{figure}">|; 

$key = q/{displaymath}alpha_1=frac{3}{4},m+frac{255}{32},m^2+{calO}(m^3).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2981.png"
 ALT="\begin{displaymath}
\alpha_1 = \frac{3}{4} m+ \frac{255}{32} m^2 + {\cal O}(m^3).
\end{displaymath}">|; 

$key = q/displaystyleddot{x},dot{x}-2,omega,dot{x},dot{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2549.png"
 ALT="$\displaystyle \ddot{x} \dot{x} - 2 \omega \dot{x} \dot{y}$">|; 

$key = q/dot{~}equivdslashdT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2771.png"
 ALT="$\dot{&nbsp;}\equiv d/dT$">|; 

$key = q/gamma_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2851.png"
 ALT="$\gamma_j$">|; 

$key = q/{displaymath}-esimeqfrac{-(225slash16),m^3,e}{-(3slash2),m^2-2,c,m^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2951.png"
 ALT="\begin{displaymath}
-e \simeq \frac{ - (225/16) m^3 e}{-(3/2) m^2-2 c m^2},
\end{displaymath}">|; 

$key = q/{displaymath}K=frac{1}{2},frac{p^{,2}}{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1860.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} \frac{p^{ 2}}{m}.
\end{displaymath}">|; 

$key = q/displaystyle(2,m)^{-1slash2},(1,-1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2043.png"
 ALT="$\displaystyle (2 m)^{-1/2} (1,-1),$">|; 

$key = q/{displaymath}m,frac{d^2{{bf{r}}{dt^2}={{bf{f}=-k,{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img527.png"
 ALT="\begin{displaymath}
m \frac{d^2{\bf r}}{dt^2} = {\bf f} = - k {\bf r}.
\end{displaymath}">|; 

$key = q/displaystyle-(xi_1,dot{xi}+eta_1,dot{eta})+omega,(xi,eta_1-eta,xi_1)+xi,dot{xi}+eta,dot{eta}+zeta,dot{zeta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="411" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2516.png"
 ALT="$\displaystyle -(\xi_1 \dot{\xi}+\eta_1 \dot{\eta}) + \omega (\xi \eta_1-\eta \xi_1) + \xi \dot{\xi}
+ \eta \dot{\eta} + \zeta \dot{\zeta},$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter05slashfig5.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="365" HEIGHT="251" BORDER="0"
 SRC="|."$dir".q|img682.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter05/fig5.02.eps}}
\end{figure}">|; 

$key = q/{{bf{a}times({{bf{b}times{{bf{c});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3350.png"
 ALT="${\bf a}\times({\bf b} \times {\bf c})$">|; 

$key = q/2{calF};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1857.png"
 ALT="$2{\cal F}$">|; 

$key = q/F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1736.png"
 ALT="$F$">|; 

$key = q/rho<rho_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img506.png"
 ALT="$\rho&lt;\rho_0$">|; 

$key = q/A=pi,a,b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img754.png"
 ALT="$A=\pi a b$">|; 

$key = q/M_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1901.png"
 ALT="$M_{ij}$">|; 

$key = q/{displaymath}frac{dsigma}{d{mitOmega}}=frac{b}{sintheta}left|frac{db}{dtheta}right|{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img978.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d{\mit\Omega}} = \frac{b}{\sin\theta}\left\vert\frac{db}{d\theta}\right\vert
\end{displaymath}">|; 

$key = q/K=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img242.png"
 ALT="$K=0$">|; 

$key = q/displaystylex';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$\displaystyle x'$">|; 

$key = q/{displaymath}p_x=frac{partialL}{partialdot{x}}=M,dot{x}+m,(dot{x}+dot{x}',costheta){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1669.png"
 ALT="\begin{displaymath}
p_x = \frac{\partial L}{\partial \dot{x}} = M \dot{x} + m (\dot{x}+\dot{x}' \cos\theta)
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{c}={{bf{a}+{{bf{b}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3193.png"
 ALT="\begin{displaymath}
{\bf c} = {\bf a} + {\bf b}.
\end{displaymath}">|; 

$key = q/n=1,l=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3179.png"
 ALT="$n=1, l=-1$">|; 

$key = q/{{bf{r}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img911.png"
 ALT="${\bf r}_1$">|; 

$key = q/2,epsilonslash(k,r_c^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img883.png"
 ALT="$2 \epsilon/(k r_c^{ 2})$">|; 

$key = q/{displaymath}UsimeqU_0'-epsilon,alpha_m,cos(2,theta)+epsilon,beta_m,sin(2,theta),cos(varpi_n-phi),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="443" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2321.png"
 ALT="\begin{displaymath}
U \simeq U_0' - \epsilon \alpha_m \cos(2 \theta) + \epsilon \beta_m \sin(2 \theta) \cos(\varpi_n-\phi),
\end{displaymath}">|; 

$key = q/[k_1,k_2slash(k_1+k_2),m]^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img504.png"
 ALT="$[k_1 k_2/(k_1+k_2) m]^{1/2}$">|; 

$key = q/displaystyle{{bf{x}_i';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1313.png"
 ALT="$\displaystyle {\bf x}_i'$">|; 

$key = q/x_t=mslash(2,c);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img272.png"
 ALT="$x_t = m/ (2 c)$">|; 

$key = q/lambda'=lambdaslashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1767.png"
 ALT="$\lambda' = \lambda/k$">|; 

$key = q/Qsimeq1.375;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3095.png"
 ALT="$Q\simeq 1.375$">|; 

$key = q/a_pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img321.png"
 ALT="$a_\pm$">|; 

$key = q/d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img283.png"
 ALT="$d$">|; 

$key = q/5.20;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2434.png"
 ALT="$5.20$">|; 

$key = q/q_1=q_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2015.png"
 ALT="$q_1 = q_2$">|; 

$key = q/displaystylefrac{d}{dt}!left(frac{partialL}{partialdot{x}'}right)-frac{partialL}{partialx'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1638.png"
 ALT="$\displaystyle \frac{d}{dt}\!\left(\frac{\partial L}{\partial\dot{x}'}\right) - \frac{\partial L}{\partial x'}$">|; 

$key = q/{displaymath}dot{u}^{,2}=2,(E'-m,g,l,u),(1-u^2),I_perp^{-1}-(L_phi-L_psi,u)^2,I_perp^{-2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1502.png"
 ALT="\begin{displaymath}
\dot{u}^{ 2} = 2 (E' -m g l u) (1-u^2) I_\perp^{-1} - (L_\phi-L_\psi u)^2 I_\perp^{-2},
\end{displaymath}">|; 

$key = q/displaystyleB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2655.png"
 ALT="$\displaystyle B$">|; 

$key = q/displaystylefrac{d}{dt}!left(frac{partial{calL}}{partialdot{phi}}right)-frac{partial{calL}}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2336.png"
 ALT="$\displaystyle \frac{d}{dt}\!\left(\frac{\partial {\cal L}}{\partial \dot{\phi}}\right)-\frac{\partial{\cal L}}{\partial \phi}$">|; 

$key = q/{displaymath}psisimeqpileft(1+frac{3,h^2}{c^2,r^2}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2467.png"
 ALT="\begin{displaymath}
\psi \simeq \pi\left(1+ \frac{3 h^2}{c^2 r^2}\right).
\end{displaymath}">|; 

$key = q/{displaymath}frac{dK}{dt}={{bf{f}cdot{{bf{v}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="\begin{displaymath}
\frac{d K}{dt} = {\bf f}\cdot {\bf v}.
\end{displaymath}">|; 

$key = q/displaystyle{{bf{r}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img923.png"
 ALT="$\displaystyle {\bf r}_2$">|; 

$key = q/2,m,v_0^{3slash2}slash(3,c);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img495.png"
 ALT="$2 m v_0^{3/2}/(3 c)$">|; 

$key = q/1.37575;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3116.png"
 ALT="$1.37575$">|; 

$key = q/lambda_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1545.png"
 ALT="$\lambda_0$">|; 

$key = q/displaystyleP_2(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2109.png"
 ALT="$\displaystyle P_2(x)$">|; 

$key = q/(deltadot{Psi})_{th};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2445.png"
 ALT="$(\delta\dot{\Psi})_{th}$">|; 

$key = q/{calL};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img314.png"
 ALT="${\cal L}$">|; 

$key = q/displaystyleleft(array{{cc}m<col_mark>0<row_mark>0<col_mark>marray{right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1992.png"
 ALT="$\displaystyle \left(
\begin{array}{cc}
m&amp; 0\\\\
0&amp; m\end{array}\right),$">|; 

$key = q/j=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2902.png"
 ALT="$j=0$">|; 

$key = q/{displaymath}{{bf{a}cdot{{bf{b}equiva_x,b_x+a_y,b_y+a_z,b_z={{rm{scalar~number}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img3254.png"
 ALT="\begin{displaymath}
{\bf a} \cdot {\bf b} \equiv a_x  b_x + a_y  b_y + a_z  b_z = {\rm scalar&nbsp;number}.
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{2},e^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2907.png"
 ALT="$\displaystyle \frac{1}{2} e^2,$">|; 

$key = q/1.37458;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3112.png"
 ALT="$1.37458$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter06slashfig6.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img943.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter06/fig6.01.eps}}
\end{figure}">|; 

$key = q/rightarrow;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3109.png"
 ALT="$\rightarrow$">|; 

$key = q/displaystylea,cos({mitOmega},sinlambda,t),cosleft(sqrt{frac{g}{l}},tright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1209.png"
 ALT="$\displaystyle a \cos({\mit\Omega} \sin\lambda t) \cos\left(\sqrt{\frac{g}{l}} t\right),$">|; 

$key = q/{displaymath}frac{d}{dx}!left(frac{partialF}{partialy_i'}right)-frac{partialF}{partialy_i}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1783.png"
 ALT="\begin{displaymath}
\frac{d}{dx}\!\left(\frac{\partial F}{\partial y_i'}\right)-\frac{\partial F}{\partial y_i} = 0,
\end{displaymath}">|; 

$key = q/displaystyle2,e,sin[(1+c,m^2),T-alpha_0]+frac{1}{4},e^2,sin[2,(1+c,m^2),T-2,alpha_0];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="476" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2959.png"
 ALT="$\displaystyle 2 e \sin[(1+c m^2) T-\alpha_0] + \frac{1}{4} e^2 \sin[2 (1+c m^2) T-2 \alpha_0]$">|; 

$key = q/{{bf{e}_xequiv(1,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3223.png"
 ALT="${\bf e}_x\equiv (1, 0, 0)$">|; 

$key = q/vartheta={mitTheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img957.png"
 ALT="$\vartheta={\mit\Theta}$">|; 

$key = q/{displaymath}sum_{i=1,N}!m_i,frac{d^2{{bf{r}_i}{dt^2}=sum_{i,j=1,N}^{jneqi}!{{bf{f}_{ij}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="67" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="\begin{displaymath}
\sum_{i=1,N} \!m_i \frac{d^2 {\bf r}_i}{dt^2}=\sum_{i,j=1,N}^{j\neq i}\! {\bf f}_{ij}.
\end{displaymath}">|; 

$key = q/{displaymath}UsimeqU_0-sum_{i=1,{calF}}Q_{i,0},deltaq_i-frac{1}{2}sum_{i,j=1,{calF}}G_{ij},deltaq_i,deltaq_j,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1902.png"
 ALT="\begin{displaymath}
U \simeq U_0 - \sum_{i=1,{\cal F}} Q_{i 0} \delta q_i - \frac{1}{2}\sum_{i,j=1,{\cal F}} G_{ij} \delta q_i  \delta q_j,
\end{displaymath}">|; 

$key = q/c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img185.png"
 ALT="$c$">|; 

$key = q/{displaymath}frac{x^2}{a^2}+frac{y^2}{b^2}=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img695.png"
 ALT="\begin{displaymath}
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.
\end{displaymath}">|; 

$key = q/displaystyle{{bf{e}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img678.png"
 ALT="$\displaystyle {\bf e}_r$">|; 

$key = q/2,e,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img890.png"
 ALT="$2 e a$">|; 

$key = q/displaystylefrac{partialU}{partialrho_1},frac{partialrho_1}{partialx}+frac{partialU}{partialrho_2},frac{partialrho_2}{partialx}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2567.png"
 ALT="$\displaystyle \frac{\partial U}{\partial \rho_1} \frac{\partial \rho_1}{\partial x}+\frac{\partial U}{\partial \rho_2} \frac{\partial \rho_2}{\partial x}=0,$">|; 

$key = q/{{bf{e}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1321.png"
 ALT="${\bf e}_y$">|; 

$key = q/displaystyle-frac{3}{2},m^{2},sin[2,(1-m),T],Y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2763.png"
 ALT="$\displaystyle -\frac{3}{2} m^{2} 
\sin[2 (1-m) T] Y,$">|; 

$key = q/displaystylepmfrac{sqrt{3}}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2602.png"
 ALT="$\displaystyle \pm \frac{\sqrt{3}}{2},$">|; 

$key = q/displaystylemu_1left(frac{1-rho_1^{,3}}{rho_1^{,2}}right)left(frac{x+mu_2}{rho_11-rho_2^{,3}}{rho_2^{,2}}right)left(frac{x-mu_1}{rho_2}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2570.png"
 ALT="$\displaystyle \mu_1\left(\frac{1-\rho_1^{ 3}}{\rho_1^{ 2}}\right)\left(\frac{...
...t(\frac{1-\rho_2^{ 3}}{\rho_2^{ 2}}\right)\left(\frac{x-\mu_1}{\rho_2}\right)$">|; 

$key = q/q_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1794.png"
 ALT="$q_2$">|; 

$key = q/mbox{{boldmath{{omega{}=n,{{bf{e}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2717.png"
 ALT="$\mbox{\boldmath$\omega$}= n {\bf e}_z$">|; 

$key = q/partialfslashpartialphi=-a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1821.png"
 ALT="$\partial f/\partial\phi = -a$">|; 

$key = q/displaystyle{{bf{x}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2010.png"
 ALT="$\displaystyle {\bf x}_1$">|; 

$key = q/deltaZ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2784.png"
 ALT="$\delta Z$">|; 

$key = q/{displaymath}omega^2=frac{G,M}{R^3},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2201.png"
 ALT="\begin{displaymath}
\omega^2 = \frac{G M}{R^3},
\end{displaymath}">|; 

$key = q/Qsimeq1.245;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3072.png"
 ALT="$Q\simeq 1.245$">|; 

$key = q/{{bf{a}equiv(a_x,,a_y,,a_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3222.png"
 ALT="${\bf a} \equiv (a_x, a_y, a_z)$">|; 

$key = q/{displaymath}dot{psi}=omega,cosalpha-costheta,dot{phi}=omega,cosalphaleft(1-fraca,cosalphaleft(1-frac{I_parallel}{I_perp}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="554" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1460.png"
 ALT="\begin{displaymath}
\dot{\psi} = \omega \cos\alpha-\cos\theta \dot{\phi} = \om...
...
\omega \cos\alpha\left(1-\frac{I_\parallel}{I_\perp}\right).
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.05.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="272" BORDER="0"
 SRC="|."$dir".q|img3065.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.05.eps}}
\end{figure}">|; 

$key = q/{{bf{f}_{ji};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="${\bf f}_{ji}$">|; 

$key = q/delta{{bf{q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1917.png"
 ALT="$\delta {\bf q}$">|; 

$key = q/rslasha=R=(X^2+Y^2+Z^2)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2759.png"
 ALT="$r/a=R=(X^2+Y^2+Z^2)^{1/2}$">|; 

$key = q/m_nsimeqm_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1100.png"
 ALT="$m_n\simeq m_p$">|; 

$key = q/dot{q}_i=ddot{q}_i=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1891.png"
 ALT="$\dot{q}_i=\ddot{q}_i=0$">|; 

$key = q/theta_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img750.png"
 ALT="$\theta_0=0$">|; 

$key = q/epsilon>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img882.png"
 ALT="$\epsilon&gt;0$">|; 

$key = q/{displaymath}frac{dsigma}{d{mitOmega}}=frac{a^2}{4}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img988.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d{\mit\Omega}} = \frac{a^2}{4}.
\end{displaymath}">|; 

$key = q/displaystylePhi_n(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2121.png"
 ALT="$\displaystyle \Phi_n(r)$">|; 

$key = q/displaystyle{{bf{V}_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1033.png"
 ALT="$\displaystyle {\bf V}_1'$">|; 

$key = q/{displaymath}frac{1}{2,b},lnleft(1+frac{b,v_0^{,2}}{a}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img486.png"
 ALT="\begin{displaymath}
\frac{1}{2 b} \ln\left(1+\frac{b v_0^{ 2}}{a}\right).
\end{displaymath}">|; 

$key = q/{displaymath}frac{dsigma}{d{mitOmega}}=frac{k^2}{16,E^{,2}}frac{1}{sin^4(thetaslash2)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img998.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d{\mit\Omega}} = \frac{k^2}{16 E^{ 2}}\frac{1}{\sin^4(\theta/2)}.
\end{displaymath}">|; 

$key = q/displaystylefrac{partialH}{partialp_theta}=frac{p_theta}{m,r^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1877.png"
 ALT="$\displaystyle \frac{\partial H}{\partial p_\theta} = \frac{p_\theta}{m r^2},$">|; 

$key = q/180^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$180^\circ $">|; 

$key = q/displaystyleddot{y}_1+2,n,dot{x}_1-left(n^2+n'^{,2}slash2right)y_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2746.png"
 ALT="$\displaystyle \ddot{y}_1+2 n \dot{x}_1-\left(n^2+n'^{ 2}/2\right)y_1$">|; 

$key = q/{displaymath}f(x)=-frac{dU(x)}{dx},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img206.png"
 ALT="\begin{displaymath}
f(x) = - \frac{dU(x)}{dx},
\end{displaymath}">|; 

$key = q/{displaymath}alpha=alpha_s+alpha_m.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img2333.png"
 ALT="\begin{displaymath}
\alpha=\alpha_s+\alpha_m.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2U}{dx^2}>0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img226.png"
 ALT="\begin{displaymath}
\frac{d^2 U}{dx^2}&gt;0.
\end{displaymath}">|; 

$key = q/{displaymath}Gamma=frac{A-2pmsqrt{A,(9,A-8)}}{2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img2676.png"
 ALT="\begin{displaymath}
\Gamma = \frac{A-2\pm\sqrt{A (9 A-8)}}{2},
\end{displaymath}">|; 

$key = q/1.00;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2428.png"
 ALT="$1.00$">|; 

$key = q/displaystyle-g,frac{v_x}{v_t},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img578.png"
 ALT="$\displaystyle - g \frac{v_x}{v_t},$">|; 

$key = q/Oy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3215.png"
 ALT="$Oy$">|; 

$key = q/ddot{z}'=-g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1156.png"
 ALT="$\ddot{z}'=-g$">|; 

$key = q/{displaymath}{mitOmega}=0.00327,omega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1475.png"
 ALT="\begin{displaymath}
{\mit\Omega} = 0.00327 \omega,
\end{displaymath}">|; 

$key = q/{displaymath}omega=costheta,dot{phi}+dot{psi}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2287.png"
 ALT="\begin{displaymath}
\omega = \cos\theta \dot{\phi} + \dot{\psi}
\end{displaymath}">|; 

$key = q/displaystyle-frac{1}{4},iota^2,sin[2,(1+g,m^2),T-2,gamma_0]+frac{11}{8},m^2,sin[2,(1-m),T];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="467" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2969.png"
 ALT="$\displaystyle - \frac{1}{4} \iota^2 \sin[2 (1+g m^2) T-2 \gamma_0] + \frac{11}{8} m^2 \sin[2 (1-m) T]$">|; 

$key = q/v_0slashv_t=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img614.png"
 ALT="$v_0/v_t = 0$">|; 

$key = q/lambda>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1144.png"
 ALT="$\lambda&gt;0$">|; 

$key = q/deltadot{psi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2471.png"
 ALT="$\delta\dot{\psi}$">|; 

$key = q/{displaymath}D=bar{theta}-bar{theta}',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img2997.png"
 ALT="\begin{displaymath}
D = \bar{\theta}-\bar{\theta}',
\end{displaymath}">|; 

$key = q/displaystyleleft[frac{m_1^{,2}+2,m_1,m_2,costheta+m_2^{,2}}{(m_1+m_2)^2}right]{calE},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1051.png"
 ALT="$\displaystyle \left[\frac{m_1^{ 2}+2 m_1 m_2 \cos\theta+m_2^{ 2}}{(m_1+m_2)^2}\right]{\cal E},$">|; 

$key = q/displaystyle-gleft(1+frac{v_z}{v_t}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img580.png"
 ALT="$\displaystyle - g\left(1+\frac{v_z}{v_t}\right).$">|; 

$key = q/{{bf{c}cdot({{bf{a}times{{bf{b})=({{bf{b}times{{bf{a})cdot{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3517.png"
 ALT="${\bf c}\cdot ({\bf a}\times {\bf b}) = ({\bf b}\times {\bf a})\cdot{\bf c}$">|; 

$key = q/displaystyleA,cos(omega_0,t'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img539.png"
 ALT="$\displaystyle A \cos(\omega_0 t'),$">|; 

$key = q/displaystyleint!int!intz,dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3427.png"
 ALT="$\displaystyle \int\!\int\!\int z dV$">|; 

$key = q/displaystyle-frac{3}{2},n'^{,2},cos[2,(n-n'),t],y_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2748.png"
 ALT="$\displaystyle -\frac{3}{2} n'^{ 2} 
\cos[2 (n-n') t] y_1,$">|; 

$key = q/displaystyle{{bf{b}cdot{{bf{a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3260.png"
 ALT="$\displaystyle {\bf b} \cdot {\bf a},$">|; 

$key = q/{displaymath}pileft(frac{m,b^3}{8,k}right)^{1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img492.png"
 ALT="\begin{displaymath}
\pi\left(\frac{m b^3}{8 k}\right)^{1/2}.
\end{displaymath}">|; 

$key = q/nablaf;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3486.png"
 ALT="$\nabla f$">|; 

$key = q/displaystyle2,omega.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1471.png"
 ALT="$\displaystyle 2 \omega.$">|; 

$key = q/displaystyledeltaddot{X}-2,deltadot{Y}-3,(1+m^2slash2),deltaX;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2787.png"
 ALT="$\displaystyle \delta \ddot{X}-2 \delta \dot{Y} - 3 (1+m^2/2) \delta X$">|; 

$key = q/{displaymath}iotasimeqfrac{(9slash16),m^3,iota}{(3slash2),m^2-2,g,m^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2954.png"
 ALT="\begin{displaymath}
\iota \simeq \frac{ (9/16) m^3 \iota}{(3/2) m^2-2 g m^2},
\end{displaymath}">|; 

$key = q/h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img487.png"
 ALT="$h$">|; 

$key = q/y=r,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2796.png"
 ALT="$y=r \sin\theta$">|; 

$key = q/{{bf{b}equiv(b_x,,b_y,,b_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3230.png"
 ALT="${\bf b}\equiv (b_x, b_y, b_z)$">|; 

$key = q/displaystyle2pileft(frac{t}{T}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img817.png"
 ALT="$\displaystyle 2\pi\left(\frac{t}{T}\right),$">|; 

$key = q/beta_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1955.png"
 ALT="$\beta_k$">|; 

$key = q/displaystylem,g,l,sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1485.png"
 ALT="$\displaystyle m g l \sin\theta,$">|; 

$key = q/r=(x_1^{,2}+y_1^{,2}+z_1^{,2})^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2751.png"
 ALT="$r=(x_1^{ 2}+y_1^{ 2}+z_1^{ 2})^{1/2}$">|; 

$key = q/displaystyle+frac{Aleft[(1-omega^2),cos(omega,t)+(omegaslashQ),sin(omega,t)right]}{left[(1-omega^2)^2+omega^2slashQ^2right]},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3046.png"
 ALT="$\displaystyle +\frac{A\left[(1-\omega^2) \cos(\omega  t)+ (\omega/Q) \sin(\omega  t)\right]}
{\left[(1-\omega^2)^2+\omega^2/Q^2\right]},$">|; 

$key = q/{displaymath}frac{partialU}{partialx}=frac{partialU}{partialy}=frac{partialU}{partialz}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img2561.png"
 ALT="\begin{displaymath}
\frac{\partial U}{\partial x} = \frac{\partial U}{\partial y}= \frac{\partial U}{\partial z}=0.
\end{displaymath}">|; 

$key = q/{displaymath}F(Q)=F(P)+int_P^Qleft(f,dx+g,dyright){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img3396.png"
 ALT="\begin{displaymath}
F(Q) = F(P) + \int_P^Q \left(f dx + g dy\right)
\end{displaymath}">|; 

$key = q/{displaymath}nablaV=frac{partialV}{partialr},{{bf{e}_r+frac{1}{r}frac{partialV}{{bf{e}_theta+frac{partialV}{partialz},{{bf{e}_z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3494.png"
 ALT="\begin{displaymath}
\nabla V = \frac{\partial V}{\partial r} {\bf e}_r
+ \frac{...
...a} {\bf e}_\theta
+ \frac{\partial V}{\partial z} {\bf e}_z.
\end{displaymath}">|; 

$key = q/x_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img235.png"
 ALT="$x_0=0$">|; 

$key = q/{{bf{T}=(T_{x'},,T_{y'},T_{z'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1356.png"
 ALT="${\bf T}= (T_{x'}, T_{y'},T_{z'})$">|; 

$key = q/{displaymath}F=sintheta,sintheta',cosphi'+costheta,costheta'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2085.png"
 ALT="\begin{displaymath}
F = \sin\theta \sin\theta' \cos\phi' + \cos\theta \cos\theta'.
\end{displaymath}">|; 

$key = q/v(0)slashA=0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$v(0)/A=0.5$">|; 

$key = q/stackrel{displaystylerightarrow}{CD};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3509.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{CD}$">|; 

$key = q/textstyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$\textstyle =$">|; 

$key = q/xslashr_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img853.png"
 ALT="$x/r_c$">|; 

$key = q/{{bf{e}_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1320.png"
 ALT="${\bf e}_x$">|; 

$key = q/2,sqrt{m,g,l,I_perp,costheta}slashI_parallel;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1522.png"
 ALT="$2 \sqrt{m g l I_\perp \cos\theta}/I_\parallel$">|; 

$key = q/displaystylefrac{d}{dt}!left(m,dot{r}right)-m,r,dot{theta}^{,2}+frac{dU}{dr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1615.png"
 ALT="$\displaystyle \frac{d}{dt}\!\left(m \dot{r}\right) - m r \dot{\theta}^{ 2} + \frac{dU}{dr}$">|; 

$key = q/displaystyle-frac{k'}{m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2000.png"
 ALT="$\displaystyle - \frac{k'}{m},$">|; 

$key = q/{displaymath}rho_n(r)=(n+1slash2)int_0^pirho(r,theta),P_n(costheta),sintheta,dtheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="378" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2125.png"
 ALT="\begin{displaymath}
\rho_n(r) = (n+1/2)\int_0^\pi\rho(r,\theta) P_n(\cos\theta) \sin\theta d\theta.
\end{displaymath}">|; 

$key = q/{displaymath}m,frac{d{{bf{v}}{dt}=m,{{bf{g}-c,{{bf{v},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img574.png"
 ALT="\begin{displaymath}
m \frac{d{\bf v}}{dt} = m {\bf g} - c {\bf v},
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{P}=sum_{i=1}^Nm_i,frac{d{{bf{r}_i}{dt}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img150.png"
 ALT="\begin{displaymath}
{\bf P} = \sum_{i=1}^N m_i \frac{d {\bf r}_i}{dt}.
\end{displaymath}">|; 

$key = q/delta_{n,n'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img402.png"
 ALT="$\delta_{n n'}$">|; 

$key = q/cos2D=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3010.png"
 ALT="$\cos 2D=-1$">|; 

$key = q/{{bf{v}({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3404.png"
 ALT="${\bf v}({\bf r})$">|; 

$key = q/displaystylevartheta_0^{,3}left[alpha,omega_0^{,2}-9,alpha,omega^2-(1slash24),omega_0^{,2}right]cos(3,omega,t)+{calO}(vartheta_0^{,5});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="424" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img475.png"
 ALT="$\displaystyle \vartheta_0^{ 3}\left[\alpha \omega_0^{ 2} - 9 \alpha \omega^2-
(1/24) \omega_0^{ 2}\right] \cos(3 \omega t) + {\cal O}(\vartheta_0^{ 5})$">|; 

$key = q/a_s=1.50times10^{11},{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2266.png"
 ALT="$a_s= 1.50\times 10^{11} {\rm m}$">|; 

$key = q/{{bf{a}={{bf{e}_x-2,{{bf{e}_y+{{bf{e}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3515.png"
 ALT="${\bf a} = {\bf e}_x - 2 {\bf e}_y+{\bf e}_z$">|; 

$key = q/frac{3}{2},e^2+frac{3}{4},iota^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2888.png"
 ALT="$\frac{3}{2} e^2+\frac{3}{4} \iota^2$">|; 

$key = q/{displaymath}V=-2,U=frac{2,mu_1}{rho_1}+frac{2,mu_2}{rho_2}+x^2+y^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img2606.png"
 ALT="\begin{displaymath}
V = - 2 U = \frac{2 \mu_1}{\rho_1}
+ \frac{2 \mu_2}{\rho_2} +x^2+y^2.
\end{displaymath}">|; 

$key = q/Phi_n(r)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2134.png"
 ALT="$\Phi_n(r)=0$">|; 

$key = q/{{bf{b}=(1slashsqrt{2},,-1slashsqrt{2},,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3293.png"
 ALT="${\bf b} = (1/\sqrt{2},  -1/\sqrt{2}, 0)$">|; 

$key = q/displaystyle{{bf{G},{{bf{x}_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1938.png"
 ALT="$\displaystyle {\bf G} {\bf x}_l$">|; 

$key = q/{displaymath}dot{phi}_0=sqrt{frac{g}{d}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1690.png"
 ALT="\begin{displaymath}
\dot{\phi}_0= \sqrt{\frac{g}{d}},
\end{displaymath}">|; 

$key = q/u,{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3532.png"
 ALT="$u {\bf a}$">|; 

$key = q/T=2pi,(a^3slashGM)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img809.png"
 ALT="$T= 2\pi (a^3/GM)^{1/2}$">|; 

$key = q/omega_2=2,omega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2871.png"
 ALT="$\omega_2=2 \omega_1$">|; 

$key = q/displaystylefrac{3}{2},frac{epsilon,iota_m,mu_m,T_n({{rm{yr})}{T_s({{rm{day}),[T_m({{rm{yr})]^2},costheta_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2369.png"
 ALT="$\displaystyle \frac{3}{2} \frac{\epsilon \iota_m \mu_m T_n({\rm yr})}{T_s({\rm day}) [T_m({\rm yr})]^2} \cos\theta_0,$">|; 

$key = q/R=3.84times10^8,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2241.png"
 ALT="$R=3.84\times 10^8 {\rm m}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter08slashfig8.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="317" HEIGHT="279" BORDER="0"
 SRC="|."$dir".q|img1240.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter08/fig8.01.eps}}
\end{figure}">|; 

$key = q/M_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2390.png"
 ALT="$M_i$">|; 

$key = q/varpi_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2312.png"
 ALT="$\varpi_n$">|; 

$key = q/displaystylen,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2974.png"
 ALT="$\displaystyle n t,$">|; 

$key = q/90^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img891.png"
 ALT="$90^\circ$">|; 

$key = q/f_4,f_5,f_6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1572.png"
 ALT="$f_4, f_5, f_6$">|; 

$key = q/phi=phi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1687.png"
 ALT="$\phi = \phi_0$">|; 

$key = q/displaystyleiota.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2949.png"
 ALT="$\displaystyle \iota.$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter06slashfig6.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="455" HEIGHT="283" BORDER="0"
 SRC="|."$dir".q|img949.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter06/fig6.02.eps}}
\end{figure}">|; 

$key = q/displaystyle{{bf{f},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img918.png"
 ALT="$\displaystyle {\bf f},$">|; 

$key = q/displaystyleM,ddot{x}+m,(ddot{x}+ddot{x}',costheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1652.png"
 ALT="$\displaystyle M \ddot{x}+ m (\ddot{x} + \ddot{x}' \cos\theta)$">|; 

$key = q/-frac{3}{2},x_4,z_5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2891.png"
 ALT="$-\frac{3}{2} x_4 z_5$">|; 

$key = q/{{bf{f}_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img139.png"
 ALT="${\bf f}_{ij}$">|; 

$key = q/E_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img211.png"
 ALT="$E_0$">|; 

$key = q/f(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3372.png"
 ALT="$f(x,y)$">|; 

$key = q/{displaymath}m,{{bf{a}'={{bf{f}-m,mbox{{boldmath{{Omega{}times(mbox{{boldmath{{Obf{r})-2,m,mbox{{boldmath{{Omega{}times{{bf{v}'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1119.png"
 ALT="\begin{displaymath}
m {\bf a}' = {\bf f} - m \mbox{\boldmath$\Omega$}\times(\m...
...}\times{\bf r}) -2 m \mbox{\boldmath$\Omega$}\times{\bf v}'.
\end{displaymath}">|; 

$key = q/I_{yz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1259.png"
 ALT="$I_{yz}$">|; 

$key = q/displaystyledot{L}_psi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1489.png"
 ALT="$\displaystyle \dot{L}_\psi,$">|; 

$key = q/displaystylefrac{{{bf{x}_k^{T},{{bf{M},delta{{bf{q}^{(0)}-{{bf{x}_k^{T},{{bf{M},deltadot{{bf{q}^{(0)}slashsqrt{lambda}_k}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1969.png"
 ALT="$\displaystyle \frac{{\bf x}_k^{T} {\bf M} \delta {\bf q}^{(0)} -{\bf x}_k^{T} {\bf M} \delta \dot{\bf q}^{(0)}/\sqrt{\lambda}_k}{2}.$">|; 

$key = q/{displaymath}trightarrowt'+phi_1slashomega_0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img538.png"
 ALT="\begin{displaymath}
t\rightarrow t' + \phi_1/\omega_0.
\end{displaymath}">|; 

$key = q/nu>omega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img330.png"
 ALT="$\nu&gt;\omega_0$">|; 

$key = q/U_{xx}=-1-2,A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2671.png"
 ALT="$U_{xx}=-1-2 A$">|; 

$key = q/displaystyle+frac{1}{omega_ast}left{v(0)+frac{theta(0)}{2Q}-frac{A,(1-3,omega^2)2+omega^2slashQ^2]}right}{{rm{e}^{-tslash2Q},sin(omega_ast,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="505" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3045.png"
 ALT="$\displaystyle +\frac{1}{\omega_\ast}\left\{ v(0) + \frac{\theta(0)}{2Q}
-\frac{...
...}
{[(1-\omega^2)^2+ \omega^2/Q^2]}\right\}{\rm e}^{-t/2Q} \sin(\omega_\ast t)$">|; 

$key = q/dxslashdt=v_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img513.png"
 ALT="$dx/dt = v_0$">|; 

$key = q/{displaymath}ddot{r}-frac{h^2}{r^3}=f(r),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img845.png"
 ALT="\begin{displaymath}
\ddot{r} - \frac{h^2}{r^3} = f(r),
\end{displaymath}">|; 

$key = q/U({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="$U({\bf r})$">|; 

$key = q/displaystyledot{x}_j,frac{d}{dt}!left(frac{partialx_j}{partialq_i}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1597.png"
 ALT="$\displaystyle \dot{x}_j \frac{d}{dt}\!\left(\frac{\partial x_j}{\partial q_i}\right),$">|; 

$key = q/displaystyle1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2106.png"
 ALT="$\displaystyle 1,$">|; 

$key = q/{displaymath}dot{C}=-frac{2,mu_1,dot{rho}_1}{rho_1^{,2}}-frac{2,mu_2,dot{rho}_2}xi}-2,dot{eta},ddot{eta}-2,dot{zeta},ddot{zeta}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="514" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2514.png"
 ALT="\begin{displaymath}
\dot{C} = - \frac{2 \mu_1 \dot{\rho}_1}{\rho_1^{ 2}} - \f...
...} - 2 \dot{\eta} \ddot{\eta}
- 2 \dot{\zeta} \ddot{\zeta}.
\end{displaymath}">|; 

$key = q/displaystyle+(-a_x,sintheta+a_y,costheta),(-b_x,sintheta+b_y,costheta)+a_z,b_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="451" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3258.png"
 ALT="$\displaystyle +(-a_x \sin\theta + a_y \cos\theta) (-b_x \sin \theta + b_y \cos\theta)
+a_z  b_z$">|; 

$key = q/-deltatheta_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3328.png"
 ALT="$-\delta\theta_z$">|; 

$key = q/M_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2303.png"
 ALT="$M_m$">|; 

$key = q/{{bf{r}cdot{{bf{n}=b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3527.png"
 ALT="${\bf r}\cdot{\bf n} = b$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter13slashfig13.06.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="239" BORDER="0"
 SRC="|."$dir".q|img2625.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter13/fig13.06.eps}}
\end{figure}">|; 

$key = q/displaystylen,(m,{{bf{a});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3208.png"
 ALT="$\displaystyle n (m {\bf a})$">|; 

$key = q/f(r)=-c,r^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img871.png"
 ALT="$f(r) = -c r^n$">|; 

$key = q/displaystyle{{bf{v}_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1029.png"
 ALT="$\displaystyle {\bf v}_2'$">|; 

$key = q/{displaymath}{{bf{A}=frac{{{bf{b}times{{bf{c}}{{{bf{a}cdot{{bf{b}times{{bf{c}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img3542.png"
 ALT="\begin{displaymath}
{\bf A} = \frac{{\bf b}\times {\bf c}}{{\bf a}\cdot{\bf b}\times {\bf c}},
\end{displaymath}">|; 

$key = q/displaystylex_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2732.png"
 ALT="$\displaystyle x_1'$">|; 

$key = q/E^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1001.png"
 ALT="$E^{-2}$">|; 

$key = q/L_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1463.png"
 ALT="$L_z$">|; 

$key = q/{displaymath}E'=E-frac{1}{2}frac{L_psi^{,2}}{I_parallel},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1500.png"
 ALT="\begin{displaymath}
E' = E - \frac{1}{2}\frac{L_\psi^{ 2}}{I_\parallel},
\end{displaymath}">|; 

$key = q/displaystylea,(e,coshH-1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img833.png"
 ALT="$\displaystyle a (e \cosh H - 1),$">|; 

$key = q/{displaymath}ddot{{bf{r}_M=-n^{,2},a^{,3},frac{({{bf{r}_M-{{bf{r}_E)}{|{{bf{r}_M^{,2},a'^{,3},frac{{{bf{r}_M}{|{{bf{r}_M|^{,3}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2701.png"
 ALT="\begin{displaymath}
\ddot{\bf r}_M = - n^{ 2} a^{ 3} \frac{({\bf r}_M-{\bf r...
...^{ 2} a'^{ 3} \frac{{\bf r}_M}{\vert{\bf r}_M\vert^{ 3}},
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{A}=A_r,{{bf{e}_r+A_theta,{{bf{e}_theta+A_z,{{bf{e}_z,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img3488.png"
 ALT="\begin{displaymath}
{\bf A} = A_r {\bf e}_r+ A_\theta {\bf e}_\theta + A_z {\bf e}_z,
\end{displaymath}">|; 

$key = q/displaystylea;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img718.png"
 ALT="$\displaystyle a$">|; 

$key = q/displaystyle-xsintheta+ycostheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3236.png"
 ALT="$\displaystyle -x\sin\theta + y\cos\theta,$">|; 

$key = q/{{bf{L}=tilde{{bf{I},mbox{{boldmath{{omega{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1325.png"
 ALT="${\bf L} = \tilde{\bf I} \mbox{\boldmath$\omega$}$">|; 

$key = q/displaystylefrac{8pi,G,gamma,a^5,epsilon}{15}=frac{2}{5},G,M,a^2,epsilon,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2160.png"
 ALT="$\displaystyle \frac{8\pi G \gamma a^5 \epsilon}{15} = \frac{2}{5} G M a^2 \epsilon,$">|; 

$key = q/alpha_0+gamma_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2862.png"
 ALT="$\alpha_0+\gamma_0$">|; 

$key = q/displaystyle-g-2,{mitOmega},coslambda,dot{y}'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1153.png"
 ALT="$\displaystyle -g - 2 {\mit\Omega} \cos\lambda \dot{y}'.$">|; 

$key = q/dslashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img686.png"
 ALT="$d/dt$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter05slashfig5.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="250" BORDER="0"
 SRC="|."$dir".q|img702.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter05/fig5.04.eps}}
\end{figure}">|; 

$key = q/|{{bf{r}'-{{bf{r}|^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2088.png"
 ALT="$\vert{\bf r}'-{\bf r}\vert^{-1}$">|; 

$key = q/displaystylev_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1015.png"
 ALT="$\displaystyle v_2'$">|; 

$key = q/{{bf{G};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1914.png"
 ALT="${\bf G}$">|; 

$key = q/displaystyle-frac{partialU}{partialz},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2547.png"
 ALT="$\displaystyle -\frac{\partial U}{\partial z},$">|; 

$key = q/displaystyleJ_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2157.png"
 ALT="$\displaystyle J_0$">|; 

$key = q/2.04;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2448.png"
 ALT="$2.04$">|; 

$key = q/{displaymath}alpha_2=frac{1}{sqrt{1-e}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img786.png"
 ALT="\begin{displaymath}
\alpha_2 = \frac{1}{\sqrt{1-e}}.
\end{displaymath}">|; 

$key = q/{{bf{x}_i^T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1305.png"
 ALT="${\bf x}_i^T$">|; 

$key = q/m_1ggm_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1097.png"
 ALT="$m_1\gg m_2$">|; 

$key = q/{displaymath}frac{m_2}{m_1+m_2}<0.0385.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img2691.png"
 ALT="\begin{displaymath}
\frac{m_2}{m_1+ m_2} &lt; 0.0385.
\end{displaymath}">|; 

$key = q/e=0.05488;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2722.png"
 ALT="$e=0.05488$">|; 

$key = q/l<R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1706.png"
 ALT="$l&lt; R$">|; 

$key = q/r>a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2133.png"
 ALT="$r&gt;a$">|; 

$key = q/E=pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img801.png"
 ALT="$E=\pi$">|; 

$key = q/omega_0^{,2},X_1,exp(-{{rm{i},omega,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img366.png"
 ALT="$\omega_0^{ 2} X_1 \exp(-{\rm i} \omega t)$">|; 

$key = q/{{bf{a}cdot{{bf{c}=a_x,c_x+a_y,c_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3358.png"
 ALT="${\bf a}\cdot {\bf c} = a_x  c_x + a_y  c_y$">|; 

$key = q/gamma_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2814.png"
 ALT="$\gamma_0$">|; 

$key = q/{{bf{e}_{y'''};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1430.png"
 ALT="${\bf e}_{y'''}$">|; 

$key = q/displaystyle(-a_x,sintheta+a_y,costheta),b_z-a_z,(-b_x,sintheta+b_y,costheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="437" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3297.png"
 ALT="$\displaystyle (-a_x  \sin\theta + a_y \cos\theta) b_z - a_z (-b_x  \sin\theta + b_y \cos\theta)$">|; 

$key = q/r=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3414.png"
 ALT="$r=$">|; 

$key = q/{displaymath}1-costheta=2,sin^2(thetaslash2)=frac{2,(1+e),sin^2(Eslash2)}{1-e,cosE}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="374" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img814.png"
 ALT="\begin{displaymath}
1-\cos\theta = 2 \sin^2(\theta/2) = \frac{2 (1+e) \sin^2 (E/2)}{1-e \cos E}.
\end{displaymath}">|; 

$key = q/{displaymath}deltaint_{t_1}^{t_2}L,dt=int_{t_1}^{t_2}left{left[frac{partialL}{pa{partialdot{q}_2}right)right]deltaq_2right}dt=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="617" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1798.png"
 ALT="\begin{displaymath}
\delta\int_{t_1}^{t_2} L dt = \int_{t_1}^{t_2}\left\{\left[...
... L}{\partial \dot{q}_2}\right)\right]\delta q_2 \right\} dt=0.
\end{displaymath}">|; 

$key = q/displaystyleddot{y},dot{y}+2,omega,dot{x},dot{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2551.png"
 ALT="$\displaystyle \ddot{y} \dot{y} + 2 \omega \dot{x} \dot{y}$">|; 

$key = q/{displaymath}{{bf{L}=tilde{{bf{I},mbox{{boldmath{{omega{}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1264.png"
 ALT="\begin{displaymath}
{\bf L} = \tilde{\bf I} \mbox{\boldmath$\omega$}.
\end{displaymath}">|; 

$key = q/displaystyleI_{zz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1247.png"
 ALT="$\displaystyle I_{zz}$">|; 

$key = q/{displaymath}E=frac{mu,h^2}{2}left[left(frac{du}{dvartheta}right)^2+u^2right]+U(u),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img961.png"
 ALT="\begin{displaymath}
E = \frac{\mu h^2}{2}\left[\left(\frac{du}{d\vartheta}\right)^2 + u^2\right] + U(u),
\end{displaymath}">|; 

$key = q/T=365.24;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img892.png"
 ALT="$T=365.24$">|; 

$key = q/deltaJ=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1752.png"
 ALT="$\delta J = 0$">|; 

$key = q/H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img835.png"
 ALT="$H$">|; 

$key = q/displaystyle-frac{X}{R^3}+frac{3}{2},m^{2},cos[2,(1-m),T],X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2762.png"
 ALT="$\displaystyle -\frac{X}{R^3}+\frac{3}{2} m^{2} \cos[2 (1-m) T] X$">|; 

$key = q/{calF}times{calF};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1915.png"
 ALT="${\cal F}\times {\cal F}$">|; 

$key = q/I=m,l^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img453.png"
 ALT="$I=m l^2$">|; 

$key = q/displaystyleA-B-1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2664.png"
 ALT="$\displaystyle A - B - 1,$">|; 

$key = q/displaystylea_x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="$\displaystyle a_x'$">|; 

$key = q/I_parallel;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img203.png"
 ALT="$I_\parallel$">|; 

$key = q/displaystyle({{bf{a}times{{bf{b})^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3309.png"
 ALT="$\displaystyle ({\bf a}\times{\bf b})^2$">|; 

$key = q/deltatheta_0=deltav_0=10^{-6};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$\delta \theta _0=\delta v_0 = 10^{-6}$">|; 

$key = q/{displaymath}{calE}=2,E.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1070.png"
 ALT="\begin{displaymath}
{\cal E} = 2 E.
\end{displaymath}">|; 

$key = q/{displaymath}deltaddot{theta}+dot{phi}_0^{,2},(1+3,cos^2theta_0),deltathetasimeq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1693.png"
 ALT="\begin{displaymath}
\delta\ddot{\theta} + \dot{\phi}_0^{ 2} (1+ 3 \cos^2\theta_0) \delta\theta \simeq 0.
\end{displaymath}">|; 

$key = q/2cdot2^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3166.png"
 ALT="$2\cdot 2^n$">|; 

$key = q/L_{z'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1464.png"
 ALT="$L_{z'}$">|; 

$key = q/displaystyleleftlangle|{{bf{r}'-{{bf{r}|^{-1}rightrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2099.png"
 ALT="$\displaystyle \left\langle \vert{\bf r}'-{\bf r}\vert^{-1}\right\rangle$">|; 

$key = q/0.2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1472.png"
 ALT="$0.2$">|; 

$key = q/omega_2simeq2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2905.png"
 ALT="$\omega_2\simeq 2$">|; 

$key = q/displaystyle(costheta,,sintheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img679.png"
 ALT="$\displaystyle (\cos\theta,  \sin\theta),$">|; 

$key = q/E=E_1+E_2=E_1'+E_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1041.png"
 ALT="$E=E_1+E_2=E_1'+E_2'$">|; 

$key = q/{displaymath}left.frac{df}{dx}right|_{x_0}>0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img225.png"
 ALT="\begin{displaymath}
\left.\frac{df}{dx}\right\vert _{x_0} &gt;0
\end{displaymath}">|; 

$key = q/{displaymath}deltaAsimeqfrac{1}{2},r^2,deltatheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img740.png"
 ALT="\begin{displaymath}
\delta A \simeq \frac{1}{2} r^2 \delta\theta,
\end{displaymath}">|; 

$key = q/I_{x'x'}=I_{y'y'}=I_perp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1375.png"
 ALT="$I_{x'x'} = I_{y'y'} = I_\perp$">|; 

$key = q/tan^{-1}(1slashsqrt{8})simeq19.5^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1556.png"
 ALT="$\tan^{-1}(1/\sqrt{8})\simeq19.5^\circ$">|; 

$key = q/{displaymath}int_P^Qleft[y,dx+x^3,dyright]=left.int_0^1x^3,dyright|_{x=1}+left.int_1^2y,dxright|_{y=1}=frac{7}{4}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="421" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img3392.png"
 ALT="\begin{displaymath}
\int_P^Q \left[ y dx + x^3 dy\right]= \left.\int_0^1 x^3 ...
..._{x=1} + \left.\int_1^2 y dx\right\vert _{y=1} = \frac{7}{4}.
\end{displaymath}">|; 

$key = q/m_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1599.png"
 ALT="$m_j$">|; 

$key = q/{{bf{e}_requiv{{bf{r}slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img674.png"
 ALT="${\bf e}_r\equiv {\bf r}/r$">|; 

$key = q/Q=1.387976;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$Q=1.387976$">|; 

$key = q/a>b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img566.png"
 ALT="$a&gt;b$">|; 

$key = q/{displaymath}M_{ij}=m_{ij}(q_{1,0},q_{2,0},cdots,q_{{calF},0}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1899.png"
 ALT="\begin{displaymath}
M_{ij} = m_{ij}(q_{1 0},q_{2 0},\cdots,q_{{\cal F} 0}),
\end{displaymath}">|; 

$key = q/({{bf{a}times{{bf{b})cdot{{bf{d}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3521.png"
 ALT="$({\bf a}\times {\bf b})\cdot {\bf d} = 0$">|; 

$key = q/r=r_0,exp(k,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img900.png"
 ALT="$r=r_0 \exp(k \theta)$">|; 

$key = q/displaystyledot{v}_{y}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1168.png"
 ALT="$\displaystyle \dot{v}_{y}'$">|; 

$key = q/{displaymath}frac{partialF}{partialy_k'}={{rm{const}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1785.png"
 ALT="\begin{displaymath}
\frac{\partial F}{\partial y_k'} = {\rm const}.
\end{displaymath}">|; 

$key = q/{displaymath}lambda=lambda_0,cos({mitOmega}_{x'},t'-alpha).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1544.png"
 ALT="\begin{displaymath}
\lambda = \lambda_0  \cos({\mit\Omega}_{x'} t' - \alpha).
\end{displaymath}">|; 

$key = q/displaystyleeta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2019.png"
 ALT="$\displaystyle \eta_1$">|; 

$key = q/L_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2610.png"
 ALT="$L_i$">|; 

$key = q/displaystylez_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2736.png"
 ALT="$\displaystyle z_1'$">|; 

$key = q/phi(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3366.png"
 ALT="$\phi(t)$">|; 

$key = q/partialfslashpartialr=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1832.png"
 ALT="$\partial f/\partial r =1$">|; 

$key = q/2-2,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2863.png"
 ALT="$2-2 m$">|; 

$key = q/deltaK=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1754.png"
 ALT="$\delta K= 0$">|; 

$key = q/rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$\rho $">|; 

$key = q/v_0,sinthetallv_t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img599.png"
 ALT="$v_0 \sin\theta \ll v_t$">|; 

$key = q/p=m,dot{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1657.png"
 ALT="$p=m \dot{x}$">|; 

$key = q/displaystyle-frac{T}{l,m},x'+2,{mitOmega},sinlambda,dot{y}',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1188.png"
 ALT="$\displaystyle - \frac{T}{l m} x' + 2 {\mit\Omega} \sin\lambda \dot{y}',$">|; 

$key = q/iotall1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2824.png"
 ALT="$\iota\ll 1$">|; 

$key = q/{displaymath}epsilon=frac{I_{parallel}-I_perp}{I_parallel}=0.00335{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2284.png"
 ALT="\begin{displaymath}
\epsilon = \frac{I_{\parallel}-I_\perp}{I_\parallel}=0.00335
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}'={{bf{a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="\begin{displaymath}
{\bf a}' = {\bf a}.
\end{displaymath}">|; 

$key = q/Q=1.387977;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$Q=1.387977$">|; 

$key = q/Phi(r)=Phi_0(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2139.png"
 ALT="$\Phi(r) = \Phi_0(r)$">|; 

$key = q/displaystylesqrt{frac{m}{2}},(q_1+q_2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2020.png"
 ALT="$\displaystyle \sqrt{\frac{m}{2}} (q_1 + q_2),$">|; 

$key = q/{displaymath}tau=frac{2pi}{omega}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3059.png"
 ALT="\begin{displaymath}
\tau = \frac{2\pi}{\omega}
\end{displaymath}">|; 

$key = q/{{bf{e}_zequiv(0,,0,,1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3225.png"
 ALT="${\bf e}_z\equiv
(0, 0, 1)$">|; 

$key = q/{{bf{a}={{bf{q}+lambda,{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3537.png"
 ALT="${\bf a}={\bf q} + \lambda {\bf r}$">|; 

$key = q/displaystylefrac{dv}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3040.png"
 ALT="$\displaystyle \frac{dv}{dt}$">|; 

$key = q/{displaymath}frac{1}{sqrt{a,b}},tan^{-1}left(sqrt{frac{b}{a}},v_0right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img485.png"
 ALT="\begin{displaymath}
\frac{1}{\sqrt{a b}} \tan^{-1}\left(\sqrt{\frac{b}{a}} v_0\right),
\end{displaymath}">|; 

$key = q/-gamma_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2868.png"
 ALT="$-\gamma_0$">|; 

$key = q/{displaymath}Phi(a_theta,theta)=-frac{G,M}{a_theta}+frac{2}{5}frac{G,M,a^2}{a_tha^{,3}},epsilon,P_2(costheta)+{calO}(epsilon^2),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="416" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2162.png"
 ALT="\begin{displaymath}
\Phi(a_\theta, \theta) = - \frac{G M}{a_\theta} +\frac{2}{5...
...heta^{ 3}} \epsilon P_2(\cos\theta) + {\cal O}(\epsilon^2),
\end{displaymath}">|; 

$key = q/{{bf{e}_{z'''};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1431.png"
 ALT="${\bf e}_{z'''}$">|; 

$key = q/a=384,399,{{rm{km};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2703.png"
 ALT="$a=384,399 {\rm km}$">|; 

$key = q/{displaymath}{{bf{T}=frac{d{{bf{L}}{dt},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1353.png"
 ALT="\begin{displaymath}
{\bf T} = \frac{d{\bf L}}{dt},
\end{displaymath}">|; 

$key = q/{displaymath}z=frac{v_t}{g},left(v_0,sintheta+v_tright)-v_t,t.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img596.png"
 ALT="\begin{displaymath}
z = \frac{v_t}{g} \left(v_0 \sin\theta + v_t\right) - v_t t.
\end{displaymath}">|; 

$key = q/v_t=r,dot{theta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img779.png"
 ALT="$v_t=r \dot{\theta}$">|; 

$key = q/n>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img410.png"
 ALT="$n&gt;0$">|; 

$key = q/psi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1457.png"
 ALT="$\psi=0$">|; 

$key = q/displaystyle-frac{9}{8},m^2,e,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2927.png"
 ALT="$\displaystyle -\frac{9}{8} m^2 e,$">|; 

$key = q/{{bf{b}equiv(b_x,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3354.png"
 ALT="${\bf b} \equiv (b_x, 0, 0)$">|; 

$key = q/I_{x'x'}^{-1}:I_{y'y'}^{-1}:I_{z'z'}^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1415.png"
 ALT="$I_{x'x'}^{-1}:I_{y'y'}^{-1}:I_{z'z'}^{-1}$">|; 

$key = q/dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3283.png"
 ALT="$dt$">|; 

$key = q/omega=2,omega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img446.png"
 ALT="$\omega = 2 \omega_0$">|; 

$key = q/{displaymath}K=frac{1}{2}sum_{i,j=1,{calF}}m_{ij}(q_1,q_2,cdots,q_{calF}),dot{q}_i,dot{q}_j.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1887.png"
 ALT="\begin{displaymath}
K = \frac{1}{2}\sum_{i,j=1,{\cal F}} m_{ij}(q_1,q_2,\cdots, q_{\cal F}) \dot{q}_i \dot{q}_j.
\end{displaymath}">|; 

$key = q/5.18times10^{-5};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2441.png"
 ALT="$5.18\times 10^{-5}$">|; 

$key = q/displaystylefrac{dv_y'}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img630.png"
 ALT="$\displaystyle \frac{d v_y'}{dt}$">|; 

$key = q/c>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img254.png"
 ALT="$c&gt;0$">|; 

$key = q/displaystyleddot{Y}+2,dot{X}-(1+m^2slash2),Y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2764.png"
 ALT="$\displaystyle \ddot{Y}+2 \dot{X}-(1+m^2/2) Y$">|; 

$key = q/displaystylesinpsi,sintheta,dot{phi}+cospsi,dot{theta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1447.png"
 ALT="$\displaystyle \sin\psi \sin\theta \dot{\phi} +\cos\psi \dot{\theta},$">|; 

$key = q/deltay(t)=deltay_0,exp(gamma,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2650.png"
 ALT="$\delta y(t) = \delta y_0 \exp(\gamma t)$">|; 

$key = q/1slashr^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img884.png"
 ALT="$1/r^4$">|; 

$key = q/{displaymath}W=int_{infty}^0frac{-y,dy}{(a^2+y^2)^{3slash2}}=left[frac{1}{(y^2+a^2)^{1slash2}}right]^0_infty=frac{1}{a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="376" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img3416.png"
 ALT="\begin{displaymath}
W = \int_{\infty}^0 \frac{-y dy}{(a^2 + y^2)^{3/2}} = \left[\frac{1}{(y^2+a^2)^{1/2}}
\right]^0_\infty = \frac{1}{a}.
\end{displaymath}">|; 

$key = q/r_2=x+d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img706.png"
 ALT="$r_2 = x+d$">|; 

$key = q/{displaymath}{{bf{c}={{bf{a}+{{bf{b}={{bf{b}+{{bf{a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3198.png"
 ALT="\begin{displaymath}
{\bf c} = {\bf a} + {\bf b} = {\bf b} + {\bf a}.
\end{displaymath}">|; 

$key = q/{{bf{g};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1125.png"
 ALT="${\bf g}$">|; 

$key = q/{displaymath}dot{theta}^{,2}=frac{2,g}{a},(costheta-1)+frac{v_0^{,2}}{a^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1838.png"
 ALT="\begin{displaymath}
\dot{\theta}^{ 2} = \frac{2 g}{a} (\cos\theta-1) + \frac{v_0^{ 2}}{a^2}.
\end{displaymath}">|; 

$key = q/K=(1slash2),m,v^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img209.png"
 ALT="$K=(1/2) m v^2$">|; 

$key = q/thetarightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1007.png"
 ALT="$\theta\rightarrow 0$">|; 

$key = q/{displaymath}a=frac{h^2}{(1-e^2),G,M}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img937.png"
 ALT="\begin{displaymath}
a = \frac{h^2}{(1-e^2) G M}.
\end{displaymath}">|; 

$key = q/deltaI;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1729.png"
 ALT="$\delta I$">|; 

$key = q/displaystylefrac{d}{dt}!left(frac{partial{calL}}{partialdot{theta}}right)-frac{partial{calL}}{partialtheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2335.png"
 ALT="$\displaystyle \frac{d}{dt}\!\left(\frac{\partial {\cal L}}{\partial \dot{\theta}}\right)-\frac{\partial{\cal L}}{\partial \theta}$">|; 

$key = q/displaystyleC_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2615.png"
 ALT="$\displaystyle C_2$">|; 

$key = q/{displaymath}frac{dsigma(psi)}{d{mitOmega}'}=a^2,cospsi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1086.png"
 ALT="\begin{displaymath}
\frac{d\sigma(\psi)}{d{\mit\Omega}'} = a^2 \cos\psi.
\end{displaymath}">|; 

$key = q/{displaymath}ddot{eta}_k=lambda_k,eta_k,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1982.png"
 ALT="\begin{displaymath}
\ddot{\eta}_k = \lambda_k \eta_k,
\end{displaymath}">|; 

$key = q/270^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3005.png"
 ALT="$270^\circ$">|; 

$key = q/{displaymath}int_a^bleft[frac{partialF}{partialy}-frac{d}{dx}!left(frac{partialF}{partialy'}right)right]deltay,dx=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1734.png"
 ALT="\begin{displaymath}
\int_a^b\left[\frac{\partial F}{\partial y}- \frac{d}{dx}\!\left(\frac{\partial F}{\partial y'}\right)\right]\delta y dx =0.
\end{displaymath}">|; 

$key = q/{displaymath}y(x)=-h,[lambda'+cosh(-xslashh+c)].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1772.png"
 ALT="\begin{displaymath}
y(x) =-h [\lambda' + \cosh(-x/h + c)].
\end{displaymath}">|; 

$key = q/{displaymath}|{{bf{a}times{{bf{b}|=|{{bf{a}|,|{{bf{b}|,sintheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3314.png"
 ALT="\begin{displaymath}
\vert{\bf a}\times{\bf b}\vert = \vert{\bf a}\vert \vert{\bf b}\vert \sin\theta,
\end{displaymath}">|; 

$key = q/{displaymath}{mitTheta}=frac{pi}{2}-sin^{-1}left(frac{alpha}{sqrt{4+alpha^2}}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img994.png"
 ALT="\begin{displaymath}
{\mit\Theta} = \frac{\pi}{2} - \sin^{-1}\left(\frac{\alpha}{\sqrt{4+\alpha^2}}\right).
\end{displaymath}">|; 

$key = q/displaystylez_1slasha,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2758.png"
 ALT="$\displaystyle z_1/a,$">|; 

$key = q/displaystyle(m_1slashm_2),v_1(-costheta,,-sintheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1030.png"
 ALT="$\displaystyle (m_1/m_2) v_1(-\cos\theta, -\sin\theta),$">|; 

$key = q/displaystyledot{p}_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1880.png"
 ALT="$\displaystyle \dot{p}_\theta$">|; 

$key = q/m_1slashm_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img945.png"
 ALT="$m_1/m_2$">|; 

$key = q/{{bf{f}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="${\bf f}_2$">|; 

$key = q/{displaymath}frac{d}{dt}!left(frac{partialK}{partialdot{eta}_k}right)-frac{partiK}{partialeta_k}+frac{partialU}{partialeta_k}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1981.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial K}{\partial \dot{\eta}_k}...
...l K}{\partial \eta_k}+ \frac{\partial U}{\partial \eta_k} = 0,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.18.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="274" BORDER="0"
 SRC="|."$dir".q|img3089.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.18.eps}}
\end{figure}">|; 

$key = q/displaystyleleft[a^2,z^2slash2-2,a,z^3slash3+z^4slash4right]_0^a=frac{1}{12},a^4.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3429.png"
 ALT="$\displaystyle \left[a^2 z^2/2-2 a z^3/3+z^4/4\right]_0^a= \frac{1}{12} a^4.$">|; 

$key = q/psi>pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1063.png"
 ALT="$\psi&gt;\pi/2$">|; 

$key = q/{displaymath}frac{d}{dt}!left(frac{1}{2},frac{partialdot{x}_j^{,2}}{partialdot{qac{1}{2},frac{partialdot{x}_j^{,2}}{partialq_i}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1598.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\frac{1}{2} \frac{\partial \dot{x}_j^{\...
... + \frac{1}{2} \frac{\partial \dot{x}_j^{ 2}}{\partial q_i}.
\end{displaymath}">|; 

$key = q/stackrel{displaystylerightarrow}{DA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2221.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{DA}$">|; 

$key = q/alpha={mitTheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img959.png"
 ALT="$\alpha={\mit\Theta}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter06slashfig6.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="454" HEIGHT="283" BORDER="0"
 SRC="|."$dir".q|img952.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter06/fig6.03.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{a,(1-e^2)}{1+e,cos(theta-alpha_0)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2819.png"
 ALT="$\displaystyle \frac{a (1-e^2)}{1+e \cos(\theta-\alpha_0)},$">|; 

$key = q/{{bf{e}_z!cdot!{{bf{e}_{z'}=costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1445.png"
 ALT="${\bf e}_z\!\cdot\!{\bf e}_{z'} =\cos\theta$">|; 

$key = q/displaystyledot{p}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1878.png"
 ALT="$\displaystyle \dot{p}_r$">|; 

$key = q/{displaymath}nabla(phi,psi)=phi,nablapsi+psi,nablaphi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3479.png"
 ALT="\begin{displaymath}
\nabla(\phi  \psi) = \phi  \nabla\psi + \psi  \nabla \phi.
\end{displaymath}">|; 

$key = q/{displaymath}J_nsimeq-frac{2pi,G,gamma,a^{3+n}}{(3+n)}int_0^piP_n(costheta)left[3+n),epsilon,P_2(costheta)right]sintheta,dtheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="614" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2155.png"
 ALT="\begin{displaymath}
J_n \simeq -\frac{2\pi G \gamma a^{3+n}}{(3+n)}\int_0^\pi...
...} (3+n) \epsilon P_2(\cos\theta)\right]\sin\theta d\theta,
\end{displaymath}">|; 

$key = q/dhslashdr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3434.png"
 ALT="$dh/dr$">|; 

$key = q/{displaymath}int_P^Q{{bf{A}cdotd{{bf{r}=V(Q),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img3467.png"
 ALT="\begin{displaymath}
\int_P^Q {\bf A}\cdot d{\bf r} = V(Q),
\end{displaymath}">|; 

$key = q/displaystyleleft(frac{1-costheta}{2}right){calE}=sin^2(thetaslash2),,{calE}=sin^2psi,,{calE}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1073.png"
 ALT="$\displaystyle \left(\frac{1-\cos\theta}{2}\right){\cal E} = \sin^2(\theta/2)  {\cal E} = \sin^2\psi  {\cal E}.$">|; 

$key = q/{displaymath}|{{bf{G}-lambda,{{bf{M}|=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1921.png"
 ALT="\begin{displaymath}
\vert{\bf G} -\lambda {\bf M}\vert = 0.
\end{displaymath}">|; 

$key = q/T_n({{rm{yr})=n_sslashOmega_n=18.6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2371.png"
 ALT="$T_n({\rm yr}) = n_s/\Omega_n = 18.6$">|; 

$key = q/{displaymath}W=int_{infty}^aleft(-frac{1}{x^2}right),dx=left[frac{1}{x}right]_{infty}^a=frac{1}{a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img3413.png"
 ALT="\begin{displaymath}
W = \int_{\infty}^a \left(-\frac{1}{x^2}\right) dx = \left[\frac{1}{x}\right]_{\infty}^a
=\frac{1}{a}.
\end{displaymath}">|; 

$key = q/displaystylebeta_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1968.png"
 ALT="$\displaystyle \beta_k$">|; 

$key = q/x(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img437.png"
 ALT="$x(t)$">|; 

$key = q/R_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2261.png"
 ALT="$R_c$">|; 

$key = q/displaystylelambda_l,{{bf{x}_k^T,{{bf{M},{{bf{x}_l.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1943.png"
 ALT="$\displaystyle \lambda_l {\bf x}_k^T {\bf M} {\bf x}_l.$">|; 

$key = q/{displaymath}dF(x,y)=f(x,y),dx+g(x,y),dy.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3400.png"
 ALT="\begin{displaymath}
dF(x,y) = f(x,y) dx + g(x,y) dy.
\end{displaymath}">|; 

$key = q/displaystylefrac{mu_1}{rho_1^{,3}}+frac{mu_2}{rho_2^{,3}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2654.png"
 ALT="$\displaystyle \frac{\mu_1}{\rho_1^{ 3}} + \frac{\mu_2}{\rho_2^{ 3}},$">|; 

$key = q/displaystylea_0+sum_{j>0}a_j,cos(omega_j,T-alpha_j),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2831.png"
 ALT="$\displaystyle a_0 + \sum_{j&gt;0}a_j \cos(\omega_j T-\alpha_j),$">|; 

$key = q/displaystyleQ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3033.png"
 ALT="$\displaystyle Q$">|; 

$key = q/{displaymath}frac{dsigma}{d{mitOmega}}=frac{q^4}{16,(4,pi,epsilon_0)^2,E^{,2}}frac{1}{sin^4(thetaslash2)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1005.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d{\mit\Omega}} = \frac{q^4}{16 (4 \pi \epsilon_0)^2 E^{ 2}}\frac{1}{\sin^4(\theta/2)}.
\end{displaymath}">|; 

$key = q/{displaymath}U=-frac{G,M_s,M}{a_s}+frac{G,M_s,(I_parallel-I_perp)}{2,a_s^{,3}},left(frac{3}{2},sin^2theta-1right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2280.png"
 ALT="\begin{displaymath}
U = - \frac{G M_s M}{a_s} + \frac{G M_s (I_\parallel-I_\perp)}{2 a_s^{ 3}} \left(\frac{3}{2} \sin^2\theta -1\right)
\end{displaymath}">|; 

$key = q/{displaymath}lnleft(frac{v_t+v_z}{v_t+v_{z,0}}right)=-frac{g}{v_t},t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img589.png"
 ALT="\begin{displaymath}
\ln\left(\frac{v_t+v_z}{v_t+v_{z 0}}\right) = - \frac{g}{v_t} t,
\end{displaymath}">|; 

$key = q/displaystylea_xcostheta+a_ysintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3241.png"
 ALT="$\displaystyle a_x\cos\theta + a_y\sin \theta,$">|; 

$key = q/F(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3398.png"
 ALT="$F(x,y)$">|; 

$key = q/dW={{bf{f}cdotd{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3284.png"
 ALT="$dW={\bf f}\cdot d{\bf r}$">|; 

$key = q/db(theta)slashdtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img975.png"
 ALT="$db(\theta)/d\theta$">|; 

$key = q/{displaymath}tilde{{bf{I}=left(array{{ccc}I_{xx}&0&00&I_{yy}&00&0&I_{zz}array{right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="76" BORDER="0"
 SRC="|."$dir".q|img1324.png"
 ALT="\begin{displaymath}
\tilde{\bf I} =
\left(\begin{array}{ccc}
I_{xx}&amp;0&amp;0\\\\
0&amp;I_{yy}&amp;0\\\\
0&amp;0&amp;I_{zz}
\end{array}\right).
\end{displaymath}">|; 

$key = q/displaystyleX;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2753.png"
 ALT="$\displaystyle X$">|; 

$key = q/x=-d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img703.png"
 ALT="$x=-d$">|; 

$key = q/langle1rangle=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2091.png"
 ALT="$\langle 1\rangle =1$">|; 

$key = q/m,{{bf{g};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1178.png"
 ALT="$m {\bf g}$">|; 

$key = q/{displaymath}int!int!int_Vf(x,y,z),dV,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img3418.png"
 ALT="\begin{displaymath}
\int\!\int\!\int_V f(x,y,z) dV,
\end{displaymath}">|; 

$key = q/m_1=m_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1064.png"
 ALT="$m_1=m_2$">|; 

$key = q/I_{ij}^{,ast}=I_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1267.png"
 ALT="$I_{ij}^{ \ast} = I_{ij}$">|; 

$key = q/{displaymath}{{bf{x}_k^{ast,T},{{bf{G},{{bf{x}_k=lambda_k^ast,{{bf{x}_k^{ast,T},{{bf{M},{{bf{x}_k.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1928.png"
 ALT="\begin{displaymath}
{\bf x}_k^{\ast T} {\bf G} {\bf x}_k = \lambda_k^\ast 
{\bf x}_k^{\ast T} {\bf M} {\bf x}_k.
\end{displaymath}">|; 

$key = q/displaystyletheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2816.png"
 ALT="$\displaystyle \theta$">|; 

$key = q/displaystyley,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$\displaystyle y,$">|; 

$key = q/{mitDelta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img544.png"
 ALT="${\mit\Delta}$">|; 

$key = q/{{bf{L}=sum_{i=1,N}m_i,{{bf{r}_itimesd{{bf{r}_islashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1236.png"
 ALT="${\bf L}=\sum_{i=1,N} m_i {\bf r}_i\times d{\bf r}_i/dt$">|; 

$key = q/29.53;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3012.png"
 ALT="$29.53$">|; 

$key = q/-epsilon=2.1times10^{-7};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2244.png"
 ALT="$-\epsilon=2.1\times 10^{-7}$">|; 

$key = q/{displaymath}thetasimeqtheta_0+deltatheta_0,cos({mitOmega},t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1694.png"
 ALT="\begin{displaymath}
\theta \simeq \theta_0 + \delta\theta_0 \cos({\mit\Omega} t),
\end{displaymath}">|; 

$key = q/{displaymath}dT=frac{partialT}{partialx},dx+frac{partialT}{partialy},dy+frac{partialT}{partialz},dz.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img3457.png"
 ALT="\begin{displaymath}
dT = \frac{\partial T}{\partial x} dx +\frac{\partial T}{\partial y} dy+
\frac{\partial T}{\partial z} dz.
\end{displaymath}">|; 

$key = q/s_+=s_-=aslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1207.png"
 ALT="$s_+=s_-=a/2$">|; 

$key = q/1.000;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2427.png"
 ALT="$1.000$">|; 

$key = q/displaystyledeltaZ-deltaX,deltaZ.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2802.png"
 ALT="$\displaystyle \delta Z-\delta X \delta Z.$">|; 

$key = q/{{bf{V}_1={{bf{v}_1-{{bf{v}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1018.png"
 ALT="${\bf V}_1= {\bf v}_1-{\bf v}_2$">|; 

$key = q/1.881;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2430.png"
 ALT="$1.881$">|; 

$key = q/{displaymath}K=frac{1}{2}sum_{k=1,{calF}}dot{eta}_k^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1977.png"
 ALT="\begin{displaymath}
K = \frac{1}{2}\sum_{k=1,{\cal F}}\dot{\eta}_k^{ 2}.
\end{displaymath}">|; 

$key = q/{displaymath}U=U_0-epsilon,alpha_s,cos(2,theta),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2282.png"
 ALT="\begin{displaymath}
U = U_0 - \epsilon \alpha_s \cos(2 \theta),
\end{displaymath}">|; 

$key = q/k_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img501.png"
 ALT="$k_1$">|; 

$key = q/cos^2theta+sin^2thetaequiv1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img306.png"
 ALT="$\cos^2\theta+\sin^2\theta \equiv 1$">|; 

$key = q/{{bf{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1284.png"
 ALT="${\bf x}$">|; 

$key = q/U_{xy}=mpsqrt{27slash16},(1-2,mu_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2687.png"
 ALT="$U_{xy} = \mp\sqrt{27/16} (1-2 \mu_2)$">|; 

$key = q/mu_1+mu_2=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2564.png"
 ALT="$\mu_1+\mu_2=1$">|; 

$key = q/z'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1419.png"
 ALT="$z''$">|; 

$key = q/{displaymath}X_0=frac{1}{T}int_0^TX(t),dt.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img405.png"
 ALT="\begin{displaymath}
X_0 = \frac{1}{T}\int_0^T X(t)  dt.
\end{displaymath}">|; 

$key = q/displaystyle+frac{3pi}{2}sum_{j>i}left(frac{M_j}{M_0}right)left(frac{R_i}{R_j}riright)^2+frac{175}{64}left(frac{R_i}{R_j}right)^4+cdotsright];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="487" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2412.png"
 ALT="$\displaystyle + \frac{3\pi}{2}\sum_{j&gt;i}\left(\frac{M_j}{M_0}\right)
\left(\fra...
..._i}{R_j}\right)^2 + \frac{175}{64}\left(\frac{R_i}{R_j}
\right)^4+\cdots\right]$">|; 

$key = q/T_r=2pislashomega_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img351.png"
 ALT="$T_r=2\pi/\omega_r$">|; 

$key = q/1+c,m^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2852.png"
 ALT="$1+c m^2$">|; 

$key = q/displaystyleddot{{bf{r}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2714.png"
 ALT="$\displaystyle \ddot{\bf r}'$">|; 

$key = q/z=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img597.png"
 ALT="$z=0$">|; 

$key = q/{displaymath}bar{z}=left.frac{1}{12},a^4rightslashfrac{1}{3},a^3=frac{1}{4},a.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3430.png"
 ALT="\begin{displaymath}
\bar{z} = \left.\frac{1}{12} a^4\right/\frac{1}{3} a^3 = \frac{1}{4} a.
\end{displaymath}">|; 

$key = q/costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3439.png"
 ALT="$\cos\theta$">|; 

$key = q/nu_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img334.png"
 ALT="$\nu_-$">|; 

$key = q/{displaymath}s=a,{{rm{e}^{-{{rm{i},{mitOmega},sinlambda,t},cosleft(sqrt{frac{g}{l}},tright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1208.png"
 ALT="\begin{displaymath}
s = a {\rm e}^{-{\rm i} {\mit\Omega} \sin\lambda t}  \cos\left(\sqrt{\frac{g}{l}} t\right).
\end{displaymath}">|; 

$key = q/displaystylea_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2940.png"
 ALT="$\displaystyle a_1$">|; 

$key = q/{displaymath}q_3=-frac{m}{M},(q_1+q_2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img2032.png"
 ALT="\begin{displaymath}
q_3 = - \frac{m}{M} (q_1+q_2).
\end{displaymath}">|; 

$key = q/{displaymath}L=frac{1}{2},m,(dot{r}^{,2}+r^2,dot{theta}^{,2})-U(r).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1607.png"
 ALT="\begin{displaymath}
L = \frac{1}{2} m (\dot{r}^{ 2} + r^2 \dot{\theta}^{ 2}) - U(r).
\end{displaymath}">|; 

$key = q/{{bf{k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3556.png"
 ALT="${\bf k}$">|; 

$key = q/a,bsintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3317.png"
 ALT="$a b \sin\theta$">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{Chapter13slashfig13.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="290" BORDER="0"
 SRC="|."$dir".q|img2604.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{Chapter13/fig13.03.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{Chapter03slashfig3.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="501" HEIGHT="224" BORDER="0"
 SRC="|."$dir".q|img228.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{Chapter03/fig3.02.eps}}
\end{figure}">|; 

$key = q/k_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img180.png"
 ALT="$k_i$">|; 

$key = q/{displaymath}theta_0=vartheta_0+alpha,vartheta_0^{,3}=vartheta_0-frac{omega_0^{,2}}{192},vartheta_0^{,3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img481.png"
 ALT="\begin{displaymath}
\theta_0 = \vartheta_0 + \alpha \vartheta_0^{ 3} = \vartheta_0-\frac{\omega_0^{ 2}}{192} \vartheta_0^{ 3}.
\end{displaymath}">|; 

$key = q/displaystyleomega_{y'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1387.png"
 ALT="$\displaystyle \omega_{y'}$">|; 

$key = q/displaystylefrac{e,r_c}{1-e^2}=e,a.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img723.png"
 ALT="$\displaystyle \frac{e r_c}{1-e^2}= e a.$">|; 

$key = q/u_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1509.png"
 ALT="$u_3$">|; 

$key = q/{displaymath}frac{(x-x_c)^2}{a^2}+frac{y^2}{b^2}=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img716.png"
 ALT="\begin{displaymath}
\frac{(x-x_c)^2}{a^2} + \frac{y^2}{b^2} = 1,
\end{displaymath}">|; 

$key = q/H=H(q_i,p_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1851.png"
 ALT="$H=H(q_i,p_i)$">|; 

$key = q/displaystyle-frac{mu_1,z}{rho_1^{,3}}-frac{mu_2,z}{rho_2^{,3}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2544.png"
 ALT="$\displaystyle - \frac{\mu_1 z}{\rho_1^{ 3}}- \frac{\mu_2 z}{\rho_2^{ 3}},$">|; 

$key = q/4,{mitOmega},v_0^{,3},sinlambda,sin^2alpha,cosalphaslashg^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1220.png"
 ALT="$4 {\mit\Omega} v_0^{ 3} \sin\lambda \sin^2\alpha \cos\alpha/g^2$">|; 

$key = q/{{bf{r}_1=mu_2,(-1,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2533.png"
 ALT="${\bf r}_1=\mu_2 (-1, 0, 0)$">|; 

$key = q/m,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img251.png"
 ALT="$m g$">|; 

$key = q/{{bf{x}_i^{ast,T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1292.png"
 ALT="${\bf x}_i^{\ast T}$">|; 

$key = q/displaystylea,ddot{phi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1825.png"
 ALT="$\displaystyle a \ddot{\phi}$">|; 

$key = q/{bf{a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3274.png"
 ALT="$\bf a$">|; 

$key = q/aslasha'=0.00257;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2741.png"
 ALT="$a/a'=0.00257$">|; 

$key = q/{displaymath}x_j=x_j(q_1,q_2,cdots,q_{calF},t){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1566.png"
 ALT="\begin{displaymath}
x_j = x_j(q_1,q_2,\cdots,q_{\cal F}, t)
\end{displaymath}">|; 

$key = q/{displaymath}chi=omega^2,rho,z{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2212.png"
 ALT="\begin{displaymath}
\chi = \omega^2 \rho z
\end{displaymath}">|; 

$key = q/g_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2057.png"
 ALT="$g_y$">|; 

$key = q/f(R_i)=-dPhi(R_i)slashdr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2397.png"
 ALT="$f(R_i) = - d\Phi(R_i)/dr$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.29.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="274" BORDER="0"
 SRC="|."$dir".q|img3147.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.29.eps}}
\end{figure}">|; 

$key = q/displaystyle+left.left(frac{r'}{r}right)^2left(frac{3}{2}cos^2theta-frac{1}{2}ritheta'-frac{1}{2}right)+{calO}left(frac{r'}{r}right)^3right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="451" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2101.png"
 ALT="$\displaystyle +\left.\left(\frac{r'}{r}\right)^2\left(\frac{3}{2}\cos^2\theta-\...
...}\cos^2\theta'-\frac{1}{2}\right)
+ {\cal O}\left(\frac{r'}{r}\right)^3\right].$">|; 

$key = q/{displaymath}epsilon=frac{5-alpha}{4},frac{Omega^2,a^3}{G,M}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2475.png"
 ALT="\begin{displaymath}
\epsilon = \frac{5-\alpha}{4} \frac{\Omega^2 a^3}{G M}.
\end{displaymath}">|; 

$key = q/v_t=(m,gslashc)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img256.png"
 ALT="$v_t=(m g/c)^{1/2}$">|; 

$key = q/displaystyle(2,m)^{-1slash2},(1+2,alpha)^{-1slash2},(1,,,,,1).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2044.png"
 ALT="$\displaystyle (2 m)^{-1/2} (1+2 \alpha)^{-1/2} (1,    1).$">|; 

$key = q/m_1geqm_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2559.png"
 ALT="$m_1\geq m_2$">|; 

$key = q/q_2(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1809.png"
 ALT="$q_2(t)$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter05slashfig5.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="317" HEIGHT="252" BORDER="0"
 SRC="|."$dir".q|img699.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter05/fig5.03.eps}}
\end{figure}">|; 

$key = q/{displaymath}frac{x_1(omega)}{x_1(omega=omega_0)}simeqfrac{nu}{[(omega-omega_0)^2+nu^2]^{1slash2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img385.png"
 ALT="\begin{displaymath}
\frac{x_1(\omega)}{x_1(\omega=\omega_0)}\simeq \frac{\nu}{[(\omega-\omega_0)^2 + \nu^2]^{1/2}},
\end{displaymath}">|; 

$key = q/I_{z'z'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1361.png"
 ALT="$I_{z'z'}$">|; 

$key = q/{displaymath}phisimeq-Omega_phi,t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2296.png"
 ALT="\begin{displaymath}
\phi \simeq- \Omega_\phi t,
\end{displaymath}">|; 

$key = q/displaystyleE_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1046.png"
 ALT="$\displaystyle E_1$">|; 

$key = q/displaystyle-sum_{i=1,N}y_i,z_i,m_i=-inty,z,dm,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1252.png"
 ALT="$\displaystyle - \sum_{i=1,N}y_i z_i  m_i= -\int y z dm,$">|; 

$key = q/displaystyle-x',sin(n,t)+y',cos(n,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2735.png"
 ALT="$\displaystyle -x' \sin(n t) + y' \cos(n t),$">|; 

$key = q/U=(1slash2),k,x^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img234.png"
 ALT="$U=(1/2) k x^2$">|; 

$key = q/U;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="$U$">|; 

$key = q/displaystyledeltaX;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2807.png"
 ALT="$\displaystyle \delta X$">|; 

$key = q/i=1,N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2063.png"
 ALT="$i=1, N$">|; 

$key = q/{displaymath}psi=frac{pi}{(3+n)^{1slash2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img872.png"
 ALT="\begin{displaymath}
\psi = \frac{\pi}{(3+n)^{1/2}}.
\end{displaymath}">|; 

$key = q/{displaymath}m^2,lambda^2+2,m,(k+k'),lambda+k',(k'+2,k)=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1998.png"
 ALT="\begin{displaymath}
m^2 \lambda^2 + 2 m (k+k') \lambda + k' (k'+2 k) = 0.
\end{displaymath}">|; 

$key = q/n=-3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img862.png"
 ALT="$n =-3$">|; 

$key = q/displaystyle-frac{4,alpha,costheta_0}{I_parallel,omega}-frac{2,beta_m,cos(2,theta_0)}{I_parallel,omega,sintheta_0},cos(Omega_n,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2349.png"
 ALT="$\displaystyle -\frac{4 \alpha \cos\theta_0}{I_\parallel \omega}-\frac{2 \beta_m \cos(2 \theta_0)}{I_\parallel \omega \sin\theta_0} \cos(\Omega_n t),$">|; 

$key = q/55,000,000;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img886.png"
 ALT="$55,000,000$">|; 

$key = q/displaystylefrac{d}{dt}!left(frac{partialL}{partialdot{x}}right)-frac{partialL}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1637.png"
 ALT="$\displaystyle \frac{d}{dt}\!\left(\frac{\partial L}{\partial\dot{x}}\right) - \frac{\partial L}{\partial x}$">|; 

$key = q/{displaymath}frac{dsigma}{dOmega}=frac{k}{E,pi},frac{1}{sintheta},frac{(1-thetaslashpi)}{(thetaslashpi)^2,(2-thetaslashpi)^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1103.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d\Omega } = \frac{k}{E \pi}  \frac{1}{\sin\theta} 
\frac{(1-\theta/\pi)}{(\theta/\pi)^2 (2-\theta/\pi)^2}.
\end{displaymath}">|; 

$key = q/{displaymath}Phi(r,theta)=sum_{n=0,infty}J_n,frac{P_n(costheta)}{r^{n+1}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2151.png"
 ALT="\begin{displaymath}
\Phi(r,\theta) = \sum_{n=0,\infty} J_n \frac{P_n(\cos\theta)}{r^{n+1}},
\end{displaymath}">|; 

$key = q/{{bf{p}_i=m_i,d{{bf{r}_islashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.png"
 ALT="${\bf p}_i = m_i d{\bf r}_i/dt$">|; 

$key = q/{{bf{grad},hequiv(partialhslashpartialx,,partialhslashpartialy);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3450.png"
 ALT="${\bf grad} h \equiv (\partial h/\partial x,  \partial h/\partial y)$">|; 

$key = q/{displaymath}omega^2-2,{mitOmega},sinlambda,omega-frac{g}{l}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1201.png"
 ALT="\begin{displaymath}
\omega^2 - 2 {\mit\Omega} \sin\lambda \omega - \frac{g}{l}=0.
\end{displaymath}">|; 

$key = q/{displaymath}tilde{Q}_i=lambda(t),frac{partialf}{partialq_i}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1816.png"
 ALT="\begin{displaymath}
\tilde{Q}_i = \lambda(t) \frac{\partial f}{\partial q_i}
\end{displaymath}">|; 

$key = q/r_1=r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img707.png"
 ALT="$r_1=r$">|; 

$key = q/displaystylefrac{1}{2},sin^2theta,sin^2theta'+cos^2theta,cos^2theta';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2097.png"
 ALT="$\displaystyle \frac{1}{2} \sin^2\theta \sin^2\theta'
+ \cos^2\theta \cos^2\theta'$">|; 

$key = q/{displaymath}theta=theta_0+deltatheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1692.png"
 ALT="\begin{displaymath}
\theta = \theta_0 + \delta\theta.
\end{displaymath}">|; 

$key = q/displaystylez'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3481.png"
 ALT="$\displaystyle z'.$">|; 

$key = q/nabla;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3485.png"
 ALT="$\nabla$">|; 

$key = q/theta>alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1408.png"
 ALT="$\theta&gt;\alpha$">|; 

$key = q/F(t)=-F_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img520.png"
 ALT="$F(t)=-F_0$">|; 

$key = q/omega_0=sqrt{gslashl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img462.png"
 ALT="$\omega_0 = \sqrt{g/l}$">|; 

$key = q/{displaymath}m,{{bf{a}={{bf{f}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1118.png"
 ALT="\begin{displaymath}
m {\bf a} = {\bf f}.
\end{displaymath}">|; 

$key = q/{{bf{e}_{z'}cdot{{bf{L}timesmbox{{boldmath{{omega{}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1405.png"
 ALT="${\bf e}_{z'}\cdot{\bf L}\times \mbox{\boldmath$\omega$}=0$">|; 

$key = q/displaystylefrac{domega_{z'}}{dt'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1380.png"
 ALT="$\displaystyle \frac{d\omega_{z'}}{dt'}$">|; 

$key = q/dfslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3431.png"
 ALT="$df/dx$">|; 

$key = q/{displaymath}r=r_0,theta^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img840.png"
 ALT="\begin{displaymath}
r = r_0 \theta^{ 2}.
\end{displaymath}">|; 

$key = q/r_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img710.png"
 ALT="$r_2$">|; 

$key = q/{displaymath}{{bf{x}_k^{ast,T},{{bf{G}^{ast,T},{{bf{x}_k=lambda_k^ast,{{bf{x}_k^{ast,T},{{bf{M}^{ast,T},{{bf{x}_k.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1925.png"
 ALT="\begin{displaymath}
{\bf x}_k^{\ast T} {\bf G}^{\ast T} {\bf x}_k = \lambda_k^\ast 
{\bf x}_k^{\ast T} {\bf M}^{\ast T} {\bf x}_k.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.20.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="287" BORDER="0"
 SRC="|."$dir".q|img3092.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.20.eps}}
\end{figure}">|; 

$key = q/{displaymath}chi=frac{G,m'}{R}frac{r}{R},P_1(costheta).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2214.png"
 ALT="\begin{displaymath}
\chi = \frac{G m'}{R}\frac{r}{R} P_1(\cos\theta).
\end{displaymath}">|; 

$key = q/{displaymath}psi=frac{theta}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1068.png"
 ALT="\begin{displaymath}
\psi = \frac{\theta}{2}.
\end{displaymath}">|; 

$key = q/{displaymath}m,l,frac{d^2theta}{dt^2}+m,g,sintheta=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img3024.png"
 ALT="\begin{displaymath}
m l \frac{d^2\theta}{dt^2} + m g \sin\theta = 0,
\end{displaymath}">|; 

$key = q/{{bf{a}equiv(1slashsqrt{2},,1slashsqrt{2},,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3251.png"
 ALT="${\bf a} \equiv (1/\sqrt{2},  1/\sqrt{2},  0)$">|; 

$key = q/{displaymath}z=-frac{x}{h}+c,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1771.png"
 ALT="\begin{displaymath}
z =-\frac{x}{h} + c,
\end{displaymath}">|; 

$key = q/{displaymath}{calL}=frac{1}{2}left(I_perp,dot{theta}^2+I_perp,sin^2theta,dot{phia)-epsilon,beta_m,sin(2,theta),cos(varpi_n-phi),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="638" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2334.png"
 ALT="\begin{displaymath}
{\cal L}= \frac{1}{2}\left(I_\perp \dot{\theta}^2 + I_\perp...
...eta)- \epsilon \beta_m \sin(2 \theta) \cos(\varpi_n-\phi),
\end{displaymath}">|; 

$key = q/x'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1417.png"
 ALT="$x''$">|; 

$key = q/{displaymath}frac{partial{calL}}{partialtheta}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2292.png"
 ALT="\begin{displaymath}
\frac{\partial {\cal L}}{\partial\theta} = 0.
\end{displaymath}">|; 

$key = q/stackrel{displaystylerightarrow}{PQ};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3183.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{PQ}$">|; 

$key = q/U_{xy}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2673.png"
 ALT="$U_{xy}=0$">|; 

$key = q/displaystylefrac{3}{2},m^2,cos(omega_4,T-alpha_4)+frac{3}{2},m^2,cos(omega_4,T-alpha_4),deltaX;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2878.png"
 ALT="$\displaystyle \frac{3}{2} m^2 \cos(\omega_4 T-\alpha_4)+\frac{3}{2} m^2 \cos(\omega_4 T-\alpha_4) \delta X$">|; 

$key = q/nu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img308.png"
 ALT="$\nu$">|; 

$key = q/r_2=1-r_1=mu_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2497.png"
 ALT="$r_2=1-r_1=\mu_1$">|; 

$key = q/0.2";MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1479.png"
 ALT="$0.2''$">|; 

$key = q/y';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img550.png"
 ALT="$y'$">|; 

$key = q/displaystyleddot{zeta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2507.png"
 ALT="$\displaystyle \ddot{\zeta}$">|; 

$key = q/{displaymath}alpha=frac{partialh}{partialx},~~~~~beta=frac{partialh}{partialy},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img3446.png"
 ALT="\begin{displaymath}
\alpha = \frac{\partial h}{\partial x},&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\beta= \frac{\partial h}{\partial y},
\end{displaymath}">|; 

$key = q/0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2781.png"
 ALT="$0)$">|; 

$key = q/R_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img983.png"
 ALT="$R_2$">|; 

$key = q/EslashB<v_0<2,EslashB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img650.png"
 ALT="$E/B &lt; v_0 &lt; 2 E/B$">|; 

$key = q/mu_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$\mu _2$">|; 

$key = q/I_{x'x'}^{-1slash2}:I_{y'y'}^{-1slash2}:I_{z'z'}^{-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1416.png"
 ALT="$I_{x'x'}^{-1/2}:I_{y'y'}^{-1/2}:I_{z'z'}^{-1/2}$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter07slashfig7.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="335" BORDER="0"
 SRC="|."$dir".q|img1195.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter07/fig7.04.eps}}
\end{figure}">|; 

$key = q/displaystyle2,m,nu,x_0^{,2}int_0^{T_r}{{rm{e}^{-2,nu,t}left[nu,cos(omega_r,t-phi_0)+omega_r,sin(omega_r,t-phi_0)right]^2dt,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="506" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img350.png"
 ALT="$\displaystyle 2 m \nu x_0^{ 2}\int_0^{T_r}{\rm e}^{-2 \nu t}\left[\nu \cos(\omega_r t-\phi_0) + \omega_r \sin(\omega_r t-\phi_0)\right]^2 dt,$">|; 

$key = q/Q=1.387978;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$Q=1.387978$">|; 

$key = q/ABABABcdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3087.png"
 ALT="$ABABAB\cdots$">|; 

$key = q/{displaymath}{mitOmega}=frac{q,B}{m}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img633.png"
 ALT="\begin{displaymath}
{\mit\Omega} = \frac{q B}{m}
\end{displaymath}">|; 

$key = q/Tslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img865.png"
 ALT="$T/2$">|; 

$key = q/deltaq_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1973.png"
 ALT="$\delta q_k$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{AppendixAslashfigA.05.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="195" BORDER="0"
 SRC="|."$dir".q|img3308.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.05.eps}}
\end{figure}">|; 

$key = q/{{bf{T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1179.png"
 ALT="${\bf T}$">|; 

$key = q/displaystyleC;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2657.png"
 ALT="$\displaystyle C$">|; 

$key = q/displaystylefrac{omega_j,a_j-2,b_j}{omega_j,(1-3,m^2slash2-omega_j^{,2})},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2842.png"
 ALT="$\displaystyle \frac{\omega_j a_j-2 b_j}{\omega_j (1-3 m^2/2-\omega_j^{ 2})},$">|; 

$key = q/{displaymath}int_P^Qleft(3,x^2,y,dx+x^3,dyright)=left[x^3,yright]_P^Q{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img3403.png"
 ALT="\begin{displaymath}
\int_P^Q \left(3 x^2 y dx + x^3 dy\right) = \left[x^3 y\right]_P^Q
\end{displaymath}">|; 

$key = q/pislash(2,k,a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img499.png"
 ALT="$\pi/(2 k a)$">|; 

$key = q/g_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2058.png"
 ALT="$g_z$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{AppendixAslashfigA.07.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="254" BORDER="0"
 SRC="|."$dir".q|img3323.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{AppendixA/figA.07.eps}}
\end{figure}">|; 

$key = q/omega_0=sqrt{kslashm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img532.png"
 ALT="$\omega_0 = \sqrt{k/m}$">|; 

$key = q/displaystyle{{bf{a}cdot{{bf{a}times{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3305.png"
 ALT="$\displaystyle {\bf a}\cdot {\bf a}\times{\bf b}$">|; 

$key = q/{displaymath}(lambda_k-lambda_l),{{bf{x}_k^T,{{bf{M},{{bf{x}_l=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1944.png"
 ALT="\begin{displaymath}
(\lambda_k-\lambda_l) {\bf x}_k^T {\bf M} {\bf x}_l = 0.
\end{displaymath}">|; 

$key = q/displaystyleddot{X}-2,dot{Y}-(1+m^2slash2),X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2761.png"
 ALT="$\displaystyle \ddot{X} -2 \dot{Y}-(1+m^2/2) X$">|; 

$key = q/{displaymath}Phi(r)=-frac{2pi,G,gamma}{3},(3,a^2-r^2)=-G,M,frac{(3,a^2-r^2)}{2,a^3}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="392" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2135.png"
 ALT="\begin{displaymath}
\Phi(r) = - \frac{2\pi G \gamma}{3} (3 a^2-r^2) = - G M \frac{(3 a^2-r^2)}{2 a^3}
\end{displaymath}">|; 

$key = q/{displaymath}frac{dU}{dx}=frac{d^2U}{dx^2}=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img227.png"
 ALT="\begin{displaymath}
\frac{dU}{dx}=\frac{d^2 U}{dx^2}=0
\end{displaymath}">|; 

$key = q/{mitDelta}t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3151.png"
 ALT="${\mit\Delta} t$">|; 

$key = q/R=4000,{{rm{mi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img909.png"
 ALT="$R=4000 {\rm mi}$">|; 

$key = q/{displaymath}psisimeqpileft(3-2,[1+epsilonslash(k,r_c^{,2})]right)^{-1slash2}=pi^{-1slash2}simeqpi,[1+epsilonslash(k,r_c^{,2})].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="605" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img881.png"
 ALT="\begin{displaymath}
\psi\simeq \pi\left(3-2 [1+\epsilon/(k r_c^{ 2})]\right)^...
...^{ 2})\right]^{-1/2} \simeq
\pi [1+\epsilon/(k r_c^{ 2})].
\end{displaymath}">|; 

$key = q/{{bf{L};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1265.png"
 ALT="${\bf L}$">|; 

$key = q/displaystyleI_perp,ddot{theta}-frac{1}{2},I_perp,sin(2,theta),dot{phi}^{,2}+I_{parallel},sintheta,omega,dot{phi}+2,epsilon,alpha,sin(2,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="440" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2337.png"
 ALT="$\displaystyle I_\perp \ddot{\theta} - \frac{1}{2} I_\perp \sin(2 \theta) \...
...parallel} \sin\theta \omega \dot{\phi}
+2 \epsilon \alpha \sin(2 \theta)$">|; 

$key = q/displaystylea_y';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$\displaystyle a_y'$">|; 

$key = q/{displaymath}int_{-1}^1P_n(x),P_m(x),dx=int_0^piP_n(costheta),P_m(costheta),sintheta,dtheta=frac{delta_{nm}}{n+1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="525" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2111.png"
 ALT="\begin{displaymath}
\int_{-1}^1 P_n(x) P_m(x) dx = \int_0^\pi P_n(\cos\theta)\...
...(\cos\theta) \sin\theta d\theta = \frac{\delta_{nm}}{n+1/2}.
\end{displaymath}">|; 

$key = q/{displaymath}L=K-U.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1604.png"
 ALT="\begin{displaymath}
L = K - U.
\end{displaymath}">|; 

$key = q/displaystylefrac{m_2}{m_1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2490.png"
 ALT="$\displaystyle \frac{m_2}{m_1},$">|; 

$key = q/nu=omega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img328.png"
 ALT="$\nu=\omega_0$">|; 

$key = q/lambda_kneqlambda_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1946.png"
 ALT="$\lambda_k\neq \lambda_l$">|; 

$key = q/displaystyleV_0,costheta-2,{mitOmega},V_0,sinlambda,sintheta,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1171.png"
 ALT="$\displaystyle V_0 \cos\theta -2 {\mit\Omega} V_0 \sin\lambda \sin\theta t,$">|; 

$key = q/|dot{phi}|llomega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2294.png"
 ALT="$\vert\dot{\phi}\vert\ll \omega$">|; 

$key = q/{{bf{C};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3541.png"
 ALT="${\bf C}$">|; 

$key = q/displaystyleeta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2021.png"
 ALT="$\displaystyle \eta_2$">|; 

$key = q/{displaymath}{{bf{g}_g=-nablaPhi',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2216.png"
 ALT="\begin{displaymath}
{\bf g}_g = -\nabla\Phi',
\end{displaymath}">|; 

$key = q/v_0,sinthetaggv_t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img601.png"
 ALT="$v_0 \sin\theta \gg v_t$">|; 

$key = q/L_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$L_3$">|; 

$key = q/displaystylefrac{r_c}{sqrt{1-e^2}}=sqrt{1-e^2},a,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img721.png"
 ALT="$\displaystyle \frac{r_c}{\sqrt{1-e^2}}=\sqrt{1-e^2} a,$">|; 

$key = q/displaystyle{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2707.png"
 ALT="$\displaystyle {\bf r}$">|; 

$key = q/displaystylea_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="$\displaystyle a_x,$">|; 

$key = q/d{mitOmega}=2pi,sintheta,dtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1079.png"
 ALT="$d{\mit\Omega} =2\pi \sin\theta d\theta$">|; 

$key = q/deltaq_1(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1803.png"
 ALT="$\delta q_1(t)$">|; 

$key = q/{{bf{b}=2,{{bf{e}_x+{{bf{e}_y+{{bf{e}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3516.png"
 ALT="${\bf b} = 2 {\bf e}_x+ {\bf e}_y+{\bf e}_z$">|; 

$key = q/dot{p}_i=partialLslashpartialq_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1849.png"
 ALT="$\dot{p}_i = \partial L/\partial q_i$">|; 

$key = q/displaystyle-frac{3}{2},m^2,sin(omega_4,T-alpha_4)-frac{3}{2},m^2,sin(omega_4,T-alpha_4),deltaX;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="405" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2880.png"
 ALT="$\displaystyle - \frac{3}{2} m^2 \sin(\omega_4 T-\alpha_4)-\frac{3}{2} m^2 \sin(\omega_4 T-\alpha_4) \delta X$">|; 

$key = q/{displaymath}frac{d^2x}{dt^2}+2,nu,frac{dx}{dt}+omega_0^{,2},x=omega_0^{,2},X_1,cos(omega,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img365.png"
 ALT="\begin{displaymath}
\frac{d^2 x}{dt^2} + 2 \nu \frac{dx}{dt} + \omega_0^{ 2} x
= \omega_0^{ 2} X_1 \cos(\omega t).
\end{displaymath}">|; 

$key = q/stackrel{displaystylerightarrow}{PR};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3191.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{PR}$">|; 

$key = q/displaystylem,(ddot{x}'+ddot{x},costheta)-m,g,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1653.png"
 ALT="$\displaystyle m (\ddot{x}'+ \ddot{x} \cos\theta) - m g \sin\theta$">|; 

$key = q/+lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1219.png"
 ALT="$+\lambda$">|; 

$key = q/T_i=2pislashomega_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2413.png"
 ALT="$T_i = 2\pi/\omega_i$">|; 

$key = q/{{bf{a}times{{bf{a}={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3315.png"
 ALT="${\bf a}\times{\bf a} = {\bf0}$">|; 

$key = q/{displaymath}{{bf{x}_i^{ast,T},{{bf{A}=lambda_i^{,ast},{{bf{x}_i^{ast,T},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1289.png"
 ALT="\begin{displaymath}
{\bf x}_i^{\ast T} {\bf A} = \lambda_i^{ \ast} {\bf x}_i^{\ast T},
\end{displaymath}">|; 

$key = q/{displaymath}left(1+frac{1}{4pi^2,n^2}right)^{1slash2}simeq1+frac{1}{8pi^2,n^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img511.png"
 ALT="\begin{displaymath}
\left(1+\frac{1}{4\pi^2 n^2}\right)^{1/2}\simeq 1 + \frac{1}{8\pi^2 n^2}.
\end{displaymath}">|; 

$key = q/(1slash2),cos(varpi_m-phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2324.png"
 ALT="$(1/2) \cos(\varpi_m-\phi)$">|; 

$key = q/sqrt{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img787.png"
 ALT="$\sqrt{2}$">|; 

$key = q/{displaymath}F_n=frac{r_{n-1}-r_{n-2}}{r_{n}-r_{n-1}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img3124.png"
 ALT="\begin{displaymath}
F_n =\frac{r_{n-1}-r_{n-2}}{r_{n}-r_{n-1}}
\end{displaymath}">|; 

$key = q/displaystyle-{mitOmega},v_x',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img631.png"
 ALT="$\displaystyle -{\mit\Omega} v_x',$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter09slashfig9.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="339" BORDER="0"
 SRC="|."$dir".q|img1622.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter09/fig9.01.eps}}
\end{figure}">|; 

$key = q/n_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2313.png"
 ALT="$n_m$">|; 

$key = q/M_EslashM_Msimeq81;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2704.png"
 ALT="$M_E/M_M\simeq 81$">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{Chapter05slashfig5.06.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="411" HEIGHT="169" BORDER="0"
 SRC="|."$dir".q|img737.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{Chapter05/fig5.06.eps}}
\end{figure}">|; 

$key = q/r=u^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img744.png"
 ALT="$r = u^{-1}$">|; 

$key = q/{displaymath}r_a=frac{r_c}{1-e}=a,(1+e).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img758.png"
 ALT="\begin{displaymath}
r_a = \frac{r_c}{1-e} = a (1+e).
\end{displaymath}">|; 

$key = q/Oz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$Oz$">|; 

$key = q/displaystyleE_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1048.png"
 ALT="$\displaystyle E_2$">|; 

$key = q/lambda(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1751.png"
 ALT="$\lambda(x)$">|; 

$key = q/19.63^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2993.png"
 ALT="$19.63^\circ$">|; 

$key = q/{displaymath}frac{d^2theta}{dt^2}+omega_0^{,2},(theta-theta^{,3}slash6)simeq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img467.png"
 ALT="\begin{displaymath}
\frac{d^2\theta}{dt^2} + \omega_0^{ 2} (\theta-\theta^{ 3}/6)\simeq 0.
\end{displaymath}">|; 

$key = q/{mitDelta}=phi_2-phi_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img541.png"
 ALT="${\mit\Delta}=\phi_2-\phi_1$">|; 

$key = q/{displaymath}p_k={{rm{const.}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1666.png"
 ALT="\begin{displaymath}
p_k = {\rm const.}
\end{displaymath}">|; 

$key = q/{{bf{x}_k^{ast,T},{{bf{M},{{bf{x}_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1932.png"
 ALT="${\bf x}_k^{\ast T} {\bf M} {\bf x}_k$">|; 

$key = q/displaystyle-frac{15}{8},m,e,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2935.png"
 ALT="$\displaystyle -\frac{15}{8} m e,$">|; 

$key = q/T_m({{rm{yr})=0.081;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2372.png"
 ALT="$T_m({\rm yr})=0.081$">|; 

$key = q/{displaymath}theta=frac{1}{2},tan^{-1}left(frac{2,A,B,cos{mitDelta}}{A^2-B^2}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img559.png"
 ALT="\begin{displaymath}
\theta = \frac{1}{2} \tan^{-1}\left(\frac{2 A B \cos{\mit\Delta}}{A^2-B^2}\right).
\end{displaymath}">|; 

$key = q/omega_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1276.png"
 ALT="$\omega_z$">|; 

$key = q/erightarrow1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img762.png"
 ALT="$e\rightarrow 1$">|; 

$key = q/displaystylea_5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2926.png"
 ALT="$\displaystyle a_5$">|; 

$key = q/Q=10;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$Q=10$">|; 

$key = q/{displaymath}ddot{r}-r,dot{theta}^{,2}=-frac{G,M}{r^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img693.png"
 ALT="\begin{displaymath}
\ddot{r}-r \dot{\theta}^{ 2} = - \frac{G M}{r^2},
\end{displaymath}">|; 

$key = q/e,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img889.png"
 ALT="$e a$">|; 

$key = q/{displaymath}Deltaa=a_e-a_p=epsilon,a=27.3,{{rm{km}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2183.png"
 ALT="\begin{displaymath}
\Delta a = a_e-a_p = \epsilon a = 27.3 {\rm km}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{h}={{bf{r}times{{bf{v},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img669.png"
 ALT="\begin{displaymath}
{\bf h} = {\bf r}\times {\bf v},
\end{displaymath}">|; 

$key = q/b(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img971.png"
 ALT="$b(\theta)$">|; 

$key = q/gamma^2=lambda_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1956.png"
 ALT="$\gamma^2=\lambda_k$">|; 

$key = q/displaystylez',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2737.png"
 ALT="$\displaystyle z',$">|; 

$key = q/phi_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img422.png"
 ALT="$\phi_n$">|; 

$key = q/d{{bf{r}slashdt';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1109.png"
 ALT="$d{\bf r}/dt'$">|; 

$key = q/{displaymath}Deltaa=a_+-a_-=-epsilon,a=1.34,{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2245.png"
 ALT="\begin{displaymath}
\Delta a = a_+-a_- = -\epsilon a = 1.34 {\rm m}.
\end{displaymath}">|; 

$key = q/{displaymath}dT={{bf{grad},Tcdotd{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img3458.png"
 ALT="\begin{displaymath}
dT = {\bf grad} T \cdot d{\bf r}.
\end{displaymath}">|; 

$key = q/{displaymath}X(t+T)=X(t){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img395.png"
 ALT="\begin{displaymath}
X(t+T) = X(t)
\end{displaymath}">|; 

$key = q/displaystyleddot{x}-2,omega,dot{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2539.png"
 ALT="$\displaystyle \ddot{x} - 2 \omega \dot{y}$">|; 

$key = q/{displaymath}T_phisimeq27,600,{{rm{years}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2365.png"
 ALT="\begin{displaymath}
T_\phi\simeq 27,600 {\rm years}.
\end{displaymath}">|; 

$key = q/{displaymath}F_n=frac{Q_{n-1}-Q_{n-2}}{Q_{n}-Q_{n-1}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img3100.png"
 ALT="\begin{displaymath}
F_n = \frac{Q_{n-1}-Q_{n-2}}{Q_{n}-Q_{n-1}}
\end{displaymath}">|; 

$key = q/{displaymath}dr=frac{r_p}{1-e},e,sinE,dE,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img802.png"
 ALT="\begin{displaymath}
dr = \frac{r_p}{1-e} e \sin E dE,
\end{displaymath}">|; 

$key = q/{displaymath}sintheta=theta-frac{theta^{,3}}{3!}+frac{theta^{,5}}{5!}+cdots.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img466.png"
 ALT="\begin{displaymath}
\sin\theta = \theta - \frac{\theta^{ 3}}{3!} + \frac{\theta^{ 5}}{5!} +\cdots.
\end{displaymath}">|; 

$key = q/Z=zslash(v_0^{,2}slashg);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img613.png"
 ALT="$Z=z/(v_0^{ 2}/g)$">|; 

$key = q/{{bf{f}={{bf{f}_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img919.png"
 ALT="${\bf f} = {\bf f}_{21}$">|; 

$key = q/displaystyle-frac{k,(1+2alpha)}{m}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2040.png"
 ALT="$\displaystyle - \frac{k (1+2\alpha)}{m}.$">|; 

$key = q/displaystylefrac{m,g,l}{I_parallel,{mitOmega}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1527.png"
 ALT="$\displaystyle \frac{m g l}{I_\parallel {\mit\Omega}},$">|; 

$key = q/nullomega_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img352.png"
 ALT="$\nu\ll \omega_r$">|; 

$key = q/oint{{bf{A}cdotd{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3549.png"
 ALT="$\oint {\bf A}\cdot d{\bf r}$">|; 

$key = q/P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="$P$">|; 

$key = q/nu<omega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img322.png"
 ALT="$\nu &lt; \omega_0$">|; 

$key = q/q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img619.png"
 ALT="$q$">|; 

$key = q/displaystyle(2,m)^{-1slash2},(1,-1).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2013.png"
 ALT="$\displaystyle (2 m)^{-1/2} (1,-1).$">|; 

$key = q/displaystylev_1,(costheta+m_1slashm_2,,sintheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1034.png"
 ALT="$\displaystyle v_1 (\cos\theta+m_1/m_2, \sin\theta),$">|; 

$key = q/{displaymath}Phi_0(r)=-frac{4pi,G,gamma}{r}int_0^ar'^{,2},dr'{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2132.png"
 ALT="\begin{displaymath}
\Phi_0(r) = - \frac{4\pi G \gamma}{r}\int_0^a r'^{ 2} dr'
\end{displaymath}">|; 

$key = q/E_y=E_0,sin(omega,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img653.png"
 ALT="$E_y=E_0 \sin(\omega t)$">|; 

$key = q/{displaymath}frac{d}{dt}!left(frac{partialK}{partialdot{q}_i}right)=-frac{partialU}{partialq_i}+frac{partialK}{partialq_i}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1602.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial K}{\partial \dot{q}_i}\ri...
...rac{\partial U}{\partial q_i}+\frac{\partial K}{\partial q_i}.
\end{displaymath}">|; 

$key = q/{displaymath}gamma^4+(4+U_{xx}+U_{yy}),gamma^2+(U_{xx},U_{yy}-U_{xy}^{,2})=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img2652.png"
 ALT="\begin{displaymath}
\gamma^4 + (4+U_{xx}+U_{yy}) \gamma^2 + (U_{xx} U_{yy}-U_{xy}^{ 2}) = 0.
\end{displaymath}">|; 

$key = q/C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$C$">|; 

$key = q/omega_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1537.png"
 ALT="$\omega_{x'}$">|; 

$key = q/{displaymath}{{bf{l}_i=m_i,{{bf{r}_itimesfrac{d{{bf{r}_i}{dt}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img154.png"
 ALT="\begin{displaymath}
{\bf l}_i = m_i {\bf r}_i\times \frac{d{\bf r}_i}{dt}
\end{displaymath}">|; 

$key = q/X_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img364.png"
 ALT="$X_1$">|; 

$key = q/{displaymath}sinhu=frac{v^2}{4,g,l}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1708.png"
 ALT="\begin{displaymath}
\sinh u = \frac{v^2}{4 g l}.
\end{displaymath}">|; 

$key = q/deltaq_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1799.png"
 ALT="$\delta q_1$">|; 

$key = q/{displaymath}K=frac{1}{2},m,(dot{r}^{,2}+r^2,dot{theta}^{,2}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1866.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} m (\dot{r}^{ 2} + r^2 \dot{\theta}^{ 2}),
\end{displaymath}">|; 

$key = q/{displaymath}frac{r_1}{r_2}=frac{sqrt{x^2+y^2}}{x+d}=frac{r}{r,costheta+d}=e,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img711.png"
 ALT="\begin{displaymath}
\frac{r_1}{r_2} = \frac{\sqrt{x^2+y^2}}{x+d}= \frac{r}{r \cos\theta+d}=e,
\end{displaymath}">|; 

$key = q/displaystyle{{bf{r}_M-{{bf{r}_E,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2708.png"
 ALT="$\displaystyle {\bf r}_M-{\bf r}_E,$">|; 

$key = q/displaystyle{{bf{V}_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1035.png"
 ALT="$\displaystyle {\bf V}_2'$">|; 

$key = q/{displaymath}E=frac{1}{2},I_perp,dot{theta}^{,2}+frac{(L_phi-L_psi,costheta)^2}{}{2}frac{L_psi^{,2}}{I_parallel}+m,g,l,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="435" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1499.png"
 ALT="\begin{displaymath}
E = \frac{1}{2} I_\perp \dot{\theta}^{ 2} + \frac{(L_\phi...
...c{1}{2}\frac{L_\psi^{ 2}}{I_\parallel} + m g l \cos\theta.
\end{displaymath}">|; 

$key = q/{displaymath}K=frac{1}{2}sum_{i=1,N}m_i,(mbox{{boldmath{{omega{}times{{bf{r}_i)c}_itimes(mbox{{boldmath{{omega{}times{{bf{r}_i).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="521" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1270.png"
 ALT="\begin{displaymath}
K = \frac{1}{2}\sum_{i=1,N} m_i (\mbox{\boldmath$\omega$}\t...
...i {\bf r}_i\times (\mbox{\boldmath$\omega$}\times {\bf r}_i).
\end{displaymath}">|; 

$key = q/omega=1.2,omega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img435.png"
 ALT="$\omega = 1.2 \omega_0$">|; 

$key = q/{displaymath}tan(2,alpha)=frac{2,I_{xy}}{I_{xx}-I_{yy}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1342.png"
 ALT="\begin{displaymath}
\tan (2 \alpha) = \frac{2 I_{xy}}{I_{xx} - I_{yy}}.
\end{displaymath}">|; 

$key = q/{{bf{r}_2=(xi_2,,eta_2,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2499.png"
 ALT="${\bf r}_2=(\xi_2, \eta_2, 0)$">|; 

$key = q/displaystyleI_{z'z'},dot{omega}_{z'}-(I_{x'x'}-I_{y'y'}),omega_{x'},omega_{y'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1367.png"
 ALT="$\displaystyle I_{z'z'} \dot{\omega}_{z'} - (I_{x'x'}-I_{y'y'}) \omega_{x'} \omega_{y'},$">|; 

$key = q/{displaymath}left.frac{df}{dx}right|_{x_0}<0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img222.png"
 ALT="\begin{displaymath}
\left.\frac{df}{dx}\right\vert _{x_0} &lt;0
\end{displaymath}">|; 

$key = q/Qsimeq1.3755;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3097.png"
 ALT="$Q\simeq 1.3755$">|; 

$key = q/{displaymath}g_x=G,m',frac{(x'-x)}{[(x'-x)^2+(y'-y)^2+(z'-z)^2]^{,3slash2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2052.png"
 ALT="\begin{displaymath}
g_x = G m' \frac{(x'-x)}{[(x'-x)^2+(y'-y)^2+(z'-z)^2]^{ 3/2}},
\end{displaymath}">|; 

$key = q/{{bf{T}_{hz}=-T,sintheta,{{bf{r}'slashr'simeq-T,{{bf{r}'slashl;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1186.png"
 ALT="${\bf T}_{hz}= - T \sin\theta {\bf r}'/r'\simeq
-T {\bf r}'/l$">|; 

$key = q/displaystylex,'sintheta+y',costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3480.png"
 ALT="$\displaystyle x '\sin\theta +y' \cos\theta,$">|; 

$key = q/{displaymath}({{bf{G}-lambda,{{bf{M}),{{bf{x}={{bf{0},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1995.png"
 ALT="\begin{displaymath}
({\bf G} -\lambda {\bf M}) {\bf x} = {\bf0},
\end{displaymath}">|; 

$key = q/t=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$t=0$">|; 

$key = q/displaystyle-frac{1}{4},iota^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2917.png"
 ALT="$\displaystyle -\frac{1}{4} \iota^2.$">|; 

$key = q/displaystyle1-e,cos[(1+c,m^2),T-alpha_0]+frac{1}{2},e^2-frac{1}{6},m^2-frac{1}{2},e^2,cos[2,(1+c,m^2),T-2,alpha_0];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="619" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2966.png"
 ALT="$\displaystyle 1 - e \cos[(1+c m^2) T-\alpha_0] +\frac{1}{2} e^2 - \frac{1}{6} m^2- \frac{1}{2} e^2 \cos[2 (1+c m^2) T-2 \alpha_0]$">|; 

$key = q/{displaymath}F=bar{theta}-gamma,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2996.png"
 ALT="\begin{displaymath}
F = \bar{\theta}-\gamma,
\end{displaymath}">|; 

$key = q/M_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2705.png"
 ALT="$M_E$">|; 

$key = q/{{bf{d}=lambda,{{bf{a}+mu,{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3520.png"
 ALT="${\bf d} = \lambda {\bf a} + \mu {\bf b}$">|; 

$key = q/{displaymath}dh={{bf{grad},hcdotd{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img3451.png"
 ALT="\begin{displaymath}
dh = {\bf grad} h \cdot d{\bf r}.
\end{displaymath}">|; 

$key = q/U_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img188.png"
 ALT="$U_0$">|; 

$key = q/displaystylefrac{a,(1-e^2)}{1-e,costheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img764.png"
 ALT="$\displaystyle \frac{a (1-e^2)}{1-e \cos\theta},$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.12.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="287" BORDER="0"
 SRC="|."$dir".q|img3079.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.12.eps}}
\end{figure}">|; 

$key = q/t+deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img739.png"
 ALT="$t+\delta t$">|; 

$key = q/theta=gamma_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2823.png"
 ALT="$\theta=\gamma_0$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter15slashfig15.22.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="461" HEIGHT="345" BORDER="0"
 SRC="|."$dir".q|img3098.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter15/fig15.22.eps}}
\end{figure}">|; 

$key = q/f(x_1,x_2,cdots,x_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img173.png"
 ALT="$f(x_1,x_2,\cdots,x_n)$">|; 

$key = q/displaystylez;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img648.png"
 ALT="$\displaystyle z$">|; 

$key = q/displaystyleomega_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1385.png"
 ALT="$\displaystyle \omega_{x'}$">|; 

$key = q/{displaymath}frac{v_t}{v_c}=sqrt{1-e},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img781.png"
 ALT="\begin{displaymath}
\frac{v_t}{v_c} = \sqrt{1-e},
\end{displaymath}">|; 

$key = q/{displaymath}ddot{x}-frac{h^2}{(r_c+x)^3}=f(r_c+x).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img851.png"
 ALT="\begin{displaymath}
\ddot{x} - \frac{h^2}{(r_c+x)^3} = f(r_c+x).
\end{displaymath}">|; 

$key = q/{displaymath}nablaequivleft(frac{partial}{partialx},,frac{partial}{partialy},,frac{partial}{partialz}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3476.png"
 ALT="\begin{displaymath}
\nabla \equiv \left( \frac{\partial}{\partial x},  \frac{\partial}{\partial y}, 
\frac{\partial }{\partial z}\right),
\end{displaymath}">|; 

$key = q/{displaymath}x(t)=a,sin(omega_0,t-phi_0).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img294.png"
 ALT="\begin{displaymath}
x(t) = a \sin(\omega_0 t -\phi_0).
\end{displaymath}">|; 

$key = q/{displaymath}dh=frac{partialh}{partialx},dx+frac{partialh}{partialy},dy.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img3449.png"
 ALT="\begin{displaymath}
dh = \frac{\partial h}{\partial x}  dx +\frac{\partial h}
{\partial y}  dy.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d{{bf{L}}{dt}=sum_{i,j=1,N}^{ineqj}{{bf{r}_itimes{{bf{f}_{ij}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="67" BORDER="0"
 SRC="|."$dir".q|img156.png"
 ALT="\begin{displaymath}
\frac{d {\bf L}}{dt} = \sum_{i,j = 1,N}^{i\neq j} {\bf r}_i\times {\bf f}_{ij}.
\end{displaymath}">|; 

$key = q/displaystylelangleF^2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2096.png"
 ALT="$\displaystyle \langle F^2\rangle$">|; 

$key = q/deltay(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1722.png"
 ALT="$\delta y(x)$">|; 

$key = q/{displaymath}{{bf{a}=(ddot{r}-r,dot{theta}^{,2}),{{bf{e}_r+(r,ddot{theta}+2,dot{r},dot{theta}),{{bf{e}_theta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img691.png"
 ALT="\begin{displaymath}
{\bf a} = (\ddot{r}-r \dot{\theta}^{ 2}) {\bf e}_r + (r \ddot{\theta} + 2 \dot{r} \dot{\theta}) {\bf e}_\theta.
\end{displaymath}">|; 

$key = q/displaystylem,r^2,ddot{theta}+m,g,r,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1836.png"
 ALT="$\displaystyle m r^2 \ddot{\theta} + m g r \sin\theta$">|; 

$key = q/displaystyle-a_xsintheta+a_ycostheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3243.png"
 ALT="$\displaystyle -a_x\sin\theta + a_y\cos\theta,$">|; 

$key = q/displaystyle3left[frac{mu_1}{rho_1^{,5}}+frac{mu_2}{rho_2^{,5}}right]y^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2656.png"
 ALT="$\displaystyle 3\left[ \frac{\mu_1}{\rho_1^{ 5}} + \frac{\mu_2}{\rho_2^{ 5}}\right]y^2,$">|; 

$key = q/{displaymath}T=frac{A}{(dAslashdt)}=frac{2pi,a,b}{h}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img755.png"
 ALT="\begin{displaymath}
T = \frac{A}{(dA/dt)} = \frac{2\pi a b}{h}.
\end{displaymath}">|; 

$key = q/v_t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img268.png"
 ALT="$v_t$">|; 

$key = q/a_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2304.png"
 ALT="$a_m$">|; 

$key = q/{calE}_2'=(1slash2),m_2,V_2'^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1043.png"
 ALT="${\cal E}_2'=(1/2) m_2 V_2'^{ 2}$">|; 

$key = q/{displaymath}int_P^Q{{bf{A}cdotd{{bf{r}=int_P^Q(A_x,dx+A_y,dy+A_z,dz).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img3408.png"
 ALT="\begin{displaymath}
\int_P^Q {\bf A}\cdot d{\bf r} = \int_P^Q (A_x dx+A_y dy + A_z dz).
\end{displaymath}">|; 

$key = q/{displaymath}sinthetasimeqtheta{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img3042.png"
 ALT="\begin{displaymath}
\sin\theta \simeq \theta
\end{displaymath}">|; 

$key = q/{displaymath}left(array{{c}x'y'z'array{right)=left(array{{ccc}cospsi&sinpsi&0-siay{right)left(array{{c}x'''y'''z'''array{right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="76" BORDER="0"
 SRC="|."$dir".q|img1437.png"
 ALT="\begin{displaymath}
\left(\begin{array}{c}x'\ y'\ z'\end{array}\right)=
\left(...
...ht)\left(\begin{array}{c}x'''\ y'''\ z'''\end{array}\right).
\end{displaymath}">|; 

$key = q/a>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img904.png"
 ALT="$a&gt;0$">|; 

$key = q/{displaymath}K=frac{1}{2}sum_{j=1,{calF}}m_j,dot{x}_j^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1584.png"
 ALT="\begin{displaymath}
K = \frac{1}{2}\sum_{j=1,{\cal F}} m_j \dot{x}_j^{ 2}.
\end{displaymath}">|; 

$key = q/K;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="$K$">|; 

$key = q/epsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2146.png"
 ALT="$\epsilon$">|; 

$key = q/d{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="$d{\bf r}$">|; 

$key = q/x_0=v_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img250.png"
 ALT="$x_0=v_0=0$">|; 

$key = q/m_2ggm_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1066.png"
 ALT="$m_2\gg m_1$">|; 

$key = q/mu_1=G,m_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2494.png"
 ALT="$\mu_1=G m_1$">|; 

$key = q/A=1.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$A=1.5$">|; 

$key = q/omega_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1275.png"
 ALT="$\omega_y$">|; 

$key = q/gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1332.png"
 ALT="$\gamma$">|; 

$key = q/partialfslashpartialtheta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1833.png"
 ALT="$\partial f/\partial\theta = 0$">|; 

$key = q/displaystylev_{y'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1172.png"
 ALT="$\displaystyle v_{y'}$">|; 

$key = q/{displaymath}L(q_1,q_2,cdots,q_{calF},dot{q}_1,dot{q}_2,cdots,dot{q}_{calF},t)=K-U.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1791.png"
 ALT="\begin{displaymath}
L(q_1,q_2,\cdots,q_{\cal F}, \dot{q}_1,\dot{q}_2,\cdots,\dot{q}_{\cal F},t) = K - U.
\end{displaymath}">|; 

$key = q/2times2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3171.png"
 ALT="$2\times 2$">|; 

$key = q/j=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2946.png"
 ALT="$j=1$">|; 

$key = q/10.75;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2449.png"
 ALT="$10.75$">|; 

$key = q/{displaymath}Phi(r,theta)=-2pi,Gint_0^inftyint_0^pir'^{,2},rho(r',theta'),sinthelangle|{{bf{r}-{{bf{r}'|^{-1}rangle,dr',dtheta',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="484" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2081.png"
 ALT="\begin{displaymath}
\Phi(r,\theta) = - 2\pi G\int_0^\infty\int_0^\pi
r'^{ 2} ...
...,\langle\vert{\bf r}-{\bf r}'\vert^{-1}\rangle dr' d\theta',
\end{displaymath}">|; 

$key = q/{displaymath}frac{2^{1slash2}}{3,pi}left(frac{2,p}{a}+1right)left(1-frac{p}{a}right)^{1slash2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img896.png"
 ALT="\begin{displaymath}
\frac{2^{1/2}}{3 \pi}\left(\frac{2 p}{a}+1\right)\left(1-\frac{p}{a}\right)^{1/2}
\end{displaymath}">|; 

$key = q/{{bf{f}=m,{{bf{g};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1124.png"
 ALT="${\bf f}= m {\bf g}$">|; 

$key = q/displaystylev_parallel,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img636.png"
 ALT="$\displaystyle v_\parallel,$">|; 

$key = q/0.3%;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1142.png"
 ALT="$0.3\%$">|; 

$key = q/{displaymath}sum_{j=1,{calF}}(G_{ij}-gamma^2,M_{ij}),deltaq_j=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1912.png"
 ALT="\begin{displaymath}
\sum_{j=1,{\cal F}}(G_{ij}- \gamma^2 M_{ij}) \delta q_j = 0,
\end{displaymath}">|; 

$key = q/displaystyley_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2734.png"
 ALT="$\displaystyle y_1'$">|; 

$key = q/Q=Q_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3104.png"
 ALT="$Q=Q_\infty$">|; 

$key = q/displaystyleleft(frac{G,M}{2,r_p^{,3}}right)^{1slash2}t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img825.png"
 ALT="$\displaystyle \left(\frac{G M}{2 r_p^{ 3}}\right)^{1/2} t,$">|; 

$key = q/displaystyleleft(frac{1+costheta}{2}right){calE}=cos^2(thetaslash2),,{calE}=cos^2psi,,{calE},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1072.png"
 ALT="$\displaystyle \left(\frac{1+\cos\theta}{2}\right){\cal E} = \cos^2(\theta/2)  {\cal E} = \cos^2\psi  {\cal E},$">|; 

$key = q/{displaymath}frac{d{{bf{L}}{dt}={{bf{0}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img162.png"
 ALT="\begin{displaymath}
\frac{d {\bf L}}{dt} = {\bf0}.
\end{displaymath}">|; 

$key = q/u_3>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1510.png"
 ALT="$u_3&gt;1$">|; 

$key = q/displaystyledot{phi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1470.png"
 ALT="$\displaystyle \dot{\phi}$">|; 

$key = q/j>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2847.png"
 ALT="$j&gt;0$">|; 

$key = q/displaystyle+g,(m_1-m_2-m_3),x+g,(m_2-m_3),x'+{{rm{const}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1636.png"
 ALT="$\displaystyle + g (m_1-m_2-m_3) x+g (m_2-m_3) x' + {\rm const}.$">|; 

$key = q/{displaymath}(x+mu_2)^2+y^2=(x-1+mu_2)^2+y^2=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2599.png"
 ALT="\begin{displaymath}
(x+\mu_2)^2 + y^2 = (x-1+\mu_2)^2 + y^2 = 1,
\end{displaymath}">|; 

$key = q/q_3=-2,alpha,q_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2046.png"
 ALT="$q_3=-2 \alpha q_1$">|; 

$key = q/theta_0(t_0)+deltatheta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3132.png"
 ALT="$\theta_0(t_0)+\delta\theta_0$">|; 

$key = q/displaystylev_perp,cos({mitOmega},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img635.png"
 ALT="$\displaystyle v_\perp \cos({\mit\Omega} t)$">|; 

$key = q/displaystyle-frac{3}{2},m^2,sin[2,(1-m),T],deltaY-3,deltaX^{,2}+frac{3}{2},(deltaY^{,2}+deltaZ^{,2}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2789.png"
 ALT="$\displaystyle -\frac{3}{2} m^2 \sin[2 (1-m) T] \delta Y-3 \delta X^{ 2} + \frac{3}{2} (\delta Y^{ 2} + \delta Z^{ 2}),$">|; 

$key = q/displaystyle-omega_0^{,2},y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img531.png"
 ALT="$\displaystyle - \omega_0^{ 2} y,$">|; 

$key = q/{displaymath}mu_m=frac{M_m}{M}=0.0123{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2330.png"
 ALT="\begin{displaymath}
\mu_m=\frac{M_m}{M} = 0.0123
\end{displaymath}">|; 

$key = q/{displaymath}m_i,{{bf{r}_itimesfrac{d^2{{bf{r}_i}{dt^2}=sum_{j=1,N}^{jneqi}{{bf{r}_itimes{{bf{f}_{ij}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="67" BORDER="0"
 SRC="|."$dir".q|img152.png"
 ALT="\begin{displaymath}
m_i {\bf r}_i\times \frac{d^2{\bf r}_i }{dt^2}= \sum_{j=1,N}^{j\neq i} {\bf r}_i\times {\bf f}_{ij}.
\end{displaymath}">|; 

$key = q/tggv_tslashg;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img264.png"
 ALT="$t\gg v_t/g$">|; 

$key = q/{{bf{r}_2=mu_1,(1,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2534.png"
 ALT="${\bf r}_2=\mu_1 (1, 0, 0)$">|; 

$key = q/df(0)slashdx=-m,omega_0^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img291.png"
 ALT="$df(0)/dx = -m \omega_0^{ 2}$">|; 

$key = q/X(t)=A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img425.png"
 ALT="$X(t) = A$">|; 

$key = q/{{bf{g}={{bf{g}_g+{{bf{g}_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2164.png"
 ALT="${\bf g} = {\bf g}_g + {\bf g}_c$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter10slashfig10.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="280" BORDER="0"
 SRC="|."$dir".q|img1719.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter10/fig10.01.eps}}
\end{figure}">|; 

$key = q/{displaymath}frac{d^2x}{dt^2}+omega_0^{,2},xsimeq0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img292.png"
 ALT="\begin{displaymath}
\frac{d^2 x}{dt^2}+ \omega_0^{ 2} x\simeq 0,
\end{displaymath}">|; 

$key = q/delta_{nm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2112.png"
 ALT="$\delta_{nm}$">|; 

$key = q/{displaymath}U=U_0+frac{1}{2},U_{xx},(deltax)^2+U_{xy},deltax,deltay+frac{1}{2},U_{yy},(deltay)^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2644.png"
 ALT="\begin{displaymath}
U = U_0 + \frac{1}{2} U_{xx} (\delta x)^2+ U_{xy} \delta x \delta y
+ \frac{1}{2} U_{yy} (\delta y)^2.
\end{displaymath}">|; 

$key = q/displaystyle+frac{G}{R_i^{,2}}sum_{j>i}M_j,left(frac{R_i}{R_j}right)!left[frac{1}right)^2+frac{9}{16}left(frac{R_i}{R_j}right)^4+cdotsright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="405" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2400.png"
 ALT="$\displaystyle +\frac{G}{R_i^{ 2}} \sum_{j&gt; i}M_j \left(\frac{R_i}{R_j}\right)...
...c{R_i}{R_j}\right)^2+ \frac{9}{16}\left(\frac{R_i}{R_j}\right)^4+\cdots\right].$">|; 

$key = q/t>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img175.png"
 ALT="$t&gt;0$">|; 

$key = q/displaystyle-U_{xx},deltax-U_{xy},deltay,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2646.png"
 ALT="$\displaystyle - U_{xx} \delta x -U_{xy} \delta y,$">|; 

$key = q/{displaymath}a_n=frac{omega_0^{,2}}{omega_0^{,2}-n^2,omega^2-{{rm{i},2,nu,n,omega}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img418.png"
 ALT="\begin{displaymath}
a_n = \frac{\omega_0^{ 2}}{\omega_0^{ 2} - n^2 \omega^2 -
{\rm i} 2 \nu n \omega}.
\end{displaymath}">|; 

$key = q/{displaymath}zeta=frac{pi}{2}-frac{theta}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1056.png"
 ALT="\begin{displaymath}
\zeta = \frac{\pi}{2}- \frac{\theta}{2}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{(x-x_c)^2}{a^2}-frac{y^2}{b^2}=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img728.png"
 ALT="\begin{displaymath}
\frac{(x-x_c)^2}{a^2} - \frac{y^2}{b^2} = 1,
\end{displaymath}">|; 

$key = q/u_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1512.png"
 ALT="$u_2$">|; 

$key = q/deltadot{{bf{q}^{(0)}=deltadot{{bf{q}(t=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1963.png"
 ALT="$\delta \dot{\bf q}^{(0)} = \delta \dot{\bf q}(t=0)$">|; 

$key = q/{{bf{a}(t+deltat);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3337.png"
 ALT="${\bf a}(t+\delta t)$">|; 

$key = q/F=sqrt{1+y'^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1745.png"
 ALT="$F = \sqrt{1+y'^{ 2}}$">|; 

$key = q/{displaymath}nabla_{x'}=costheta,nabla_x+sintheta,nabla_y.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img3484.png"
 ALT="\begin{displaymath}
\nabla_{x'} = \cos\theta  \nabla_x + \sin\theta  \nabla_y.
\end{displaymath}">|; 

$key = q/Omega_1=1+g,m^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2953.png"
 ALT="$\Omega_1=1+g m^2$">|; 

$key = q/{displaymath}Ksimeqfrac{1}{2}sum_{i,j=1,{calF}}M_{ij},deltadot{q}_i,deltadot{q}_j,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1898.png"
 ALT="\begin{displaymath}
K \simeq \frac{1}{2}\sum_{i,j=1,{\cal F}} M_{ij} \delta\dot{q}_i \delta\dot{q}_j,
\end{displaymath}">|; 

$key = q/{displaymath}K=frac{1}{2},m,v^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} m v^2.
\end{displaymath}">|; 

$key = q/displaystyle-left.intr^2,gamma,P_2(costheta),d^{3}{{bf{r}rightslashI_0=-left.fra^2,gamma,left(3,cos^2theta-1right),d^{3}{{bf{r}rightslashI_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="520" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2192.png"
 ALT="$\displaystyle - \left.\int r^2 \gamma P_2(\cos\theta) d^{3}{\bf r}\right/ I_...
...{2}\int r^2 \gamma \left(3 \cos^2\theta - 1\right) d^{3}{\bf r}\right/ I_0,$">|; 

$key = q/displaystyleomega_0,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3030.png"
 ALT="$\displaystyle \omega_0 t,$">|; 

$key = q/m,k,(v^3+a^2,v);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img497.png"
 ALT="$m k (v^3+a^2 v)$">|; 

$key = q/{displaymath}{mitOmega}=frac{m,g,l}{I_parallel,dot{phi}}+frac{I_perp}{I_parallel},costheta,dot{phi}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1521.png"
 ALT="\begin{displaymath}
{\mit\Omega}= \frac{m g l}{I_\parallel \dot{\phi}} + \frac{I_\perp}{I_\parallel} \cos\theta \dot{\phi}.
\end{displaymath}">|; 

$key = q/rho_1=1+rho_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2582.png"
 ALT="$\rho_1=1+\rho_2$">|; 

$key = q/{displaymath}frac{dot{p}_r}{m}=ddot{r}=frac{h^2}{r^3}-frac{partialV}{partialr},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1883.png"
 ALT="\begin{displaymath}
\frac{\dot{p}_r}{m} = \ddot{r} = \frac{h^2}{r^3} - \frac{\partial V}{\partial r},
\end{displaymath}">|; 

$key = q/{displaymath}d_{east}=frac{{mitOmega}}{3},coslambda,left(frac{8,h^3}{g}right)^{1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1162.png"
 ALT="\begin{displaymath}
d_{east} = \frac{{\mit\Omega}}{3} \cos\lambda \left(\frac{8 h^3}{g}\right)^{1/2}.
\end{displaymath}">|; 

$key = q/{displaymath}J_n=-frac{2pi,G,gamma}{(3+n)}int_0^piP_n(costheta),a_theta^{3+n}(theta),sintheta,dtheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="362" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2154.png"
 ALT="\begin{displaymath}
J_n = -\frac{2\pi G \gamma}{(3+n)}\int_0^\pi P_n(\cos\theta) a_\theta^{3+n}(\theta) \sin\theta d\theta,
\end{displaymath}">|; 

$key = q/{displaymath}lnleft(frac{v_x}{v_{x,0}}right)=-frac{g}{v_t},t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img584.png"
 ALT="\begin{displaymath}
\ln\left(\frac{v_x}{v_{x 0}}\right) = - \frac{g}{v_t} t,
\end{displaymath}">|; 

$key = q/U_0';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2325.png"
 ALT="$U_0'$">|; 

$key = q/1.52;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2431.png"
 ALT="$1.52$">|; 

$key = q/{{bf{B};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img199.png"
 ALT="${\bf B}$">|; 

$key = q/displaystyle(n+m),{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3210.png"
 ALT="$\displaystyle (n+m) {\bf a}$">|; 

$key = q/dl=sqrt{dx^2+dy^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3374.png"
 ALT="$dl=\sqrt{dx^2+dy^2}$">|; 

$key = q/frac{3}{4},m^2,x_5-frac{3}{4},m^2,y_5-3,x_4,x_5+frac{3}{2},y_4,y_5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2889.png"
 ALT="$\frac{3}{4} m^2 x_5 - \frac{3}{4} m^2 y_5-3 x_4 x_5+\frac{3}{2} y_4 y_5$">|; 

$key = q/{displaymath}frac{8}{9}leqAleq1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2677.png"
 ALT="\begin{displaymath}
\frac{8}{9}\leq A \leq 1.
\end{displaymath}">|; 

$key = q/pislash2<omega,t<3pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img521.png"
 ALT="$\pi/2&lt;\omega t&lt;3\pi/2$">|; 

$key = q/'equivdslashdr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2404.png"
 ALT="$'\equiv d/dr$">|; 

$key = q/a=b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1351.png"
 ALT="$a=b$">|; 

$key = q/d{{bf{r}equiv(dx,,dy,,dz);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3407.png"
 ALT="$d{\bf r} \equiv (dx, dy, dz)$">|; 

$key = q/Omega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2869.png"
 ALT="$\Omega_1$">|; 

$key = q/zeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1022.png"
 ALT="$\zeta$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter15slashfig15.38.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="345" BORDER="0"
 SRC="|."$dir".q|img3181.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter15/fig15.38.eps}}
\end{figure}">|; 

$key = q/{{bf{x}_i^{,ast}cdot{{bf{x}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1298.png"
 ALT="${\bf x}_i^{ \ast}\cdot{\bf x}_i$">|; 

$key = q/q_3=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2045.png"
 ALT="$q_3=0$">|; 

$key = q/{displaymath}{mitDelta}K=int_{t_1}^{t_2}{{bf{f}cdot{{bf{v},dt=int_P^Q{{bf{f}cdotd{{bf{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="\begin{displaymath}
{\mit\Delta}K = \int_{t_1}^{t_2} {\bf f}\cdot{\bf v} dt
=\int_P^Q {\bf f}\cdot d{\bf r},
\end{displaymath}">|; 

$key = q/displaystylex_1,sin(n,t)+y_1,cos(n,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2727.png"
 ALT="$\displaystyle x_1 \sin(n t)+y_1 \cos(n t),$">|; 

$key = q/displaystyle(dot{phi})_{{rm{slow};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1526.png"
 ALT="$\displaystyle (\dot{\phi})_{\rm slow}$">|; 

$key = q/{displaymath}{{bf{r}_{cm}=frac{m_1,{{bf{r}_1+m_2,{{bf{r}_2}{m_1+m_2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img920.png"
 ALT="\begin{displaymath}
{\bf r}_{cm} = \frac{m_1 {\bf r}_1+ m_2 {\bf r}_2}{m_1 + m_2}.
\end{displaymath}">|; 

$key = q/dot{r}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img777.png"
 ALT="$\dot{r}=0$">|; 

$key = q/tsimeqTslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1216.png"
 ALT="$t\simeq T/2$">|; 

$key = q/{displaymath}int_P^Qleft(f,dx+g,dyright)=F(Q)-F(P){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img3394.png"
 ALT="\begin{displaymath}
\int_P^Q \left(f dx + g dy\right) = F(Q) - F(P)
\end{displaymath}">|; 

$key = q/iota_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2310.png"
 ALT="$\iota_m$">|; 

$key = q/{figure}{epsfysize{=1.25in{centerline{epsffile{AppendixAslashfigA.06.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="317" HEIGHT="133" BORDER="0"
 SRC="|."$dir".q|img3319.png"
 ALT="\begin{figure}
\epsfysize =1.25in
\centerline{\epsffile{AppendixA/figA.06.eps}}
\end{figure}">|; 

$key = q/{displaymath}{{bf{v}'={{bf{v}-{{bf{u}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="\begin{displaymath}
{\bf v}' = {\bf v} - {\bf u}.
\end{displaymath}">|; 

$key = q/displaystyleI_perp,omega,sinalpha,sin({mitOmega},t'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1401.png"
 ALT="$\displaystyle I_\perp \omega \sin\alpha \sin({\mit\Omega} t'),$">|; 

$key = q/{displaymath}delta{{bf{q}_k(t)=left(alpha_k,{{rm{e}^{+gamma_k,t}+beta_k,{{rm{e}^{-gamma_k,t}right){{bf{x}_k,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img1958.png"
 ALT="\begin{displaymath}
\delta {\bf q}_k(t) = \left(\alpha_k  {\rm e}^{+\gamma_k t}+\beta_k {\rm e}^{-\gamma_k t}\right)
{\bf x}_k,
\end{displaymath}">|; 

$key = q/{displaymath}omega_0=sqrt{frac{g}{l}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3028.png"
 ALT="\begin{displaymath}
\omega_0 = \sqrt{\frac{g}{l}}.
\end{displaymath}">|; 

$key = q/h^2slash(R-h);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img488.png"
 ALT="$h^2/(R-h)$">|; 

$key = q/displaystyleI_perp,frac{domega_{x'}}{dt'}+(I_{parallel}-I_{perp}),omega_{z'},omega_{y'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1378.png"
 ALT="$\displaystyle I_\perp \frac{d\omega_{x'}}{dt'} + (I_{\parallel}-I_{\perp}) \omega_{z'} \omega_{y'}$">|; 

$key = q/{displaymath}Phi=-frac{G,m'}{|{{bf{r}'-{{bf{r}|}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2060.png"
 ALT="\begin{displaymath}
\Phi = - \frac{G m'}{\vert{\bf r}'-{\bf r}\vert}
\end{displaymath}">|; 

$key = q/L_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$L_1$">|; 

$key = q/{displaymath}alpha,{{bf{a}+beta,{{bf{b}+gamma,{{bf{c}={{bf{0},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img3531.png"
 ALT="\begin{displaymath}
\alpha {\bf a} + \beta {\bf b} + \gamma {\bf c} = {\bf0},
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{r}_itimes{{bf{f}_{ij}+{{bf{r}_jtimes{{bf{f}_{ji}={{bf{0},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img161.png"
 ALT="\begin{displaymath}
{\bf r}_i\times {\bf f}_{ij}+{\bf r}_j\times {\bf f}_{ji} = {\bf0},
\end{displaymath}">|; 

$key = q/E=K+U;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1496.png"
 ALT="$E=K+U$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter04slashfig4.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="322" BORDER="0"
 SRC="|."$dir".q|img610.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter04/fig4.02.eps}}
\end{figure}">|; 

$key = q/s_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1205.png"
 ALT="$s_+$">|; 

$key = q/{calO}(m^2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2987.png"
 ALT="${\cal O}(m^2)$">|; 

$key = q/{displaymath}int_P^QdT=int_P^Q{{bf{grad},Tcdotd{{bf{r}=T(Q)-T(P).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img3463.png"
 ALT="\begin{displaymath}
\int_P^Q dT = \int_P^Q {\bf grad} T\cdot d{\bf r} = T(Q)-T(P).
\end{displaymath}">|; 

$key = q/-1lequleq+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1513.png"
 ALT="$-1\leq u\leq +1$">|; 

$key = q/alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img473.png"
 ALT="$\alpha$">|; 

$key = q/displaystyle-(I_{y'y'}-I_{z'z'}),omega_{y'},omega_{z'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1370.png"
 ALT="$\displaystyle - (I_{y'y'}-I_{z'z'}) \omega_{y'} \omega_{z'},$">|; 

$key = q/r_0<a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img905.png"
 ALT="$r_0&lt;a$">|; 

$key = q/{displaymath}frac{partialLslashpartialq_1-(dslashdt),(partialLslashpartialdot{q}slashpartialdot{q}_2)}{partialfslashpartialq_2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="472" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1804.png"
 ALT="\begin{displaymath}
\frac{\partial L/\partial q_1-(d/dt) (\partial L/\partial \...
...t) (\partial L/\partial \dot{q}_2)}{\partial f/\partial q_2}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{r,f'}{f}=-2left(1+frac{3,h^2}{c^2,r^2}+cdotsright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2466.png"
 ALT="\begin{displaymath}
\frac{r f'}{f} = - 2\left(1+ \frac{3 h^2}{c^2 r^2}+ \cdots\right).
\end{displaymath}">|; 

$key = q/3cdot2^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3165.png"
 ALT="$3\cdot 2^n$">|; 

$key = q/dot{x}slasha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1623.png"
 ALT="$\dot{x}/a$">|; 

$key = q/4,g,l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1709.png"
 ALT="$4 g l$">|; 

$key = q/{displaymath}left|array{{cc}-k-k'-lambda,m&k[0.5ex]k&-k-k'-lambda,marray{right|=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1997.png"
 ALT="\begin{displaymath}
\left\vert\begin{array}{cc}
-k-k'-\lambda m&amp; k\ [0.5ex]
k&amp;-k-k'-\lambda m
\end{array}\right\vert = 0,
\end{displaymath}">|; 

$key = q/displaystylesum_{i=1,N}(x_i^{,2}+z_i^{,2}),m_i=int(x^2+z^2),dm,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1246.png"
 ALT="$\displaystyle \sum_{i=1,N}(x_i^{ 2}+z_i^{ 2})  m_i= \int(x^2+ z^2) dm,$">|; 

$key = q/displaystyleddot{y}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1150.png"
 ALT="$\displaystyle \ddot{y}'$">|; 

$key = q/displaystyle-Omega_phi,t-deltaphi,sin(Omega_n,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2352.png"
 ALT="$\displaystyle - \Omega_\phi t - \delta\phi \sin(\Omega_n t),$">|; 

$key = q/U_0=U(q_{1,0},q_{2,0},cdots,q_{{calF},0});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1903.png"
 ALT="$U_0=U(q_{1 0},q_{2 0},\cdots, q_{{\cal F} 0})$">|; 

$key = q/U=E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img241.png"
 ALT="$U=E$">|; 

$key = q/displaystylefrac{(x'-x)}{[(x'-x)^2+(y'-y)^2+(z'-z)^2]^{,3slash2}}equivfrac{parti!left(frac{1}{[(x'-x)^2+(y'-y)^2+(z'-z)^2]^{,1slash2}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="684" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2055.png"
 ALT="$\displaystyle \frac{(x'-x)}{[(x'-x)^2+(y'-y)^2+(z'-z)^2]^{ 3/2}}\equiv
\frac{...
...tial}{\partial x}\!\left(\frac{1}{[(x'-x)^2+(y'-y)^2+(z'-z)^2]^{ 1/2}}\right).$">|; 

$key = q/{displaymath}x_0=-frac{1}{2},e^2-frac{1}{4},iota^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2903.png"
 ALT="\begin{displaymath}
x_0 = -\frac{1}{2} e^2 -\frac{1}{4} \iota^{ 2}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.19.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="274" BORDER="0"
 SRC="|."$dir".q|img3090.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.19.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter12slashfig12.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="254" BORDER="0"
 SRC="|."$dir".q|img2150.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter12/fig12.01.eps}}
\end{figure}">|; 

$key = q/{displaymath}deltapsisimeqfrac{6pi,G,M_0}{c^2,r}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2468.png"
 ALT="\begin{displaymath}
\delta\psi \simeq \frac{6\pi G M_0}{c^2  r}
\end{displaymath}">|; 

$key = q/displaystylefrac{3}{2},m^2,iota.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2931.png"
 ALT="$\displaystyle \frac{3}{2} m^2 \iota.$">|; 

$key = q/{displaymath}theta(t+n,tau)=theta(t)+2pi,l{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3177.png"
 ALT="\begin{displaymath}
\theta(t+n \tau) = \theta(t) + 2\pi l
\end{displaymath}">|; 

$key = q/{displaymath}epsilon=-frac{15}{4},frac{m'}{m}left(frac{a}{R}right)^3{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2236.png"
 ALT="\begin{displaymath}
\epsilon = -\frac{15}{4} \frac{m'}{m}\left(\frac{a}{R}\right)^3
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2u}{dtheta^2}+u=-frac{1}{h^2}frac{dV}{du},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img839.png"
 ALT="\begin{displaymath}
\frac{d^2 u}{d\theta^2} + u = - \frac{1}{h^2}\frac{dV}{du},
\end{displaymath}">|; 

$key = q/{{bf{a}cdot{{bf{a}times{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3304.png"
 ALT="${\bf a}\cdot {\bf a}\times{\bf b}$">|; 

$key = q/x_0=x(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img325.png"
 ALT="$x_0 = x(0)$">|; 

$key = q/displaystylefrac{1}{2},(3,x^2-1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2110.png"
 ALT="$\displaystyle \frac{1}{2} (3 x^2-1),$">|; 

$key = q/{displaymath}frac{v_0,v_t}{sqrt{v_0^{,2}+v_t^{,2}}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img496.png"
 ALT="\begin{displaymath}
\frac{v_0 v_t}{\sqrt{v_0^{ 2} + v_t^{ 2}}},
\end{displaymath}">|; 

$key = q/{displaymath}I_{xy},(1-tan^2alpha)=(I_{xx}-I_{yy}),tanalpha.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1340.png"
 ALT="\begin{displaymath}
I_{xy}  (1-\tan^2\alpha) = (I_{xx}-I_{yy}) \tan\alpha.
\end{displaymath}">|; 

$key = q/theta<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3073.png"
 ALT="$\theta&lt;0$">|; 

$key = q/{displaymath}a=sqrt{a_x^{,2}+a_y^{,2}+a_z^{,2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="37" BORDER="0"
 SRC="|."$dir".q|img3229.png"
 ALT="\begin{displaymath}
a = \sqrt{a_x^{ 2} + a_y^{ 2} + a_z^{ 2}}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="267" BORDER="0"
 SRC="|."$dir".q|img3063.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.03.eps}}
\end{figure}">|; 

$key = q/beta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img958.png"
 ALT="$\beta$">|; 

$key = q/{displaymath}gsimeqfrac{3}{4}+frac{9}{32},m+{calO}(m^2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2955.png"
 ALT="\begin{displaymath}
g \simeq \frac{3}{4} + \frac{9}{32} m + {\cal O}(m^2).
\end{displaymath}">|; 

$key = q/dT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3462.png"
 ALT="$dT$">|; 

$key = q/tanhxsimeq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img266.png"
 ALT="$\tanh x\simeq 1$">|; 

$key = q/displaystyle-frac{3}{2},n'^{,2},sin[2,(n-n'),t],y_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2745.png"
 ALT="$\displaystyle -\frac{3}{2} n'^{ 2} 
\sin[2 (n-n') t] y_1,$">|; 

$key = q/alpha=1.09;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2478.png"
 ALT="$\alpha=1.09$">|; 

$key = q/{mitOmega}gg{mitOmega}_{{rm{crit};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1525.png"
 ALT="${\mit\Omega}\gg {\mit\Omega}_{\rm crit}$">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{Chapter15slashfig15.23.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="517" HEIGHT="219" BORDER="0"
 SRC="|."$dir".q|img3118.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{Chapter15/fig15.23.eps}}
\end{figure}">|; 

$key = q/m_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="$m_i$">|; 

$key = q/0.65;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2461.png"
 ALT="$0.65$">|; 

$key = q/int_P^Q{{bf{f}cdotd{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="$\int_P^Q {\bf f}\cdot d{\bf r}$">|; 

$key = q/m,g,l,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img455.png"
 ALT="$m g l 
\sin\theta$">|; 

$key = q/displaystyle{{bf{v}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1023.png"
 ALT="$\displaystyle {\bf v}_1$">|; 

$key = q/egeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img712.png"
 ALT="$e\geq 0$">|; 

$key = q/l';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1631.png"
 ALT="$l'$">|; 

$key = q/{displaymath}I_parallel=int(x^2+y^2),gamma,d^3{{bf{r}=intr^2,gamma,(1-cos^2theta),d^3{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2194.png"
 ALT="\begin{displaymath}
I_\parallel = \int (x^2+y^2) \gamma d^3{\bf r} = \int r^2 \gamma (1-\cos^2\theta) d^3{\bf r}.
\end{displaymath}">|; 

$key = q/displaystylefrac{r_c}{sqrt{e^2-1}}=sqrt{e^2-1},a,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img731.png"
 ALT="$\displaystyle \frac{r_c}{\sqrt{e^2-1}}=\sqrt{e^2-1} a,$">|; 

$key = q/30.07;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2443.png"
 ALT="$30.07$">|; 

$key = q/{displaymath}omegasimeqomega_0,sqrt{1-(1slash8),vartheta_0^{,2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img479.png"
 ALT="\begin{displaymath}
\omega \simeq \omega_0  \sqrt{1-(1/8) \vartheta_0^{ 2}},
\end{displaymath}">|; 

$key = q/9.2^{''};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2375.png"
 ALT="$9.2^{''}$">|; 

$key = q/displaystyle-2,{mitOmega},sinlambda,dot{x}'+2,{mitOmega},coslambda,dot{z}',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1151.png"
 ALT="$\displaystyle -2 {\mit\Omega} \sin\lambda \dot{x}'+2 {\mit\Omega} \cos\lambda 
\dot{z}',$">|; 

$key = q/Q=1.375;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$Q=1.375$">|; 

$key = q/displaystylefrac{11}{8},m^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2925.png"
 ALT="$\displaystyle \frac{11}{8} m^2.$">|; 

$key = q/{{bf{f};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="${\bf f}$">|; 

$key = q/displaystyle{{bf{r}_{cm}+frac{m_1}{m_1+m_2},{{bf{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img924.png"
 ALT="$\displaystyle {\bf r}_{cm} + \frac{m_1}{m_1+m_2} {\bf r},$">|; 

$key = q/displaystyle3,deltaX,deltaZ.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2794.png"
 ALT="$\displaystyle 3 \delta X \delta Z.$">|; 

$key = q/rslashr';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2118.png"
 ALT="$r/r'$">|; 

$key = q/{displaymath}r=a_theta(theta)=aleft[1-frac{2}{3},epsilon,P_2(costheta)right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2145.png"
 ALT="\begin{displaymath}
r = a_\theta(\theta) = a\left[1-\frac{2}{3} \epsilon P_2(\cos\theta)\right],
\end{displaymath}">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}=mbox{{boldmath{{omega{}_phi+mbox{{boldmath{{omega{}_theta+mbox{{boldmath{{omega{}_psi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1446.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$} = \mbox{\boldmath$\omega$}_\phi
+\mbox{\boldmath$\omega$}_\theta+\mbox{\boldmath$\omega$}_\psi,
\end{displaymath}">|; 

$key = q/{displaymath}T=frac{2pi}{{mitOmega},sinlambda}=frac{24}{sinlambda},,{{rm{hrs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1213.png"
 ALT="\begin{displaymath}
T = \frac{2\pi}{{\mit\Omega} \sin\lambda} = \frac{24}{\sin\lambda}  {\rm hrs}.
\end{displaymath}">|; 

$key = q/360,alpha_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2982.png"
 ALT="$360 \alpha_1$">|; 

$key = q/deltaX^2=deltaY^2=deltaZ^2=deltaX,deltaY=deltaX,deltaZ=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2804.png"
 ALT="$\delta X^2=\delta Y^2=\delta Z^2=\delta X \delta Y=\delta X \delta Z=0$">|; 

$key = q/{displaymath}theta(t)=vartheta_0,cos(omega,t)+alpha,vartheta_0^{,3},cos(3,omega,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img472.png"
 ALT="\begin{displaymath}
\theta(t) = \vartheta_0 \cos(\omega t) + \alpha \vartheta_0^{ 3} \cos(3 \omega t),
\end{displaymath}">|; 

$key = q/{mitDelta}=3pislash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img573.png"
 ALT="${\mit\Delta}= 3\pi/4$">|; 

$key = q/theta=90^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2379.png"
 ALT="$\theta = 90^\circ$">|; 

$key = q/lambda';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1773.png"
 ALT="$\lambda'$">|; 

$key = q/displaystyledot{lambda};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1538.png"
 ALT="$\displaystyle \dot{\lambda}$">|; 

$key = q/{{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1919.png"
 ALT="${\bf0}$">|; 

$key = q/rslashR;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2224.png"
 ALT="$r/R$">|; 

$key = q/{{bf{r}=(xi,,eta,,zeta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2502.png"
 ALT="${\bf r} = (\xi, \eta, \zeta)$">|; 

$key = q/A=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2681.png"
 ALT="$A=1$">|; 

$key = q/C_4<C<C_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$C_4&lt; C &lt; C_3$">|; 

$key = q/11.87;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2451.png"
 ALT="$11.87$">|; 

$key = q/phi=phi_0={{rm{const};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1684.png"
 ALT="$\phi=\phi_0 = {\rm const}$">|; 

$key = q/{displaymath}{calE}equivfrac{1}{2},(dot{xi}^2+dot{eta}^2+dot{zeta}^2)-left(frac{)=mbox{{boldmath{{omega{}cdot{{bf{h}-frac{C}{2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2520.png"
 ALT="\begin{displaymath}
{\cal E} \equiv \frac{1}{2} (\dot{\xi}^2+\dot{\eta}^2+\dot{...
...}\right)
= \mbox{\boldmath$\omega$}\cdot{\bf h} - \frac{C}{2},
\end{displaymath}">|; 

$key = q/{displaymath}left|{{bf{G}-lambda,{{bf{M}right|=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1996.png"
 ALT="\begin{displaymath}
\left\vert{\bf G} - \lambda {\bf M}\right\vert = 0,
\end{displaymath}">|; 

$key = q/{displaymath}omega=omega_0left[1-frac{theta_0^{,2}}{16}+{calO}(theta_0^{,4})right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img482.png"
 ALT="\begin{displaymath}
\omega = \omega_0\left[1-\frac{\theta_0^{ 2}}{16} + {\cal O}(\theta_0^{ 4})\right].
\end{displaymath}">|; 

$key = q/rho({{bf{r}');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2066.png"
 ALT="$\rho({\bf r}')$">|; 

$key = q/{displaymath}frac{d}{dt}left[frac{1}{2}left(dot{x}^2+dot{y}^2+dot{z}^2right)+Uright]=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2555.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\left[\frac{1}{2}\left(\dot{x}^2+\dot{y}^2+\dot{z}^2\right) + U\right] = 0.
\end{displaymath}">|; 

$key = q/{displaymath}{mitOmega}_{z'}=left[frac{(I_{z'z'}-I_{x'x'}),(I_{z'z'}-I_{y'y'})}{I_{x'x'},I_{y'y'}}right]^{1slash2}!omega_{z'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1548.png"
 ALT="\begin{displaymath}
{\mit\Omega}_{z'} = \left[\frac{(I_{z'z'}-I_{x'x'}) (I_{z'z'}-I_{y'y'})}{I_{x'x'} I_{y'y'}}\right]^{1/2}\! \omega_{z'}.
\end{displaymath}">|; 

$key = q/{displaymath}U=frac{k}{2},(q_3-q_1)^2+frac{k}{2},(q_2-q_3)^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2030.png"
 ALT="\begin{displaymath}
U = \frac{k}{2} (q_3-q_1)^2 + \frac{k}{2} (q_2-q_3)^2.
\end{displaymath}">|; 

$key = q/{displaymath}tilde{Q}_r=lambda,frac{partialf}{partialr}=-mleft(3,g,costheta-2,g+frac{v_0^{,2}}{a}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1840.png"
 ALT="\begin{displaymath}
\tilde{Q}_r = \lambda \frac{\partial f}{\partial r} = -m \left(3 g \cos\theta - 2 g + \frac{v_0^{ 2}}{a}\right).
\end{displaymath}">|; 

$key = q/dot{~}=dslashdt';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1368.png"
 ALT="$\dot{&nbsp;}=d/dt'$">|; 

$key = q/T(x,,y,,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3452.png"
 ALT="$T(x, y, z)$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.06.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="274" BORDER="0"
 SRC="|."$dir".q|img3066.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.06.eps}}
\end{figure}">|; 

$key = q/=41^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1218.png"
 ALT="$=41^\circ$">|; 

$key = q/U(q_1,q_2,cdots,q_{calF},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1789.png"
 ALT="$U(q_1,q_2,\cdots,q_{\cal F},t)$">|; 

$key = q/displaystyledot{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1874.png"
 ALT="$\displaystyle \dot{r}$">|; 

$key = q/{displaymath}|{{bf{r}'-{{bf{r}|^{-1}=frac{1}{r}left[1+left(frac{r'}{r}right)F+fr2(3,F^2-1)+{calO}left(frac{r'}{r}right)^3right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="492" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img2090.png"
 ALT="\begin{displaymath}
\vert{\bf r}'-{\bf r}\vert^{-1}= \frac{1}{r}\left[
1 + \left...
...ght)^2(3 F^2-1)
+ {\cal O}\left(\frac{r'}{r}\right)^3\right].
\end{displaymath}">|; 

$key = q/{{bf{v}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1010.png"
 ALT="${\bf v}_2$">|; 

$key = q/{displaymath}y'^{,2}=left(lambda'+frac{y}{h}right)^2-1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1766.png"
 ALT="\begin{displaymath}
y'^{ 2} = \left(\lambda' + \frac{y}{h}\right)^2 - 1,
\end{displaymath}">|; 

$key = q/{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="${\bf r}$">|; 

$key = q/dm=rho,dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1262.png"
 ALT="$dm = \rho dV$">|; 

$key = q/theta_0(t_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3130.png"
 ALT="$\theta_0(t_0)$">|; 

$key = q/displaystyleddot{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2712.png"
 ALT="$\displaystyle \ddot{\bf r}$">|; 

$key = q/displaystyle-frac{partialH}{partialx}=-k,x.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1865.png"
 ALT="$\displaystyle - \frac{\partial H}{\partial x} = - k x.$">|; 

$key = q/displaystylefrac{G,M}{R^3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2488.png"
 ALT="$\displaystyle \frac{G M}{R^3},$">|; 

$key = q/|{{bf{a}|=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3272.png"
 ALT="$\vert{\bf a}\vert=0$">|; 

$key = q/displaystyle-frac{1}{4},m,a,b.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1349.png"
 ALT="$\displaystyle - \frac{1}{4} m a b.$">|; 

$key = q/omega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2014.png"
 ALT="$\omega_1$">|; 

$key = q/{displaymath}L=frac{1}{2},m,(dot{r}^{,2}+r^2,dot{theta}^{,2})+m,g,r,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1834.png"
 ALT="\begin{displaymath}
L = \frac{1}{2} m (\dot{r}^{ 2} + r^2 \dot{\theta}^{ 2})
+ m g r \cos\theta.
\end{displaymath}">|; 

$key = q/{{bf{p}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img947.png"
 ALT="${\bf p}'$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.17.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="287" BORDER="0"
 SRC="|."$dir".q|img3084.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.17.eps}}
\end{figure}">|; 

$key = q/{displaymath}tanpsi_{max}=frac{m_2slashm_1}{sqrt{1-(m_2slashm_1)^2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="61" BORDER="0"
 SRC="|."$dir".q|img1061.png"
 ALT="\begin{displaymath}
\tan\psi_{max} = \frac{m_2/m_1}{\sqrt{1-(m_2/m_1)^2}},
\end{displaymath}">|; 

$key = q/displaystyle-frac{m_2}{m_1+m_2},{{bf{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img939.png"
 ALT="$\displaystyle -\frac{m_2}{m_1+m_2} {\bf r},$">|; 

$key = q/rho_1=1-rho_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2574.png"
 ALT="$\rho_1=1-\rho_2$">|; 

$key = q/({{bf{a}cdot{{bf{b})cdot{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3263.png"
 ALT="$({\bf a}\cdot{\bf b}) \cdot{\bf c}$">|; 

$key = q/+z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2231.png"
 ALT="$+z$">|; 

$key = q/displaystylefrac{dv_z'}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img632.png"
 ALT="$\displaystyle \frac{d v_z'}{dt}$">|; 

$key = q/{displaymath}mu_2<frac{1}{2}left(1-sqrt{frac{23}{27}}right)=0.0385.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img2690.png"
 ALT="\begin{displaymath}
\mu_2 &lt; \frac{1}{2}\left(1-\sqrt{\frac{23}{27}}\right) = 0.0385.
\end{displaymath}">|; 

$key = q/displaystyledot{x}_j,frac{partial}{partialq_i}!left(sum_{k=1,{calF}}frac{partialx_j}{partialq_k},dot{q}_k+frac{partialx_j}{partialt}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1595.png"
 ALT="$\displaystyle \dot{x}_j \frac{\partial}{\partial q_i}\!\left(\sum_{k=1,{\cal F...
...partial x_j}{\partial q_k} \dot{q}_k +
\frac{\partial x_j}{\partial t}\right)$">|; 

$key = q/displaystylelambda_i,{{bf{x}_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1300.png"
 ALT="$\displaystyle \lambda_i {\bf x}_i,$">|; 

$key = q/{displaymath}y',frac{partialF}{partialy'}-F={{rm{const}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1744.png"
 ALT="\begin{displaymath}
y' \frac{\partial F}{\partial y'} - F = {\rm const}.
\end{displaymath}">|; 

$key = q/{{bf{a}-3,{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3505.png"
 ALT="${\bf a}-3 {\bf b}$">|; 

$key = q/l,{{rm{e}^{-u};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1707.png"
 ALT="$l {\rm e}^{-u}$">|; 

$key = q/u_1=u_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1518.png"
 ALT="$u_1 = u_2$">|; 

$key = q/t_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$t_2$">|; 

$key = q/{{bf{e}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1322.png"
 ALT="${\bf e}_z$">|; 

$key = q/{displaymath}K=frac{1}{2},m_1,dot{x}^{,2}+frac{1}{2},m_2,dot{x}^{,2}+frac{1}{2},I,frac{dot{x}^{,2}}{a^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1624.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} m_1 \dot{x}^{ 2} + \frac{1}{2} m_2 \dot{x}^{ 2}
+ \frac{1}{2} I  \frac {\dot{x}^{ 2}}{a^2}.
\end{displaymath}">|; 

$key = q/b=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img953.png"
 ALT="$b=0$">|; 

$key = q/{displaymath}-frac{G,M}{a}left[1+frac{4}{15},epsilon,P_2(costheta)right]-frac{miOmega^2,a^2}{3}left[1-P_2(costheta)right]simeqc,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="455" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2173.png"
 ALT="\begin{displaymath}
- \frac{G M}{a} \left[1+ \frac{4}{15} \epsilon P_2(\cos\t...
...c{\mit\Omega^2 a^2}{3}\left[1-P_2(\cos\theta)\right]\simeq c,
\end{displaymath}">|; 

$key = q/urightarrowpminfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1508.png"
 ALT="$u\rightarrow\pm\infty$">|; 

$key = q/{{bf{r}_itimes{{bf{f}_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.png"
 ALT="${\bf r}_i\times {\bf f}_{ij}$">|; 

$key = q/{displaymath}{{bf{a}cdot{{bf{b}times{{bf{c}=-{{bf{b}cdot{{bf{a}times{{bf{c}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3344.png"
 ALT="\begin{displaymath}
{\bf a} \cdot {\bf b} \times{\bf c} = - {\bf b} \cdot {\bf a} \times{\bf c}.
\end{displaymath}">|; 

$key = q/deltaq_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1897.png"
 ALT="$\delta q_i$">|; 

$key = q/31.8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3014.png"
 ALT="$31.8$">|; 

$key = q/omega_2>omega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2018.png"
 ALT="$\omega_2&gt; \omega_1$">|; 

$key = q/displaystyledot{x}+dot{x}',costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1645.png"
 ALT="$\displaystyle \dot{x} + \dot{x}' \cos\theta,$">|; 

$key = q/displaystyle-frac{135}{64},m^3,e,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2941.png"
 ALT="$\displaystyle -\frac{135}{64} m^3 e,$">|; 

$key = q/ddot{r}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img847.png"
 ALT="$\ddot{r}=0$">|; 

$key = q/displaystyleU_{xy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2665.png"
 ALT="$\displaystyle U_{xy}$">|; 

$key = q/{displaymath}frac{d}{dt}!left(frac{partialL}{partialdot{x}}right)-frac{partialL}{partialx}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1627.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial L}{\partial\dot{x}}\right) - \frac{\partial L}{\partial x} = 0,
\end{displaymath}">|; 

$key = q/{displaymath}z_2simeqfrac{3}{2},e,iota.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2911.png"
 ALT="\begin{displaymath}
z_2\simeq \frac{3}{2} e \iota.
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{2},(omega_1^{,2},eta_1^{,2}+omega_2^{,2},eta_2^{,2}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2026.png"
 ALT="$\displaystyle \frac{1}{2} (\omega_1^{ 2} \eta_1^{ 2} + \omega_2^{ 2} \eta_2^{ 2}),$">|; 

$key = q/dot{~};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img685.png"
 ALT="$\dot{&nbsp;}$">|; 

$key = q/displaystyleI_perp,frac{domega_{y'}}{dt'}-(I_{parallel}-I_{perp}),omega_{z'},omega_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1379.png"
 ALT="$\displaystyle I_\perp \frac{d\omega_{y'}}{dt'} - (I_{\parallel}-I_{\perp}) 
\omega_{z'} \omega_{x'}$">|; 

$key = q/{displaymath}rho=frac{m'}{M},R,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2202.png"
 ALT="\begin{displaymath}
\rho = \frac{m'}{M} R,
\end{displaymath}">|; 

$key = q/{displaymath}U(x,y)=frac{1}{2},k,r^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img523.png"
 ALT="\begin{displaymath}
U(x,y) = \frac{1}{2} k r^2,
\end{displaymath}">|; 

$key = q/q_1(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1808.png"
 ALT="$q_1(t)$">|; 

$key = q/I_parallelsimeq(1slash3),M,R^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2481.png"
 ALT="$I_\parallel \simeq (1/3) M R^2$">|; 

$key = q/y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img543.png"
 ALT="$y$">|; 

$key = q/v(0)=-0.75;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$v(0)=-0.75$">|; 

$key = q/{displaymath}{mitOmega}>{mitOmega}_{{rm{crit}=frac{2,sqrt{m,g,l,I_perp,costheta}}{I_parallel}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1523.png"
 ALT="\begin{displaymath}
{\mit\Omega} &gt; {\mit\Omega}_{\rm crit} = \frac{2 \sqrt{m g l I_\perp \cos\theta}}{I_\parallel}
\end{displaymath}">|; 

$key = q/{displaymath}U=frac{1}{2}left[(k+k'),q_1^{,2}-2,k,q_1,q_2+(k+k'),q_2^{,2}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1990.png"
 ALT="\begin{displaymath}
U= \frac{1}{2}\left[(k+k') q_1^{ 2} -2 k q_1 q_2 + (k+k') q_2^{ 2}\right].
\end{displaymath}">|; 

$key = q/Q=1.372;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$Q=1.372$">|; 

$key = q/k_i,k_j,|{{bf{r}_i-{{bf{r}_j|^{-n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img179.png"
 ALT="$k_i k_j \vert{\bf r}_i-{\bf r}_j\vert^{-n}$">|; 

$key = q/omega=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$\omega =2$">|; 

$key = q/{displaymath}alpha=left(frac{mu_2}{3,mu_1}right)^{1slash3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img2580.png"
 ALT="\begin{displaymath}
\alpha = \left(\frac{\mu_2}{3 \mu_1}\right)^{1/3}.
\end{displaymath}">|; 

$key = q/{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1155.png"
 ALT="${\mit\Omega}$">|; 

$key = q/{displaymath}int_P^Qleft(f,dx+g,dyright)=int_P^QdF(x,y)=F(Q)-F(P).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="378" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img3399.png"
 ALT="\begin{displaymath}
\int_P^Q \left(f dx + g dy\right) = \int_P^Q dF(x,y) = F(Q) - F(P).
\end{displaymath}">|; 

$key = q/F_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3108.png"
 ALT="$F_n$">|; 

$key = q/1-2,m-c,m^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2865.png"
 ALT="$1-2 m-c m^2$">|; 

$key = q/{{bf{e}_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img677.png"
 ALT="${\bf e}_\theta$">|; 

$key = q/{displaymath}left(array{{c}x''y''z''array{right)=left(array{{ccc}cosphi&sinphi&000&0&1array{right)left(array{{c}xyzarray{right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="76" BORDER="0"
 SRC="|."$dir".q|img1423.png"
 ALT="\begin{displaymath}
\left(\begin{array}{c}x''\ y''\ z''\end{array}\right)=
\le...
...rray}\right)\left(\begin{array}{c}x\ y\ z\end{array}\right).
\end{displaymath}">|; 

$key = q/9.55times10^{-4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2432.png"
 ALT="$9.55\times 10^{-4}$">|; 

$key = q/F=4.669;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3125.png"
 ALT="$F=4.669$">|; 

$key = q/1slash8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img379.png"
 ALT="$1/8$">|; 

$key = q/{displaymath}x(t)=A,{{rm{e}^{-nu,t},cos(omega_r,t)+B,{{rm{e}^{-nu,t},sin(omega_r,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img438.png"
 ALT="\begin{displaymath}
x(t) = A {\rm e}^{-\nu t} \cos(\omega_r t) + B {\rm e}^{-\nu t} 
\sin(\omega_r t),
\end{displaymath}">|; 

$key = q/{displaymath}(lambda_i-lambda_j),{{bf{x}_i^T{{bf{x}_j=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1309.png"
 ALT="\begin{displaymath}
(\lambda_i-\lambda_j) {\bf x}_i^T{\bf x}_j = 0.
\end{displaymath}">|; 

$key = q/{displaymath}frac{dE}{dt}=-2,m,nuleft(frac{dx}{dt}right)^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img342.png"
 ALT="\begin{displaymath}
\frac{dE}{dt} = -2 m \nu\left(\frac{dx}{dt}\right)^2.
\end{displaymath}">|; 

$key = q/displaystyledot{theta}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2350.png"
 ALT="$\displaystyle \dot{\theta}_1$">|; 

$key = q/displaystyleI_{xy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1348.png"
 ALT="$\displaystyle I_{xy}$">|; 

$key = q/{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3188.png"
 ALT="${\bf b}$">|; 

$key = q/deltatheta_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3331.png"
 ALT="$\delta \theta_x$">|; 

$key = q/{displaymath}{{bf{r}=(r,costheta,,r,sintheta,,0),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img934.png"
 ALT="\begin{displaymath}
{\bf r} = (r \cos\theta, r \sin\theta,  0),
\end{displaymath}">|; 

$key = q/displaystylez';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$\displaystyle z'$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter07slashfig7.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="421" HEIGHT="287" BORDER="0"
 SRC="|."$dir".q|img1133.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter07/fig7.02.eps}}
\end{figure}">|; 

$key = q/{displaymath}frac{d^2x}{dt^2}+2,nu,frac{dx}{dt}+omega_0^{,2},x=F(t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img512.png"
 ALT="\begin{displaymath}
\frac{d^2 x}{dt^2} + 2 \nu \frac{dx}{dt} + \omega_0^{ 2}  x = F(t).
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{r}cdot{{bf{r}'=r,r',F,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img2084.png"
 ALT="\begin{displaymath}
{\bf r}\cdot{\bf r}' = r r' F,
\end{displaymath}">|; 

$key = q/displaystyleI_perp,omega,sinalpha,cos({mitOmega},t'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1399.png"
 ALT="$\displaystyle I_\perp \omega \sin\alpha \cos({\mit\Omega} t'),$">|; 

$key = q/{{bf{E}times{{bf{B};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img627.png"
 ALT="${\bf E}\times{\bf B}$">|; 

$key = q/displaystyleepsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2191.png"
 ALT="$\displaystyle \epsilon$">|; 

$key = q/{displaymath}{{bf{g}=-nablaPhi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2059.png"
 ALT="\begin{displaymath}
{\bf g} = -\nabla\Phi,
\end{displaymath}">|; 

$key = q/{displaymath}left(frac{theta}{tilde{A}}right)^2+left(frac{v}{omega,tilde{A}}right)^2=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img3057.png"
 ALT="\begin{displaymath}
\left(\frac{\theta}{\tilde{A}}\right)^2 + \left(\frac{v}{\omega \tilde{A}}\right)^2 =1,
\end{displaymath}">|; 

$key = q/{{bf{1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1286.png"
 ALT="${\bf 1}$">|; 

$key = q/{{bf{g}_g=-nablaPhi(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2165.png"
 ALT="${\bf g}_g=-\nabla\Phi(r,\theta)$">|; 

$key = q/displaystyleddot{xi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2503.png"
 ALT="$\displaystyle \ddot{\xi}$">|; 

$key = q/{displaymath}frac{2}{T}int_0^Tcos(n,omega,t),cos(n',omega,t),dt=delta_{n,n'},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img406.png"
 ALT="\begin{displaymath}
\frac{2}{T} \int_0^T \cos(n \omega t) \cos(n' \omega t)  dt= \delta_{n n'},
\end{displaymath}">|; 

$key = q/M=m_1+m_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img944.png"
 ALT="$M=m_1+m_2$">|; 

$key = q/AP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1699.png"
 ALT="$AP$">|; 

$key = q/{displaymath}frac{dsigma}{d{mitOmega}}=frac{q^4}{16,(4,pi,epsilon_0)^2,E^{,2}}frac{1}{sin^4(thetaslash2)}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1088.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d{\mit\Omega}} = \frac{q^4}{16 (4 \pi \epsilon_0)^2 E^{ 2}}\frac{1}{\sin^4(\theta/2)}
\end{displaymath}">|; 

$key = q/|{{bf{h}|^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3548.png"
 ALT="$\vert{\bf h}\vert^{-1}$">|; 

$key = q/{displaymath}tanh^{-1}left(frac{v}{v_t}right)=frac{g,t}{v_t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img258.png"
 ALT="\begin{displaymath}
\tanh^{-1}\left(\frac{v}{v_t}\right) = \frac{g t}{v_t},
\end{displaymath}">|; 

$key = q/{displaymath}x(t)=x_1,cos(omega,t-phi_1),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img372.png"
 ALT="\begin{displaymath}
x(t) = x_1 \cos(\omega t-\phi_1),
\end{displaymath}">|; 

$key = q/e=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img761.png"
 ALT="$e=0$">|; 

$key = q/mu_2=0.01;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$\mu _2=0.01$">|; 

$key = q/displaystyleint(y^2+z^2),gamma,d^3{{bf{r}=intr^2,gamma,(sin^2theta,sin^2,phi+cos^2theta),d^3{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="426" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2196.png"
 ALT="$\displaystyle \int (y^2+z^2) \gamma d^3{\bf r} = \int r^2 \gamma (\sin^2\theta \sin^2 \phi + \cos^2\theta) d^3{\bf r}$">|; 

$key = q/4.69pm0.04;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3115.png"
 ALT="$4.69\pm 0.04$">|; 

$key = q/{{bf{f}_1+{{bf{f}_2={{bf{f};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="${\bf f}_1+{\bf f}_2={\bf f}$">|; 

$key = q/displaystyle{calE}_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1052.png"
 ALT="$\displaystyle {\cal E}_2'$">|; 

$key = q/displaystylelambda_k,{{bf{x}_k^T,{{bf{M},{{bf{x}_l,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1942.png"
 ALT="$\displaystyle \lambda_k {\bf x}_k^T {\bf M} {\bf x}_l,$">|; 

$key = q/f(r)=-c,r^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img906.png"
 ALT="$f(r) = -c r^{-3}$">|; 

$key = q/V=Uslashm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1620.png"
 ALT="$V = U/m$">|; 

$key = q/v_c=sqrt{G,Mslashr_p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img780.png"
 ALT="$v_c=\sqrt{G M/r_p}$">|; 

$key = q/g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img252.png"
 ALT="$g$">|; 

$key = q/m=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2803.png"
 ALT="$m =0$">|; 

$key = q/{mitOmega}>{mitOmega}_{{rm{crit};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1524.png"
 ALT="${\mit\Omega}&gt;{\mit\Omega}_{\rm crit}$">|; 

$key = q/{displaymath}I_parallel=2,I_perp{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1465.png"
 ALT="\begin{displaymath}
I_\parallel = 2 I_\perp
\end{displaymath}">|; 

$key = q/{displaymath}x=a,sin(omega,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img237.png"
 ALT="\begin{displaymath}
x = a \sin (\omega t),
\end{displaymath}">|; 

$key = q/{displaymath}delta{{bf{q}_k(t)=left(alpha_k,{{rm{e}^{+{{rm{i},omega_k,t}+beta_k,{{rm{e}^{-{{rm{i},omega_k,t}right){{bf{x}_k,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img1960.png"
 ALT="\begin{displaymath}
\delta {\bf q}_k(t) = \left(\alpha_k  {\rm e}^{+{\rm i} \o...
...,t}+\beta_k {\rm e}^{-{\rm i} \omega_k t}\right)
{\bf x}_k,
\end{displaymath}">|; 

$key = q/{displaymath}y^2-2,r_c,(x-x_c)=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img725.png"
 ALT="\begin{displaymath}
y^2 - 2 r_c (x-x_c) = 0,
\end{displaymath}">|; 

$key = q/theta(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$\theta (0)$">|; 

$key = q/(x_0,,y_0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2633.png"
 ALT="$(x_0, y_0, 0)$">|; 

$key = q/s_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1200.png"
 ALT="$s_0$">|; 

$key = q/tilde{A};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3060.png"
 ALT="$\tilde{A}$">|; 

$key = q/{displaymath}frac{d}{dt}!left(frac{partialK}{partialdot{q}_1}right)-frac{partialartialq_1}+lambda(t),frac{partialf}{partialq_1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1811.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial K}{\partial \dot{q}_1}\ri...
...U}{\partial q_1}+ \lambda(t) \frac{\partial f}{\partial q_1}.
\end{displaymath}">|; 

$key = q/x'siml,theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1181.png"
 ALT="$x'\sim l \theta$">|; 

$key = q/-q_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1987.png"
 ALT="$-q_2$">|; 

$key = q/k=1,{calF};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1951.png"
 ALT="$k=1,{\cal F}$">|; 

$key = q/360,gamma_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2992.png"
 ALT="$360 \gamma_1$">|; 

$key = q/C=C_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$C= C_1$">|; 

$key = q/ddot{q}_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1894.png"
 ALT="$\ddot{q}_j$">|; 

$key = q/displaystyleI_perp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2195.png"
 ALT="$\displaystyle I_\perp$">|; 

$key = q/displaystylefrac{{{bf{x}_k^{T},{{bf{M},delta{{bf{q}^{(0)}+{{bf{x}_k^{T},{{bf{M},deltadot{{bf{q}^{(0)}slashsqrt{lambda}_k}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1967.png"
 ALT="$\displaystyle \frac{{\bf x}_k^{T} {\bf M} \delta{\bf q}^{(0)} +{\bf x}_k^{T} {\bf M} \delta \dot{\bf q}^{(0)}/\sqrt{\lambda}_k}{2},$">|; 

$key = q/displaystyle{{bf{e}_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img680.png"
 ALT="$\displaystyle {\bf e}_\theta$">|; 

$key = q/{displaymath}deltaW=sum_{i=1,{calF}}Q_i,deltaq_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1574.png"
 ALT="\begin{displaymath}
\delta W = \sum_{i=1,{\cal F}} Q_i \delta q_i,
\end{displaymath}">|; 

$key = q/dx(0)slashdt=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img444.png"
 ALT="$dx(0)/dt=0$">|; 

$key = q/displaystyleA,cos(omega_0,t-phi_1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img534.png"
 ALT="$\displaystyle A \cos(\omega_0 t-\phi_1),$">|; 

$key = q/D=270^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3008.png"
 ALT="$D=270^\circ$">|; 

$key = q/displaystyle-n^2,a^3,frac{x_1}{r^3}+frac{3}{2},n'^{,2},cos[2,(n-n'),t],x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2744.png"
 ALT="$\displaystyle -n^2 a^3 \frac{x_1}{r^3}+\frac{3}{2} n'^{ 2} \cos[2 (n-n') t] x_1$">|; 

$key = q/{{bf{v}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1008.png"
 ALT="${\bf v}_1$">|; 

$key = q/1slash16;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img380.png"
 ALT="$1/16$">|; 

$key = q/dot{psi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1438.png"
 ALT="$\dot{\psi}$">|; 

$key = q/r_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img186.png"
 ALT="$r_i$">|; 

$key = q/3,{{bf{a}+2,{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3504.png"
 ALT="$3 {\bf a}+2 {\bf b}$">|; 

$key = q/displaystylefrac{r_1}{r_2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2489.png"
 ALT="$\displaystyle \frac{r_1}{r_2}$">|; 

$key = q/dx=x-x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3447.png"
 ALT="$dx=x-x_0$">|; 

$key = q/F(Q);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3397.png"
 ALT="$F(Q)$">|; 

$key = q/displaystyledeltadot{psi}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2415.png"
 ALT="$\displaystyle \delta\dot{\psi}_i$">|; 

$key = q/|x|slashall1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1227.png"
 ALT="$\vert x\vert/a\ll 1$">|; 

$key = q/lambda<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1145.png"
 ALT="$\lambda &lt; 0$">|; 

$key = q/{displaymath}frac{1}{2}frac{d^2I}{dt^2}=2,K+(n-1),U,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img181.png"
 ALT="\begin{displaymath}
\frac{1}{2}\frac{d^2 I}{dt^2} = 2 K + (n-1) U,
\end{displaymath}">|; 

$key = q/(AB);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3269.png"
 ALT="$(AB)$">|; 

$key = q/displaystyleR_Y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2828.png"
 ALT="$\displaystyle R_Y,$">|; 

$key = q/{displaymath}{{bf{a}times({{bf{b}times{{bf{c})neq({{bf{a}times{{bf{b})times{{bf{c}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3303.png"
 ALT="\begin{displaymath}
{\bf a}\times({\bf b} \times{\bf c})\neq ({\bf a}\times{\bf b}) \times{\bf c}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{p_theta}{m}=r^2,dot{theta}=h,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1882.png"
 ALT="\begin{displaymath}
\frac{p_\theta}{m} =r^2 \dot{\theta} =h,
\end{displaymath}">|; 

$key = q/langlecosphi'rangle=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2092.png"
 ALT="$\langle\cos\phi'\rangle = 0$">|; 

$key = q/displaystyledeltatheta({mitDelta}t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3136.png"
 ALT="$\displaystyle \delta\theta({\mit\Delta} t)$">|; 

$key = q/{displaymath}L^2=I_{x'x'}^{,2},omega_{x'}^{,2}+I_{y'y'}^{,2},omega_{y'}^{,2}+I_{z'z'}^{,2},omega_{z'}^{,2}={{rm{constant}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="391" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img1412.png"
 ALT="\begin{displaymath}
L^2 = I_{x'x'}^{ 2} \omega_{x'}^{ 2} + I_{y'y'}^{ 2} \o...
...'}^{ 2}
+ I_{z'z'}^{ 2} \omega_{z'}^{ 2} ={\rm constant}.
\end{displaymath}">|; 

$key = q/X_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img400.png"
 ALT="$X_n$">|; 

$key = q/theta=pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img954.png"
 ALT="$\theta=\pi$">|; 

$key = q/displaystyle+frac{1}{4},iota^2,cos[2,(1+g,m^2),T-2,gamma_0]-m^2,cos[2,(1-m),T];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2957.png"
 ALT="$\displaystyle + \frac{1}{4} \iota^2 \cos[2 (1+g m^2) T-2 \gamma_0] - m^2 \cos[2 (1-m) T]$">|; 

$key = q/v(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img249.png"
 ALT="$v(x)$">|; 

$key = q/displaystyleint!int!intdV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3424.png"
 ALT="$\displaystyle \int\!\int\!\int dV$">|; 

$key = q/E_x=E_0,cos(omega,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img652.png"
 ALT="$E_x= E_0 \cos(\omega t)$">|; 

$key = q/v_0^{,2}slashg;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img605.png"
 ALT="$v_0^{ 2}/g$">|; 

$key = q/displaystyle-mu_1,frac{(eta-eta_1)}{rho_1^{,3}}-mu_2,frac{(eta-eta_2)}{rho_2^{,3}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2506.png"
 ALT="$\displaystyle -\mu_1 \frac{(\eta-\eta_1)}{\rho_1^{ 3}} - \mu_2 \frac{(\eta-\eta_2)}{\rho_2^{ 3}},$">|; 

$key = q/y(-aslash2)=y(+aslash2)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1760.png"
 ALT="$y(-a/2) = y(+a/2) = 0$">|; 

$key = q/displaystyleA,[2,phi+sin(2,phi)],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img656.png"
 ALT="$\displaystyle A [2 \phi + \sin (2 \phi)],$">|; 

$key = q/U_{xx}=partial^2U(x_0,y_0,0)slashpartialx^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2642.png"
 ALT="$U_{xx}=\partial^2 U(x_0,y_0,0)/\partial x^2$">|; 

$key = q/{displaymath}r^2=A,t^2+B,t+C,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img898.png"
 ALT="\begin{displaymath}
r^2 = A t^2+B t+C,
\end{displaymath}">|; 

$key = q/displaystylef(R_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2398.png"
 ALT="$\displaystyle f(R_i)$">|; 

$key = q/{displaymath}({{bf{A}-lambda,{{bf{1}),{{bf{x}={{bf{0},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1285.png"
 ALT="\begin{displaymath}
({\bf A} - \lambda {\bf 1}) {\bf x} = {\bf0},
\end{displaymath}">|; 

$key = q/displaystyleA-D-1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2662.png"
 ALT="$\displaystyle A - D - 1,$">|; 

$key = q/{displaymath}U=frac{1}{2}left[k',q_1^{,2}+k,(q_2-q_1)^2+k',q_2^{,2}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1989.png"
 ALT="\begin{displaymath}
U= \frac{1}{2}\left[k' q_1^{ 2} + k (q_2-q_1)^2+ k' q_2^{ 2}\right].
\end{displaymath}">|; 

$key = q/Ngg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img433.png"
 ALT="$N\gg 1$">|; 

$key = q/x=-mu_2-rho_1=mu_1-rho_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2588.png"
 ALT="$x = -\mu_2-\rho_1=\mu_1-\rho_2$">|; 

$key = q/{displaymath}U(r)=left{array{{lll}0&mbox{hspace{1cm}}&r>a[0.5ex]infty&&rleqaarray{right..{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img981.png"
 ALT="\begin{displaymath}
U(r) =
\left\{\begin{array}{lll}
0&amp;\mbox{\hspace{1cm}} &amp; r&gt;a\ [0.5ex]
\infty &amp;&amp; r\leq a
\end{array}\right..
\end{displaymath}">|; 

$key = q/vartheta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img956.png"
 ALT="$\vartheta=0$">|; 

$key = q/displaystyleC_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2613.png"
 ALT="$\displaystyle C_1$">|; 

$key = q/{displaymath}{{bf{a}'simeq{{bf{a}+deltambox{{boldmath{{theta{}times{{bf{a},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3333.png"
 ALT="\begin{displaymath}
{\bf a}' \simeq {\bf a} + \delta \mbox{\boldmath$\theta$}\times {\bf a},
\end{displaymath}">|; 

$key = q/alpha=mslashM;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2035.png"
 ALT="$\alpha =m/M$">|; 

$key = q/psi=thetaslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1083.png"
 ALT="$\psi=\theta/2$">|; 

$key = q/displaystyle-frac{m,g,sintheta}{1+m,a^2slashI}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1827.png"
 ALT="$\displaystyle -\frac{m g \sin\theta}{1+ m a^2/I}.$">|; 

$key = q/{displaymath}2,bleft|frac{db}{dtheta}right|=frac{k^2}{8,E^{,2}}frac{sintheta}{sin^4(thetaslash2)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img997.png"
 ALT="\begin{displaymath}
2  b\left\vert\frac{db}{d\theta}\right\vert = \frac{k^2}{8 E^{ 2}}\frac{\sin\theta}{\sin^4(\theta/2)}.
\end{displaymath}">|; 

$key = q/x-x_0>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img223.png"
 ALT="$x-x_0&gt;0$">|; 

$key = q/{displaymath}ddot{theta}+frac{g}{l},sintheta-sintheta,costheta,dot{phi}^{,2}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1682.png"
 ALT="\begin{displaymath}
\ddot{\theta} + \frac{g}{l} \sin\theta - \sin\theta \cos\theta \dot{\phi}^{ 2} = 0,
\end{displaymath}">|; 

$key = q/{displaymath}C=2left(frac{mu_1}{rho_1}+frac{mu_2}{rho_2}right)+2,omega,(xi,dot{e}-eta,dot{xi})-dot{xi}^2-dot{eta}^2-dot{zeta}^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2513.png"
 ALT="\begin{displaymath}
C = 2\left(\frac{\mu_1}{\rho_1}+\frac{\mu_2}{\rho_2}\right) ...
...eta}-\eta \dot{\xi})
-\dot{\xi}^2-\dot{\eta}^2-\dot{\zeta}^2.
\end{displaymath}">|; 

$key = q/displaystyle-frac{mu_1,(x+mu_2)}{rho_1^{,3}}-frac{mu_2,(x-mu_1)}{rho_2^{,3}}+omega^2,x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2540.png"
 ALT="$\displaystyle - \frac{\mu_1 (x+\mu_2)}{\rho_1^{ 3}}- \frac{\mu_2 (x-\mu_1)}{\rho_2^{ 3}}
+ \omega^2 x,$">|; 

$key = q/displaystyle-a,sin({mitOmega},sinlambda,t),cosleft(sqrt{frac{g}{l}},tright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1210.png"
 ALT="$\displaystyle -a \sin({\mit\Omega} \sin\lambda t) \cos\left(\sqrt{\frac{g}{l}} t\right).$">|; 

$key = q/omega_3=2,Omega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2872.png"
 ALT="$\omega_3=2 \Omega_1$">|; 

$key = q/{displaymath}psi_{max}=sin^{-1}left(frac{m_2}{m_1}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1062.png"
 ALT="\begin{displaymath}
\psi_{max} = \sin^{-1}\left(\frac{m_2}{m_1}\right).
\end{displaymath}">|; 

$key = q/Omega_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2850.png"
 ALT="$\Omega_j$">|; 

$key = q/m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$m$">|; 

$key = q/{displaymath}{{bf{f}_{ij}=-{{bf{f}_{ji}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="\begin{displaymath}
{\bf f}_{ij} = - {\bf f}_{ji}.
\end{displaymath}">|; 

$key = q/{displaymath}omega_pm=-{{rm{i},nupmsqrt{omega_0^{,2}-nu^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img319.png"
 ALT="\begin{displaymath}
\omega_\pm = - {\rm i} \nu \pm \sqrt{\omega_0^{ 2}-\nu^2}.
\end{displaymath}">|; 

$key = q/displaystyle-sum_{i=1,N}x_i,z_i,m_i=-intx,z,dm.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1254.png"
 ALT="$\displaystyle - \sum_{i=1,N}x_i z_i  m_i= -\int x z dm.$">|; 

$key = q/vartheta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img478.png"
 ALT="$\vartheta_0$">|; 

$key = q/{{bf{x}_k^T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1940.png"
 ALT="${\bf x}_k^T$">|; 

$key = q/deltal;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1723.png"
 ALT="$\delta l$">|; 

$key = q/{displaymath}x(t)=x_0,{{rm{e}^{-nu,t},cos(omega_r,t)+left(frac{v_0+nu,x_0}{omega_r}right){{rm{e}^{-nu,t},sin(omega_r,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="442" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img323.png"
 ALT="\begin{displaymath}
x(t) = x_0 {\rm e}^{-\nu t} \cos(\omega_r t) + \left(\fr...
...nu x_0}{\omega_r}\right){\rm e}^{-\nu t} \sin(\omega_r t),
\end{displaymath}">|; 

$key = q/{displaymath}p=frac{partialK}{partialdot{x}}=m,dot{x}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1859.png"
 ALT="\begin{displaymath}
p = \frac{\partial K}{\partial\dot{x}} = m \dot{x}.
\end{displaymath}">|; 

$key = q/displaystylesum_{i=1,N}(x_i^{,2}+y_i^{,2}),m_i=int(x^2+y^2),dm,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1248.png"
 ALT="$\displaystyle \sum_{i=1,N}(x_i^{ 2}+y_i^{ 2})  m_i= \int(x^2+ y^2) dm,$">|; 

$key = q/{{bf{a}times{{bf{c}={{bf{b}times{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3522.png"
 ALT="${\bf a}\times {\bf c} = {\bf b}\times {\bf c}$">|; 

$key = q/{displaymath}{{bf{v}=frac{{{bf{E}times{{bf{B}}{B^2}+{{bf{v}',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img624.png"
 ALT="\begin{displaymath}
{\bf v} = \frac{{\bf E}\times{\bf B}}{B^2} + {\bf v}',
\end{displaymath}">|; 

$key = q/P=dWslashdt={{bf{f}cdotd{{bf{r}slashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3285.png"
 ALT="$P = dW/dt = {\bf f}\cdot d{\bf r}/dt$">|; 

$key = q/displaystylefrac{1}{4},e^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2909.png"
 ALT="$\displaystyle \frac{1}{4} e^2.$">|; 

$key = q/displaystyle0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1190.png"
 ALT="$\displaystyle 0$">|; 

$key = q/{displaymath}frac{a_e-a_p}{a}=epsilon=frac{5}{4},frac{{mitOmega}^2,a^3}{G,M}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2177.png"
 ALT="\begin{displaymath}
\frac{a_e-a_p}{a} = \epsilon = \frac{5}{4} \frac{{\mit\Omega}^2 a^3}{G M}.
\end{displaymath}">|; 

$key = q/P_0(0)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2381.png"
 ALT="$P_0(0)=1$">|; 

$key = q/{displaymath}f(c)=-c,r^n,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img858.png"
 ALT="\begin{displaymath}
f(c) = -c r^n,
\end{displaymath}">|; 

$key = q/{displaymath}frac{dx}{dt}=-x_0,nu,{{rm{e}^{-nu,t},cos(omega_r,t-phi_0)-x_0,omega_r,{{rm{e}^{-nu,t},sin(omega_r,t-phi_0).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="496" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img347.png"
 ALT="\begin{displaymath}
\frac{dx}{dt} =- x_0 \nu {\rm e}^{-\nu t} \cos(\omega_r\...
...)-
x_0 \omega_r {\rm e}^{-\nu t} \sin(\omega_r t-\phi_0).
\end{displaymath}">|; 

$key = q/displaystyleD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2659.png"
 ALT="$\displaystyle D$">|; 

$key = q/L_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2603.png"
 ALT="$L_4$">|; 

$key = q/84.01;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2439.png"
 ALT="$84.01$">|; 

$key = q/0.723;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2425.png"
 ALT="$0.723$">|; 

$key = q/displaystyleint_0^adzint_{-(a-z)slash2}^{(a-z)slash2}dyint_{-(a-z)slash2}^{(a-z)slash2}dx=int_0^a(a-z)^2,dz=int_0^a(a^2-2,a,z+z^2),dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="577" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3425.png"
 ALT="$\displaystyle \int_0^a dz \int_{-(a-z)/2}^{(a-z)/2} dy\int_{-(a-z)/2}^{(a-z)/2} dx = \int_0^a (a-z)^2 dz=\int_0^a (a^2-2 a z+z^2) dz$">|; 

$key = q/delta_{kl}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1948.png"
 ALT="$\delta_{kl}=1$">|; 

$key = q/displaystylefrac{dv_x}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img577.png"
 ALT="$\displaystyle \frac{dv_x}{dt}$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter09slashfig9.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="372" BORDER="0"
 SRC="|."$dir".q|img1630.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter09/fig9.02.eps}}
\end{figure}">|; 

$key = q/{displaymath}Phi_0(r)=-frac{4pi,G,gamma}{r}int_0^rr'^{,2},dr'-4pi,G,gammaint_r^ar',dr'{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2130.png"
 ALT="\begin{displaymath}
\Phi_0(r) = -\frac{4\pi G \gamma}{r}\int_0^r r'^{ 2} dr' - 4\pi G \gamma
\int_r^a r' dr'
\end{displaymath}">|; 

$key = q/L_5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$L_5$">|; 

$key = q/I_parallelslashI_perp=1.00327;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1474.png"
 ALT="$I_\parallel/I_\perp=1.00327$">|; 

$key = q/{displaymath}frac{dvartheta}{du}=frac{b}{sqrt{1-b^2,u^2-U(u)slashE}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img965.png"
 ALT="\begin{displaymath}
\frac{d\vartheta}{d u } = \frac{b}{\sqrt{1 - b^2 u^2-U(u)/E}}.
\end{displaymath}">|; 

$key = q/mu_2=0.1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$\mu _2=0.1$">|; 

$key = q/displaystyle-frac{Z}{R^3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2768.png"
 ALT="$\displaystyle -\frac{Z}{R^3},$">|; 

$key = q/f_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1570.png"
 ALT="$f_j$">|; 

$key = q/e=0.2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img942.png"
 ALT="$e=0.2$">|; 

$key = q/-2,m,mbox{{boldmath{{Omega{}times{{bf{v}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1147.png"
 ALT="$-2 m \mbox{\boldmath$\Omega$}\times{\bf v}'$">|; 

$key = q/{displaymath}v=v_t,tanhleft(frac{g,t}{v_t}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img259.png"
 ALT="\begin{displaymath}
v = v_t \tanh\left(\frac{g t}{v_t}\right).
\end{displaymath}">|; 

$key = q/{{bf{x}_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1923.png"
 ALT="${\bf x}_k$">|; 

$key = q/{displaymath}m_{ij}=m_{ji}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1888.png"
 ALT="\begin{displaymath}
m_{ij} = m_{ji}.
\end{displaymath}">|; 

$key = q/{displaymath}T'=frac{2pi}{{mitOmega}}=305,,{{rm{days}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1476.png"
 ALT="\begin{displaymath}
T'= \frac{2\pi}{{\mit\Omega}} = 305  {\rm days}.
\end{displaymath}">|; 

$key = q/displaystyleomega_perp,cos({mitOmega},t'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1386.png"
 ALT="$\displaystyle \omega_\perp \cos({\mit\Omega} t'),$">|; 

$key = q/C';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$C'$">|; 

$key = q/displaystyle-frac{T}{l,m},y'-2,{mitOmega},sinlambda,dot{x}',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1189.png"
 ALT="$\displaystyle - \frac{T}{l m} y' - 2 {\mit\Omega} \sin\lambda \dot{x}',$">|; 

$key = q/U(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img207.png"
 ALT="$U(x)$">|; 

$key = q/trightarrowt+T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img821.png"
 ALT="$t\rightarrow t+T$">|; 

$key = q/135^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3001.png"
 ALT="$135^\circ$">|; 

$key = q/displaystyleC_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2619.png"
 ALT="$\displaystyle C_4$">|; 

$key = q/displaystyleleft(array{{cc}-k-k'<col_mark>k<row_mark>k<col_mark>-k-k'array{right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1994.png"
 ALT="$\displaystyle \left(\begin{array}{cc}
-k-k'&amp; k\\\\
k&amp; -k-k'\end{array}\right).$">|; 

$key = q/displaystyledeltaddot{X}-2,deltadot{Y}-3,deltaX;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2805.png"
 ALT="$\displaystyle \delta \ddot{X}-2 \delta \dot{Y} - 3 \delta X$">|; 

$key = q/-pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img798.png"
 ALT="$-\pi$">|; 

$key = q/displaystyle{calE}_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1050.png"
 ALT="$\displaystyle {\cal E}_1'$">|; 

$key = q/r_c=e,d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img714.png"
 ALT="$r_c=e d$">|; 

$key = q/{mitDelta}t=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="${\mit \Delta } t = 0$">|; 

$key = q/alpha<5slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img193.png"
 ALT="$\alpha&lt;5/2$">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}_phi=dot{phi},{{bf{e}_z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1425.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$}_\phi = \dot{\phi} {\bf e}_z.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{AppendixAslashfigA.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="191" BORDER="0"
 SRC="|."$dir".q|img3239.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.03.eps}}
\end{figure}">|; 

$key = q/v_0ggv_t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img608.png"
 ALT="$v_0\gg v_t$">|; 

$key = q/X_0=(1+m^2slash2)^{-1slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2775.png"
 ALT="$X_0=(1+m^2/2)^{-1/3}$">|; 

$key = q/deltax;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2636.png"
 ALT="$\delta x$">|; 

$key = q/i=1,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1319.png"
 ALT="$i=1, 3$">|; 

$key = q/{displaymath}x(t)=A,{{rm{e}^{-nu,t},cos(omega_r,t)+B,{{rm{e}^{-nu,t},sin(omega_r,t)+x_1,cos(omega,t-phi_1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="513" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img439.png"
 ALT="\begin{displaymath}
x(t) = A {\rm e}^{-\nu t} \cos(\omega_r t) + B {\rm e}^{-\nu t} 
\sin(\omega_r t) + x_1 \cos(\omega t-\phi_1).
\end{displaymath}">|; 

$key = q/displaystylex',cos(n,t)+y',sin(n,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2733.png"
 ALT="$\displaystyle x' \cos(n t)+y' \sin(n t),$">|; 

$key = q/{{bf{a}=(1slashsqrt{2},,1slashsqrt{2},,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3292.png"
 ALT="${\bf a} = (1/\sqrt{2},  1/\sqrt{2},  0)$">|; 

$key = q/-z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1671.png"
 ALT="$-z$">|; 

$key = q/{displaymath}frac{d{{bf{r}}{dt}=mbox{{boldmath{{Omega{}times{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1105.png"
 ALT="\begin{displaymath}
\frac{d{\bf r}}{dt} = \mbox{\boldmath$\Omega$}\times{\bf r}.
\end{displaymath}">|; 

$key = q/rho(r',theta');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2079.png"
 ALT="$\rho(r',\theta')$">|; 

$key = q/{calE}=G,M,(e-1)slash(2,r_p);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img793.png"
 ALT="${\cal E} = G M (e-1)/(2 r_p)$">|; 

$key = q/D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$D$">|; 

$key = q/n^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2752.png"
 ALT="$n^{-1}$">|; 

$key = q/{displaymath}U=-frac{G,M_s,M}{a_s}+frac{G,M_s,(I_parallel-I_perp)}{2,a_s^{,3}},(3,sin^2theta,sin^2lambda_s-1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2279.png"
 ALT="\begin{displaymath}
U = - \frac{G M_s M}{a_s} + \frac{G M_s (I_\parallel-I_\perp)}{2 a_s^{ 3}} (3 \sin^2\theta \sin^2\lambda_s -1).
\end{displaymath}">|; 

$key = q/17.2^{''};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2376.png"
 ALT="$17.2^{''}$">|; 

$key = q/{displaymath}v^2=dot{r}^{,2}+(r,dot{theta})^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1606.png"
 ALT="\begin{displaymath}
v^2 = \dot{r}^{ 2} + (r \dot{\theta})^2.
\end{displaymath}">|; 

$key = q/{displaymath}frac{partialV}{partialx}=A_x,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3470.png"
 ALT="\begin{displaymath}
\frac{\partial V}{\partial x} = A_x,
\end{displaymath}">|; 

$key = q/x=-mu_2+rho_1=mu_1-rho_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2573.png"
 ALT="$x = -\mu_2+\rho_1=\mu_1-\rho_2$">|; 

$key = q/displaystylefrac{partialH}{partialp_r}=frac{p_r}{m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1875.png"
 ALT="$\displaystyle \frac{\partial H}{\partial p_r} = \frac{p_r}{m},$">|; 

$key = q/{{bf{b}=(1,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3290.png"
 ALT="${\bf b} = (1, 0, 0)$">|; 

$key = q/{displaymath}{{bf{a}'={{bf{a}-frac{d{{bf{u}}{dt},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="\begin{displaymath}
{\bf a}' = {\bf a} - \frac{d{\bf u}}{dt},
\end{displaymath}">|; 

$key = q/{displaymath}lambda'+cosh(mpaslash2,h+c)=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1775.png"
 ALT="\begin{displaymath}
\lambda' + \cosh(\mp a/2 h + c) = 0.
\end{displaymath}">|; 

$key = q/M_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2272.png"
 ALT="$M_s$">|; 

$key = q/{{bf{a}=(0,,1,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3289.png"
 ALT="${\bf a} =
(0, 1, 0)$">|; 

$key = q/J_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2156.png"
 ALT="$J_n$">|; 

$key = q/a_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2238.png"
 ALT="$a_-$">|; 

$key = q/{bf{r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3280.png"
 ALT="$\bf r$">|; 

$key = q/n=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img494.png"
 ALT="$n=1/2$">|; 

$key = q/displaystyleleft.+frac{3}{4}sum_{j>i}left(frac{M_j}{M_0}right)left(frac{R_i}{R_j2+frac{175}{64}left(frac{R_i}{R_j}right)^4+cdotsright]right}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="501" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2409.png"
 ALT="$\displaystyle \left.+ \frac{3}{4}\sum_{j&gt;i}\left(\frac{M_j}{M_0}\right)
\left(\...
...right)^2 + \frac{175}{64}\left(\frac{R_i}{R_j}
\right)^4+\cdots\right]\right\}.$">|; 

$key = q/alpha_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2849.png"
 ALT="$\alpha_j$">|; 

$key = q/displaystyle{{bf{x}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2012.png"
 ALT="$\displaystyle {\bf x}_2$">|; 

$key = q/{displaymath}|{{bf{A}-lambda,{{bf{1}|=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1287.png"
 ALT="\begin{displaymath}
\vert{\bf A} - \lambda {\bf 1}\vert = 0.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{f}_i=sum_{j=1,N}^{jneqi}{{bf{f}_{ij}+{{bf{F}_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="67" BORDER="0"
 SRC="|."$dir".q|img163.png"
 ALT="\begin{displaymath}
{\bf f}_i = \sum_{j=1,N}^{j\neq i} {\bf f}_{ij} + {\bf F}_i,
\end{displaymath}">|; 

$key = q/{displaymath}mbox{{boldmath{{tau{}={{bf{r}times{{bf{F}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3322.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\tau$}= {\bf r}\times{\bf F}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter13slashfig13.08.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="239" BORDER="0"
 SRC="|."$dir".q|img2627.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter13/fig13.08.eps}}
\end{figure}">|; 

$key = q/{{bf{a}times({{bf{b}times{{bf{c})=({{bf{a}times{{bf{b})times{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3518.png"
 ALT="${\bf a}\times ({\bf b}\times {\bf c})= ({\bf a}\times {\bf b})\times {\bf c}$">|; 

$key = q/B=9slash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2682.png"
 ALT="$B=9/4$">|; 

$key = q/{displaymath}Gamma^2+Gamma+frac{27}{4},mu_2,(1-mu_2)=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2688.png"
 ALT="\begin{displaymath}
\Gamma^2 + \Gamma + \frac{27}{4} \mu_2 (1-\mu_2) = 0
\end{displaymath}">|; 

$key = q/{displaymath}m,l,frac{d^2theta}{dt^2}+nu,frac{dtheta}{dt}+m,g,sintheta=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img3025.png"
 ALT="\begin{displaymath}
m l \frac{d^2\theta}{dt^2}+ \nu \frac{d\theta}{dt} + m g \sin\theta = 0,
\end{displaymath}">|; 

$key = q/3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2858.png"
 ALT="$3$">|; 

$key = q/displaystylefrac{g,sintheta}{1+Islashm,a^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1824.png"
 ALT="$\displaystyle \frac{g \sin\theta}{1+ I/m a^2},$">|; 

$key = q/k=q^2slash(4pi,epsilon_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1004.png"
 ALT="$k=q^2/(4\pi \epsilon_0)$">|; 

$key = q/displaystylev_y';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$\displaystyle v_y'$">|; 

$key = q/1slashgamma_1=18.3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2994.png"
 ALT="$1/\gamma_1=18.3$">|; 

$key = q/Q=(a,0,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3412.png"
 ALT="$Q=(a,0,0)$">|; 

$key = q/-3,m,e',sinM';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3019.png"
 ALT="$-3 m e' \sin M'$">|; 

$key = q/{displaymath}r_c=frac{h^2}{G,M}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img752.png"
 ALT="\begin{displaymath}
r_c = \frac{h^2}{G M}.
\end{displaymath}">|; 

$key = q/displaystyle(x-mu_1)^2+y^2+z^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2538.png"
 ALT="$\displaystyle (x-\mu_1)^2+y^2 + z^2.$">|; 

$key = q/{displaymath}{calE}=-frac{G,M}{2,a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img776.png"
 ALT="\begin{displaymath}
{\cal E} = - \frac{G M}{2 a}.
\end{displaymath}">|; 

$key = q/x_t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img275.png"
 ALT="$x_t$">|; 

$key = q/{displaymath}eta_k={{bf{x}_k^T,{{bf{M},delta{{bf{q}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1974.png"
 ALT="\begin{displaymath}
\eta_k = {\bf x}_k^T {\bf M} \delta{\bf q}.
\end{displaymath}">|; 

$key = q/I_{zz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1323.png"
 ALT="$I_{zz}$">|; 

$key = q/deltaI=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1755.png"
 ALT="$\delta I = 0$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter08slashfig8.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="271" BORDER="0"
 SRC="|."$dir".q|img1344.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter08/fig8.02.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter08slashfig8.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="285" BORDER="0"
 SRC="|."$dir".q|img1517.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter08/fig8.04.eps}}
\end{figure}">|; 

$key = q/{displaymath}I,frac{d^2{theta}}{dt^2}=tau,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img451.png"
 ALT="\begin{displaymath}
I \frac{d^2{\theta}}{dt^2} = \tau,
\end{displaymath}">|; 

$key = q/displaystyledot{theta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1876.png"
 ALT="$\displaystyle \dot{\theta}$">|; 

$key = q/cosgamma=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1335.png"
 ALT="$\cos\gamma=0$">|; 

$key = q/2,alpha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2855.png"
 ALT="$2 \alpha_0$">|; 

$key = q/1.348;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3085.png"
 ALT="$1.348$">|; 

$key = q/displaystyle3+3^{4slash3},mu_2^{,2slash3}-10,mu_2slash3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2614.png"
 ALT="$\displaystyle 3 + 3^{4/3} \mu_2^{ 2/3}-10 \mu_2/3,$">|; 

$key = q/-frac{3}{2},m^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2898.png"
 ALT="$-\frac{3}{2} m^2$">|; 

$key = q/R_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2391.png"
 ALT="$R_i$">|; 

$key = q/alpha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2813.png"
 ALT="$\alpha_0$">|; 

$key = q/mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img928.png"
 ALT="$\mu$">|; 

$key = q/displaystylesin^2{mitDelta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img556.png"
 ALT="$\displaystyle \sin^2{\mit\Delta}.$">|; 

$key = q/(1+g,m^2),n,t-gamma_0=gamma_0+j,2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2988.png"
 ALT="$(1+g m^2) n t -\gamma_0=\gamma_0+j 2\pi$">|; 

$key = q/{displaymath}{calE}=frac{dot{r}^{,2}+r^2,dot{theta}^{,2}}{2}-frac{G,M}{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img770.png"
 ALT="\begin{displaymath}
{\cal E} = \frac {\dot{r}^{ 2} + r^2 \dot{\theta}^{ 2}}{2} - \frac{G M}{r}.
\end{displaymath}">|; 

$key = q/{displaymath}z_3simeqfrac{1}{2},e,iota.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2919.png"
 ALT="\begin{displaymath}
z_3\simeq \frac{1}{2} e \iota.
\end{displaymath}">|; 

$key = q/{displaymath}delta{{bf{q}(t)=sum_{k=1,{calF}}delta{{bf{q}_k(t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1952.png"
 ALT="\begin{displaymath}
\delta {\bf q}(t) = \sum_{k=1,{\cal F}}\delta {\bf q}_k(t),
\end{displaymath}">|; 

$key = q/{displaymath}r=frac{a,(1-e^2)}{1-e,costheta},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img935.png"
 ALT="\begin{displaymath}
r = \frac{a (1-e^2)}{1-e \cos\theta},
\end{displaymath}">|; 

$key = q/{displaymath}rho(r,theta)=sum_{n=0,infty}rho_n(r),P_n(costheta).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2124.png"
 ALT="\begin{displaymath}
\rho(r,\theta) = \sum_{n=0,\infty} \rho_n(r) P_n(\cos\theta).
\end{displaymath}">|; 

$key = q/{displaymath}-2,UgeqC,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img2558.png"
 ALT="\begin{displaymath}
-2 U \geq C,
\end{displaymath}">|; 

$key = q/{{bf{a}=d^2{{bf{r}slashdt^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1116.png"
 ALT="${\bf a}= d^2{\bf r}/dt^2$">|; 

$key = q/{displaymath}G_{ij}=G_{ji}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1909.png"
 ALT="\begin{displaymath}
G_{ij} = G_{ji}.
\end{displaymath}">|; 

$key = q/{displaymath}a_g=frac{G,m'}{R^2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2205.png"
 ALT="\begin{displaymath}
a_g = \frac{G m'}{R^2}
\end{displaymath}">|; 

$key = q/displaystylefrac{omega}{omega_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3032.png"
 ALT="$\displaystyle \frac{\omega}{\omega_0},$">|; 

$key = q/{displaymath}{calE}=frac{dot{r}^{,2}}{2}+frac{h^2}{2,r^2}-frac{G,M}{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img791.png"
 ALT="\begin{displaymath}
{\cal E} = \frac{\dot{r}^{ 2}}{2} + \frac{h^2}{2 r^2} - \frac{G M}{r},
\end{displaymath}">|; 

$key = q/{displaymath}theta(t)=vartheta_0,cos(omega,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img468.png"
 ALT="\begin{displaymath}
\theta(t) = \vartheta_0 \cos(\omega t).
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=1.in{centerline{epsffile{Chapter11slashfig11.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="484" HEIGHT="110" BORDER="0"
 SRC="|."$dir".q|img1985.png"
 ALT="\begin{figure}
\epsfysize =1.in
\centerline{\epsffile{Chapter11/fig11.01.eps}}
\end{figure}">|; 

$key = q/{displaymath}y(x)=h,[cosh(aslash2,h)-cosh(xslashh)].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1778.png"
 ALT="\begin{displaymath}
y(x) = h [\cosh(a/2 h) - \cosh(x/h)].
\end{displaymath}">|; 

$key = q/C_1>C_2>C_3>C_4=C_5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2622.png"
 ALT="$C_1&gt;C_2&gt;C_3&gt;C_4=C_5$">|; 

$key = q/{displaymath}Q_n=Q_1+(Q_2-Q_1)sum_{j=0}^{n-2}frac{1}{F^j},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img3102.png"
 ALT="\begin{displaymath}
Q_n = Q_1 + (Q_2-Q_1)\sum_{j=0}^{n-2}\frac{1}{F^j},
\end{displaymath}">|; 

$key = q/Deltaomega=2,nu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img387.png"
 ALT="$\Delta\omega = 2 \nu$">|; 

$key = q/displaystyle3left[frac{mu_1,(x+mu_2)^2}{rho_1^{,3}}+frac{mu_2,(x-mu_1)^2}{rho_2^{,3}}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2660.png"
 ALT="$\displaystyle 3\left[\frac{\mu_1 (x+\mu_2)^2}{\rho_1^{ 3}}+\frac{\mu_2 (x-\mu_1)^2}{\rho_2^{ 3}}\right],$">|; 

$key = q/{displaymath}int_a^bleft(frac{partialF}{partialy},deltay+frac{partialF}{partialy'},deltay'right),dx=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1730.png"
 ALT="\begin{displaymath}
\int_a^b\left(\frac{\partial F}{\partial y} \delta y+ \frac{\partial F}{\partial y'} \delta y'\right) dx = 0.
\end{displaymath}">|; 

$key = q/r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img192.png"
 ALT="$r$">|; 

$key = q/{{bf{r}_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2696.png"
 ALT="${\bf r}_M$">|; 

$key = q/{mitOmega}=(I_{parallel}slashI_perp-1),omega_{z'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1384.png"
 ALT="${\mit\Omega}= (I_{\parallel}/I_\perp-1) \omega_{z'}$">|; 

$key = q/stackrel{displaystylerightarrow}{PS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3196.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{PS}$">|; 

$key = q/displaystylealeft[1-e,cos(n,t-alpha_0)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2815.png"
 ALT="$\displaystyle a\left[1-e \cos(n t-\alpha_0)\right],$">|; 

$key = q/{displaymath}frac{d{{bf{a}}{dt}equivleft(frac{da_x}{dt},frac{da_y}{dt},frac{da_z}{dt}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3365.png"
 ALT="\begin{displaymath}
\frac{d {\bf a}}{dt} \equiv \left(\frac{d a_x}{dt}, \frac{d a_y}{d t}, \frac{d a_z}{ d t}\right).
\end{displaymath}">|; 

$key = q/displaystyle|{{bf{a}|^2,|{{bf{b}|^2-({{bf{a}cdot{{bf{b})^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3312.png"
 ALT="$\displaystyle \vert{\bf a}\vert^2  \vert{\bf b}\vert^2 - ({\bf a}\cdot {\bf b})^2$">|; 

$key = q/{displaymath}int_{r_p}^rfrac{r,dr}{[2,r+(e-1),r^2slashr_p-(e+1),r_p]^{1slash2}}=sqrt{G,M},,t.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img795.png"
 ALT="\begin{displaymath}
\int_{r_p}^r\frac{r dr}{[2 r + (e-1) r^2/r_p - (e+1) r_p]^{1/2}} =
\sqrt{G M}  t.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter03slashfig3.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="255" BORDER="0"
 SRC="|."$dir".q|img216.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter03/fig3.01.eps}}
\end{figure}">|; 

$key = q/1.245;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3078.png"
 ALT="$1.245$">|; 

$key = q/0<ell1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img888.png"
 ALT="$0&lt;e\ll 1$">|; 

$key = q/{displaymath}l=intds=int_{-aslash2}^{aslash2}[1+y'^{,2}]^{1slash2},dx{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1763.png"
 ALT="\begin{displaymath}
l = \int ds = \int_{-a/2}^{a/2}[1+y'^{ 2}]^{1/2} dx
\end{displaymath}">|; 

$key = q/{displaymath}Phi'=-frac{G,m'}{R'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2217.png"
 ALT="\begin{displaymath}
\Phi' = -\frac{G m'}{R'}.
\end{displaymath}">|; 

$key = q/{displaymath}sum_{j=1,{calF}}left(M_{ij},deltaddot{q}_j-G_{ij},deltaq_jright)=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1910.png"
 ALT="\begin{displaymath}
\sum_{j=1,{\cal F}}\left(M_{ij} \delta\ddot{q}_j - G_{ij} \delta q_j\right) = 0,
\end{displaymath}">|; 

$key = q/displaystyleR_Z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2834.png"
 ALT="$\displaystyle R_Z$">|; 

$key = q/j=1,{calF};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1563.png"
 ALT="$j=1,{\cal F}$">|; 

$key = q/z=r,sinbeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2797.png"
 ALT="$z=r \sin\beta$">|; 

$key = q/{displaymath}K=frac{m}{2},(dot{q}_1^{,2}+dot{q}_2^{,2})+frac{M}{2},dot{q}_3^{,2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2029.png"
 ALT="\begin{displaymath}
K = \frac{m}{2} (\dot{q}_1^{ 2} + \dot{q}_2^{ 2})+ \frac{M}{2} \dot{q}_3^{ 2},
\end{displaymath}">|; 

$key = q/{displaymath}fsimeq-frac{G,M_0}{r^2}-frac{3,G,M_0,h^2}{c^2,r^4},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2465.png"
 ALT="\begin{displaymath}
f \simeq -\frac{G M_0}{r^2}-\frac{3 G M_0 h^2}{c^2 r^4},
\end{displaymath}">|; 

$key = q/displaystylen,{{bf{a}+n,{{bf{b}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3213.png"
 ALT="$\displaystyle n {\bf a} + n {\bf b}.$">|; 

$key = q/{{bf{a}cdot{{bf{b}times{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3339.png"
 ALT="${\bf a}\cdot {\bf b}\times {\bf c}$">|; 

$key = q/{displaymath}int_P^Qx,y^2,dl=int_0^1x^3,sqrt{2},dx=frac{sqrt{2}}{4}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3379.png"
 ALT="\begin{displaymath}
\int_P^Q x y^2 dl = \int_0^1 x^3 \sqrt{2}  dx = \frac{\sqrt{2}}{4}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{V}=V_0,(costheta,-sintheta,,0).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1164.png"
 ALT="\begin{displaymath}
{\bf V} = V_0 (\cos\theta,-\sin\theta, 0).
\end{displaymath}">|; 

$key = q/displaystyle-frac{mu_1,y}{rho_1^{,3}}-frac{mu_2,y}{rho_2^{,3}}+omega^2,y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2542.png"
 ALT="$\displaystyle - \frac{\mu_1 y}{\rho_1^{ 3}}- \frac{\mu_2 y}{\rho_2^{ 3}}
+ \omega^2 y,$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter13slashfig13.05.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="239" BORDER="0"
 SRC="|."$dir".q|img2624.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter13/fig13.05.eps}}
\end{figure}">|; 

$key = q/displaystyleleft{theta(0)-frac{A,(1-omega^2)}{[(1-omega^2)^2+omega^2slashQ^2]}right}{{rm{e}^{-tslash2Q},cos(omega_ast,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="401" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3044.png"
 ALT="$\displaystyle \left\{\theta(0) - \frac{A (1-\omega^2)}{[(1-\omega^2)^2+\omega^2/Q^2]}\right\}
{\rm e}^{-t/2Q} \cos(\omega_\ast  t)$">|; 

$key = q/{displaymath}u(theta)=frac{G,M}{h^2}left[1-e,cos(theta-theta_0)right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img748.png"
 ALT="\begin{displaymath}
u(\theta) = \frac{G M}{h^2}\left[1 - e \cos(\theta-\theta_0)\right],
\end{displaymath}">|; 

$key = q/{displaymath}W=|{{bf{F}|,(|{{bf{r}|,costheta)={{bf{F}cdot{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3281.png"
 ALT="\begin{displaymath}
W = \vert{\bf F}\vert  (\vert{\bf r}\vert \cos\theta) = {\bf F}\cdot {\bf r}.
\end{displaymath}">|; 

$key = q/m_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1842.png"
 ALT="$m_{ij}$">|; 

$key = q/omega=2slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$\omega =2/3$">|; 

$key = q/{displaymath}beta=iota,sin(F+lambda)+frac{3}{8},m,iota,sin(2,D-F).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3022.png"
 ALT="\begin{displaymath}
\beta = \iota \sin(F+\lambda) + \frac{3}{8} m \iota \sin(2 D-F).
\end{displaymath}">|; 

$key = q/{displaymath}q_i=q_{i,0}+deltaq_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1896.png"
 ALT="\begin{displaymath}
q_i = q_{i 0} + \delta q_i,
\end{displaymath}">|; 

$key = q/omega=2pislashT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img398.png"
 ALT="$\omega = 2\pi/T$">|; 

$key = q/{displaymath}s=s_+,{{rm{e}^{-{{rm{i},omega_+,t}+s_-,{{rm{e}^{-{{rm{i},omega_-,t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1204.png"
 ALT="\begin{displaymath}
s = s_+  {\rm e}^{-{\rm i} \omega_+ t}+s_-  {\rm e}^{-{\rm i} 
\omega_- t},
\end{displaymath}">|; 

$key = q/epsilon=0.065;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2190.png"
 ALT="$\epsilon=0.065$">|; 

$key = q/{displaymath}Gamma^2+(2-A),Gamma+(1-A),(1+2,A)=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="317" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2674.png"
 ALT="\begin{displaymath}
\Gamma^2 + (2-A) \Gamma + (1-A) (1+2 A) = 0,
\end{displaymath}">|; 

$key = q/{displaymath}tau=-m,g,l,sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img456.png"
 ALT="\begin{displaymath}
\tau = - m g l \sin\theta.
\end{displaymath}">|; 

$key = q/displaystylev_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1170.png"
 ALT="$\displaystyle v_{x'}$">|; 

$key = q/displaystyle-mu_1,frac{(xi-xi_1)}{rho_1^{,3}}-mu_2,frac{(xi-xi_2)}{rho_2^{,3}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2504.png"
 ALT="$\displaystyle -\mu_1 \frac{(\xi-\xi_1)}{\rho_1^{ 3}} - \mu_2 \frac{(\xi-\xi_2)}{\rho_2^{ 3}},$">|; 

$key = q/x',y';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img557.png"
 ALT="$x'  y'$">|; 

$key = q/displaystyle-frac{G,M_m,M}{a_m}+frac{G,M_m,(I_parallel-I_perp)}{2,a_m^{,3}}left[3,sin^2theta,sin^2(lambda_m-phi)right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="437" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2319.png"
 ALT="$\displaystyle - \frac{G M_m M}{a_m} + \frac{G M_m (I_\parallel-I_\perp)}{2 a_m^{ 3}}\left[3 \sin^2\theta \sin^2(\lambda_m-\phi)\right.$">|; 

$key = q/displaystyle-frac{1}{2},e^2-frac{1}{4},iota^2-e,cos[(1+c,m^2),T-alpha_0]+frac{1}{2},e^2,cos[2,(1+c,m^2),T-2,alpha_0];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="590" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2956.png"
 ALT="$\displaystyle -\frac{1}{2} e^2-\frac{1}{4} \iota^2 - e \cos[(1+c m^2) T-\alpha_0] + \frac{1}{2} e^2 \cos[2 (1+c m^2) T-2 \alpha_0]$">|; 

$key = q/{{bf{F}=-{{bf{r}slash|r^3|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3409.png"
 ALT="${\bf F} = - {\bf r}/ \vert r^3\vert$">|; 

$key = q/{displaymath}omega_{pm}simeq{mitOmega},sinlambdapmsqrt{frac{g}{l}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1202.png"
 ALT="\begin{displaymath}
\omega_{\pm} \simeq {\mit\Omega} \sin\lambda \pm \sqrt{\frac{g}{l}},
\end{displaymath}">|; 

$key = q/OP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1704.png"
 ALT="$OP$">|; 

$key = q/phi_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img297.png"
 ALT="$\phi_0 = 0$">|; 

$key = q/d{{bf{L}slashdt=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1410.png"
 ALT="$d{\bf L}/dt=0$">|; 

$key = q/dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3384.png"
 ALT="$dx$">|; 

$key = q/{displaymath}z=frac{v_t}{g},left(v_0,sintheta+v_tright)left(1-{{rm{e}^{-g,tslashv_t}right)-v_t,t.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img594.png"
 ALT="\begin{displaymath}
z = \frac{v_t}{g} \left(v_0 \sin\theta + v_t\right)\left(1-{\rm e}^{-g t/v_t}\right) - v_t t.
\end{displaymath}">|; 

$key = q/{{bf{e}_yequiv(0,,1,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3224.png"
 ALT="${\bf e}_y\equiv (0, 1, 0)$">|; 

$key = q/P_n(costheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2114.png"
 ALT="$P_n(\cos\theta)$">|; 

$key = q/dsigma(psi)slashd{mitOmega}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1074.png"
 ALT="$d\sigma(\psi)/d{\mit\Omega}'$">|; 

$key = q/displaystylefrac{1}{2},frac{partialdot{x}_j^{,2}}{partialq_i}=dot{x}_j,frac{partialdot{x}_j}{partialq_i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1594.png"
 ALT="$\displaystyle \frac{1}{2} \frac{\partial \dot{x}_j^{ 2}}{\partial q_i}
= \dot{x}_j \frac{\partial \dot{x}_j}{\partial q_i}$">|; 

$key = q/-1lequleq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1514.png"
 ALT="$-1\leq u\leq 1$">|; 

$key = q/theta<alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1406.png"
 ALT="$\theta&lt;\alpha$">|; 

$key = q/Omega=1.76times10^{-4},{{rm{rad.slashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2187.png"
 ALT="$\Omega = 1.76\times 10^{-4} {\rm rad./s}$">|; 

$key = q/displaystylesqrt{-lambda_2}=sqrt{frac{2,k+k'}{m}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2006.png"
 ALT="$\displaystyle \sqrt{-\lambda_2} = \sqrt{\frac{2 k+k'}{m}}.$">|; 

$key = q/-Omega_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2298.png"
 ALT="$-\Omega_\phi$">|; 

$key = q/{displaymath}deltaH=sum_{i=1,{calF}}left(frac{partialH}{partialp_i},deltap_i+frac{partialH}{partialq_i},delta{q}_iright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1852.png"
 ALT="\begin{displaymath}
\delta H =\sum_{i=1,{\cal F}} \left(\frac{\partial H}{\parti...
...ta p_i +
\frac{\partial H}{\partial q_i} \delta{q}_i\right).
\end{displaymath}">|; 

$key = q/P_2(0)=-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2383.png"
 ALT="$P_2(0)=-1/2$">|; 

$key = q/lambda_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2268.png"
 ALT="$\lambda_s$">|; 

$key = q/-pislash2<omega,t<pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img518.png"
 ALT="$-\pi/2&lt; \omega t&lt; \pi/2$">|; 

$key = q/x=b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img490.png"
 ALT="$x=b$">|; 

$key = q/{displaymath}dot{L}_phi=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1492.png"
 ALT="\begin{displaymath}
\dot{L}_\phi = 0,
\end{displaymath}">|; 

$key = q/2.86times10^{-4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2435.png"
 ALT="$2.86\times 10^{-4}$">|; 

$key = q/{displaymath}m,{{bf{v}!cdot!frac{d{{bf{v}}{dt}=frac{m}{2}frac{d({{bf{v}cdot{{bf{dt}=frac{m}{2}frac{dv^2}{dt}={{bf{f}cdot{{bf{v}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="\begin{displaymath}
m {\bf v}\!\cdot\!\frac{d {\bf v}}{dt} =
\frac{m}{2}\frac{...
...f v})}{dt}=\frac{m}{2}\frac{d v^2}{dt} =
{\bf f}\cdot{\bf v}.
\end{displaymath}">|; 

$key = q/{{bf{r}simeq{{bf{R};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1128.png"
 ALT="${\bf r}\simeq {\bf R}$">|; 

$key = q/{displaymath}cosgamma_s=frac{mbox{{boldmath{{omega{}cdot{{bf{r}_s}{|mbox{{boldmath{{omega{}|,|{{bf{r}_s|}=sintheta,sinlambda_s,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2278.png"
 ALT="\begin{displaymath}
\cos\gamma_s =\frac{\mbox{\boldmath$\omega$}\cdot{\bf r}_s}{...
...omega$}\vert \vert{\bf r}_s\vert}= \sin\theta \sin\lambda_s,
\end{displaymath}">|; 

$key = q/displaystylem_2,frac{d^2{{bf{r}_2}{dt^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img917.png"
 ALT="$\displaystyle m_2 \frac{d^2{\bf r}_2}{dt^2}$">|; 

$key = q/q_i=q_{i,0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1890.png"
 ALT="$q_i=q_{i 0}$">|; 

$key = q/lambda_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1291.png"
 ALT="$\lambda_i$">|; 

$key = q/{displaymath}{{bf{A}=A_r,{{bf{e}_r+A_theta,{{bf{e}_theta+A_phi,{{bf{e}_phi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img3498.png"
 ALT="\begin{displaymath}
{\bf A} = A_r {\bf e}_r + A_\theta {\bf e}_\theta+ A_\phi {\bf e}_\phi,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=1.8in{centerline{epsffile{Chapter06slashfig6.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="511" HEIGHT="200" BORDER="0"
 SRC="|."$dir".q|img1021.png"
 ALT="\begin{figure}
\epsfysize =1.8in
\centerline{\epsffile{Chapter06/fig6.04.eps}}
\end{figure}">|; 

$key = q/{{bf{b}slashbeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3545.png"
 ALT="${\bf b}/\beta$">|; 

$key = q/displaystyleI_{y'y'},dot{omega}_{y'}-(I_{z'z'}-I_{x'x'}),omega_{z'},omega_{x'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1365.png"
 ALT="$\displaystyle I_{y'y'} \dot{\omega}_{y'} - (I_{z'z'}-I_{x'x'}) \omega_{z'} \omega_{x'},$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter13slashfig13.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="249" BORDER="0"
 SRC="|."$dir".q|img2535.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter13/fig13.02.eps}}
\end{figure}">|; 

$key = q/displaystyleb_5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2928.png"
 ALT="$\displaystyle b_5$">|; 

$key = q/3.23times10^{-7};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2429.png"
 ALT="$3.23\times 10^{-7}$">|; 

$key = q/displaystyledeltaddot{x}-2,deltadot{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2645.png"
 ALT="$\displaystyle \delta\ddot{x} - 2 \delta\dot{y}$">|; 

$key = q/{displaymath}X_n=frac{2}{T}int_0^TX(t),cos(n,omega,t),dt{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img409.png"
 ALT="\begin{displaymath}
X_n = \frac{2}{T} \int_0^T X(t) \cos(n \omega t) dt
\end{displaymath}">|; 

$key = q/{displaymath}frac{partial}{partialx'}=left(frac{partialx}{partialx'}right)_{y',zpartialx'}right)_{y',z'}frac{partial}{partialz},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img3482.png"
 ALT="\begin{displaymath}
\frac{\partial}{\partial x'} = \left(\frac{\partial x}{\part...
...l z}{\partial x'} \right)_{y',z'}
\frac{\partial}{\partial z},
\end{displaymath}">|; 

$key = q/{{bf{l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3324.png"
 ALT="${\bf l}$">|; 

$key = q/g=G,Mslasha^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2144.png"
 ALT="$g=G M/a^2$">|; 

$key = q/displaystylesum_{j>0}b_j,sin(omega_j,T-alpha_j),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2833.png"
 ALT="$\displaystyle \sum_{j&gt;0} b_j \sin(\omega_j T-\alpha_j),$">|; 

$key = q/ddot{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1641.png"
 ALT="$\ddot{x}$">|; 

$key = q/{displaymath}{{bf{x}_k^T,{{bf{M},{{bf{x}_l=delta_{kl},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1947.png"
 ALT="\begin{displaymath}
{\bf x}_k^T {\bf M} {\bf x}_l = \delta_{kl},
\end{displaymath}">|; 

$key = q/{{bf{v}_i=d{{bf{r}_islashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1239.png"
 ALT="${\bf v}_i = d{\bf r}_i/dt$">|; 

$key = q/displaystyledeltaphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2356.png"
 ALT="$\displaystyle \delta\phi$">|; 

$key = q/F=x^3,y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3401.png"
 ALT="$F=x^3 y$">|; 

$key = q/E_2'=(1slash2),m_2,v_2'^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1040.png"
 ALT="$E_2'=(1/2) m_2 v_2'^{ 2}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter03slashfig3.07.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="272" BORDER="0"
 SRC="|."$dir".q|img440.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter03/fig3.07.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{Chapter12slashfig12.05.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="314" BORDER="0"
 SRC="|."$dir".q|img2215.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{Chapter12/fig12.05.eps}}
\end{figure}">|; 

$key = q/lambda_k=-omega_k^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1961.png"
 ALT="$\lambda_k = -\omega_k^{ 2}$">|; 

$key = q/dot{q}_1,dot{q}_2,cdots,dot{q}_{calF},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1788.png"
 ALT="$\dot{q}_1,\dot{q}_2,\cdots,\dot{q}_{\cal F},t)$">|; 

$key = q/A=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img568.png"
 ALT="$A=2$">|; 

$key = q/trightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img500.png"
 ALT="$t\rightarrow \infty$">|; 

$key = q/U_0=U(x_0,y_0,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2640.png"
 ALT="$U_0=U(x_0,y_0,0)$">|; 

$key = q/C_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2630.png"
 ALT="$C_3$">|; 

$key = q/{displaymath}ddot{r}-frac{h^2}{r^3}=-frac{G,M}{r^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img743.png"
 ALT="\begin{displaymath}
\ddot{r} -\frac{h^2}{r^3}= - \frac{G M}{r^2}.
\end{displaymath}">|; 

$key = q/displaystyleI_{yy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1245.png"
 ALT="$\displaystyle I_{yy}$">|; 

$key = q/{displaymath}frac{d}{dt}left({{bf{a}cdot{{bf{b}right)=frac{d{{bf{a}}{dt}cdot{{bf{b}+{{bf{a}cdotfrac{d{{bf{b}}{dt},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img3370.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\left({\bf a}\cdot{\bf b}\right) = \frac{d{\bf a}}{dt}\cdot {\bf b} +{\bf a}\cdot\frac{d{\bf b}}{dt},
\end{displaymath}">|; 

$key = q/{displaymath}v=v_tleft(1-{{rm{e}^{-xslashx_t}right)^{1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img274.png"
 ALT="\begin{displaymath}
v = v_t\left(1-{\rm e}^{-x/x_t}\right)^{1/2}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter12slashfig12.08.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="351" HEIGHT="327" BORDER="0"
 SRC="|."$dir".q|img2462.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter12/fig12.08.eps}}
\end{figure}">|; 

$key = q/dot{omega}_{x'}=dot{omega}_{y'}=dot{omega}_{z'}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1369.png"
 ALT="$\dot{\omega}_{x'}=\dot{\omega}_{y'} = \dot{\omega}_{z'}=0$">|; 

$key = q/displaystylelambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1826.png"
 ALT="$\displaystyle \lambda$">|; 

$key = q/{displaymath}frac{mu_2}{3,mu_1}=frac{rho_2^{,3},(1+rho_2+rho_2^{,2}slash3)}{(1+rho_2)^2,(1-rho_2^{,3})}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2583.png"
 ALT="\begin{displaymath}
\frac{\mu_2}{3 \mu_1} = \frac{\rho_2^{ 3} (1+\rho_2+\rho_2^{ 2}/3)}{(1+\rho_2)^2 (1-\rho_2^{ 3})}.
\end{displaymath}">|; 

$key = q/displaystylea,(1-e,cosE),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img818.png"
 ALT="$\displaystyle a (1-e \cos E),$">|; 

$key = q/{displaymath}t_f=frac{2,v_0,sintheta}{g}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img598.png"
 ALT="\begin{displaymath}
t_f = \frac{2 v_0 \sin\theta}{g}
\end{displaymath}">|; 

$key = q/deltaX;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2782.png"
 ALT="$\delta X$">|; 

$key = q/displaystyle-m^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2923.png"
 ALT="$\displaystyle -m^2,$">|; 

$key = q/{displaymath}{{bf{a}times({{bf{b}times{{bf{c})equiv({{bf{a}cdot{{bf{c}),{{bf{b}-({{bf{a}cdot{{bf{b}),{{bf{c}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3352.png"
 ALT="\begin{displaymath}
{\bf a}\times({\bf b} \times {\bf c}) \equiv ({\bf a}\cdot {\bf c})  {\bf b} - ({\bf a}\cdot
{\bf b}) {\bf c}
\end{displaymath}">|; 

$key = q/{{bf{A}={{bf{grad},V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3466.png"
 ALT="${\bf A} = {\bf grad} V$">|; 

$key = q/1slash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img378.png"
 ALT="$1/4$">|; 

$key = q/j=i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="$j=i$">|; 

$key = q/1.4^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2368.png"
 ALT="$1.4^\circ$">|; 

$key = q/frac{3}{2},e,iota;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2894.png"
 ALT="$\frac{3}{2} e \iota$">|; 

$key = q/displaystyle-m^2,cos[2,(1-m),T]-frac{15}{8},m,e,cos[(1-2,m-c,m^2),T+alpha_0],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="504" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2967.png"
 ALT="$\displaystyle - m^2 \cos[2 (1-m) T]-\frac{15}{8} m e \cos[(1-2 m-c m^2) T+\alpha_0],$">|; 

$key = q/{{bf{e}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img676.png"
 ALT="${\bf e}_r$">|; 

$key = q/delta_{kl}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1950.png"
 ALT="$\delta_{kl}=0$">|; 

$key = q/lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1120.png"
 ALT="$\lambda$">|; 

$key = q/{displaymath}ddot{theta}+frac{g}{l},sintheta-h^2,frac{costheta}{sin^3theta}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1683.png"
 ALT="\begin{displaymath}
\ddot{\theta} + \frac{g}{l} \sin\theta - h^2 \frac{\cos\theta}{\sin^3\theta} = 0,
\end{displaymath}">|; 

$key = q/{displaymath}Phi(r,theta)=-frac{G,M}{r}+frac{G,(I_parallel-I_perp)}{r^3},P_2(costheta).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2200.png"
 ALT="\begin{displaymath}
\Phi(r,\theta) = - \frac{G M}{r} + \frac{G (I_\parallel - I_\perp)}{r^3} P_2(\cos\theta).
\end{displaymath}">|; 

$key = q/2pislashOmega_n=18.6,{{rm{years};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2317.png"
 ALT="$2\pi/\Omega_n=18.6 {\rm years}$">|; 

$key = q/T_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img301.png"
 ALT="$T_0$">|; 

$key = q/dl=sqrt{2},dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3378.png"
 ALT="$dl= \sqrt{2}  dx$">|; 

$key = q/{{bf{e}_phi=nablaphislash|nablaphi|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3499.png"
 ALT="${\bf e}_\phi = \nabla\phi/\vert\nabla\phi\vert$">|; 

$key = q/{displaymath}alpha=frac{1}{2}tan^{-1}left(frac{3}{2}frac{a,b}{a^2-b^2}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1350.png"
 ALT="\begin{displaymath}
\alpha = \frac{1}{2}\tan^{-1}\left(\frac{3}{2}\frac{a b}{a^2-b^2}\right).
\end{displaymath}">|; 

$key = q/{displaymath}int_{v_{x,0}}^vfrac{dv_x}{v_x}=-frac{g}{v_t},t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img582.png"
 ALT="\begin{displaymath}
\int_{v_{x 0}}^v\frac{dv_x}{v_x} = - \frac{g}{v_t} t,
\end{displaymath}">|; 

$key = q/{{bf{e}_{z''};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1422.png"
 ALT="${\bf e}_{z''}$">|; 

$key = q/displaystyle(I_{xx}-lambda),cosalpha+I_{xy},sinalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1337.png"
 ALT="$\displaystyle (I_{xx}-\lambda) \cos\alpha + I_{xy} \sin\alpha$">|; 

$key = q/dslashdtllomega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2347.png"
 ALT="$d/dt\ll \omega$">|; 

$key = q/{displaymath}r_p=frac{r_c}{1+e}=a,(1-e).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img757.png"
 ALT="\begin{displaymath}
r_p = \frac{r_c}{1+e} = a (1-e).
\end{displaymath}">|; 

$key = q/{displaymath}frac{d{{bf{P}}{dt}={{bf{F},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img167.png"
 ALT="\begin{displaymath}
\frac{d{\bf P}}{dt} = {\bf F},
\end{displaymath}">|; 

$key = q/C=C_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$C= C_2$">|; 

$key = q/{displaymath}ddot{{bf{r}_E=-n'^{,2},a'^{,3},frac{{{bf{r}_E}{|{{bf{r}_E|^{,3}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2697.png"
 ALT="\begin{displaymath}
\ddot{\bf r}_E = - n'^{ 2} a'^{ 3} \frac{{\bf r}_E}{\vert{\bf r}_E\vert^{ 3}},
\end{displaymath}">|; 

$key = q/{{bf{r}cdot{{bf{n}=c,|{{bf{r}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3528.png"
 ALT="${\bf r}\cdot{\bf n} = c \vert{\bf r}\vert$">|; 

$key = q/v_{x,0}=v_0,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img583.png"
 ALT="$v_{x 0} = v_0 \cos\theta$">|; 

$key = q/displaystyleleft(frac{1+e}{1-e}right)^{1slash2}tan(Eslash2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img820.png"
 ALT="$\displaystyle \left(\frac{1+e}{1-e}\right)^{1/2} \tan (E/2).$">|; 

$key = q/dot{theta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img867.png"
 ALT="$\dot{\theta}$">|; 

$key = q/{displaymath}int_P^Qleft[y,dx+x^3,dyright]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img3388.png"
 ALT="\begin{displaymath}
\int_P^Q \left[ y dx + x^3 dy\right]
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{A},{{bf{x}_i=lambda_i,{{bf{x}_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1288.png"
 ALT="\begin{displaymath}
{\bf A} {\bf x}_i = \lambda_i {\bf x}_i,
\end{displaymath}">|; 

$key = q/{displaymath}J_n=-2pi,G,intintr^{,2+n},rho(r,theta),P_n(costheta),sintheta,dr,dtheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="398" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2152.png"
 ALT="\begin{displaymath}
J_n = - 2\pi G \int\int r^{ 2+n} \rho(r,\theta) P_n(\cos\theta) \sin\theta dr d\theta.
\end{displaymath}">|; 

$key = q/rho_1simeqr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2524.png"
 ALT="$\rho_1\simeq r$">|; 

$key = q/{displaymath}frac{da_x}{dt}=frac{d}{dt}!left(phi,b_xright)=frac{dphi}{dt},b_x+phi,frac{db_x}{dt},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3368.png"
 ALT="\begin{displaymath}
\frac{d a_x}{dt} = \frac{d}{dt}\!\left(\phi  b_x\right) = \frac{d\phi}{dt}  b_x + \phi  
\frac{d b_x}{dt},
\end{displaymath}">|; 

$key = q/displaystyleomega,sinalpha,cos({mitOmega},t'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1393.png"
 ALT="$\displaystyle \omega \sin\alpha \cos({\mit\Omega} t'),$">|; 

$key = q/r^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img861.png"
 ALT="$r^{-3}$">|; 

$key = q/{displaymath}rho_1=rho_2=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2598.png"
 ALT="\begin{displaymath}
\rho_1 = \rho_2 =1,
\end{displaymath}">|; 

$key = q/{displaymath}F=-c,v^n,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img493.png"
 ALT="\begin{displaymath}
F = - c v^n,
\end{displaymath}">|; 

$key = q/displaystyleddot{eta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2505.png"
 ALT="$\displaystyle \ddot{\eta}$">|; 

$key = q/{displaymath}int_{t_1}^{t_2}left{left[frac{partialL}{partialq_1}-frac{d}{dt}!lef{1}{partialfslashpartialq_2}right}deltaq_1,dt=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="617" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1802.png"
 ALT="\begin{displaymath}
\int_{t_1}^{t_2}\left\{\left[
\frac{\partial L}{\partial q_1...
...ht]\frac{1}{\partial f/\partial q_2}\right\} \delta q_1 dt=0.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{g}_c=-omega^2,rho,{{bf{e}_z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2210.png"
 ALT="\begin{displaymath}
{\bf g}_c = - \omega^2 \rho {\bf e}_z.
\end{displaymath}">|; 

$key = q/displaystyleleft[3+frac{R_i,f'(R_i)}{f(R_i)}right]^{-1slash2}=1+frac{3}{4}sum_{jright)^2+frac{175}{64}left(frac{R_j}{R_i}right)^4+cdotsright];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="664" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2405.png"
 ALT="$\displaystyle \left[3 + \frac{R_i f'(R_i)}{f(R_i)}\right]^{-1/2} =1 + \frac{3}...
..._j}{R_i}\right)^2 + \frac{175}{64}\left(\frac{R_j}{R_i}
\right)^4+\cdots\right]$">|; 

$key = q/displaystyleR_i,f'(R_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2401.png"
 ALT="$\displaystyle R_i f'(R_i)$">|; 

$key = q/partialTslashpartialxequiv(partialTslashpartialx)_{y,z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3456.png"
 ALT="$\partial T/\partial x\equiv (\partial T/\partial x)_{y, z}$">|; 

$key = q/{displaymath}n>-3.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img860.png"
 ALT="\begin{displaymath}
n &gt; -3.
\end{displaymath}">|; 

$key = q/{displaymath}x=x_0,{{rm{e}^{-nu,t},cos(omega_r,t-phi_0),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img345.png"
 ALT="\begin{displaymath}
x = x_0 {\rm e}^{-\nu t} \cos(\omega_r t-\phi_0),
\end{displaymath}">|; 

$key = q/displaystyler_p,(1+P^2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img826.png"
 ALT="$\displaystyle r_p (1+P^2),$">|; 

$key = q/-pislash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img299.png"
 ALT="$-\pi/4$">|; 

$key = q/displaystyle15.3^{''}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2374.png"
 ALT="$\displaystyle 15.3^{''}.$">|; 

$key = q/r_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img902.png"
 ALT="$r_0$">|; 

$key = q/19.50;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2456.png"
 ALT="$19.50$">|; 

$key = q/x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img346.png"
 ALT="$x_0$">|; 

$key = q/{displaymath}E=frac{1}{2},m,omega_0^{,2},x_0^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img361.png"
 ALT="\begin{displaymath}
E = \frac{1}{2} m \omega_0^{ 2} x_0^{ 2}.
\end{displaymath}">|; 

$key = q/Q_i,q_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1577.png"
 ALT="$Q_i q_i$">|; 

$key = q/{displaymath}f(r_c)+frac{r_c}{3},f'(r_c)<0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img857.png"
 ALT="\begin{displaymath}
f(r_c) + \frac{r_c}{3} f'(r_c)&lt;0.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.30.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="274" BORDER="0"
 SRC="|."$dir".q|img3148.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.30.eps}}
\end{figure}">|; 

$key = q/{displaymath}{mitTheta}=int_0^{u_{max}}frac{b,du}{sqrt{1-b^2,u^2-U(u)slashE}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img966.png"
 ALT="\begin{displaymath}
{\mit\Theta} = \int_0^{u_{max}}\frac{b du}{\sqrt{1-b^2 u^2- U(u)/E}}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{sina}{A}=frac{sinb}{B}=frac{sinc}{C}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3510.png"
 ALT="\begin{displaymath}
\frac{\sin a}{A} = \frac{\sin b}{B} = \frac{\sin c}{C}
\end{displaymath}">|; 

$key = q/{calE}={calE}_1'+{calE}_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1044.png"
 ALT="${\cal E} = {\cal E}_1' + {\cal E}_2'$">|; 

$key = q/{displaymath}deltaW=sum_{j=1,{calF}}f_jsum_{i=1,{calF}}frac{partialx_j}{partialq_i},deltaq_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1573.png"
 ALT="\begin{displaymath}
\delta W = \sum_{j=1,{\cal F}} f_j\sum_{i=1,{\cal F}}\frac{\partial x_j}{\partial q_i} \delta q_i.
\end{displaymath}">|; 

$key = q/2slash3pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img897.png"
 ALT="$2/3\pi$">|; 

$key = q/t_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="$t_1$">|; 

$key = q/{displaymath}{{bf{G},{{bf{x}_k=lambda_k,{{bf{M},{{bf{x}_k.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1924.png"
 ALT="\begin{displaymath}
{\bf G} {\bf x}_k = \lambda_k {\bf M} {\bf x}_k.
\end{displaymath}">|; 

$key = q/deltatheta_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3327.png"
 ALT="$\delta \theta_z$">|; 

$key = q/sqrt{gslashl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1211.png"
 ALT="$\sqrt{g/l}$">|; 

$key = q/displaystyle{{bf{r}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2709.png"
 ALT="$\displaystyle {\bf r}'$">|; 

$key = q/z'siml,theta^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1183.png"
 ALT="$z'\sim l \theta^{ 2}$">|; 

$key = q/{displaymath}a_c=omega^2,rho{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2206.png"
 ALT="\begin{displaymath}
a_c = \omega^2 \rho
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{r}_{cm}=frac{sum_{i=1}^Nm_i,{{bf{r}_i}{sum_{i=1}^Nm_i}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img147.png"
 ALT="\begin{displaymath}
{\bf r}_{cm} = \frac{\sum_{i=1}^N m_i {\bf r}_i}{\sum_{i=1}^N m_i}.
\end{displaymath}">|; 

$key = q/M=sum_{i=1}^Nm_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="$M = \sum_{i=1}^N m_i$">|; 

$key = q/{displaymath}frac{partial}{partialx'}=costheta,frac{partial}{partialx}+sintheta,frac{partial}{partialy},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img3483.png"
 ALT="\begin{displaymath}
\frac{\partial}{\partial x'} = \cos\theta  \frac{\partial}{\partial x} +
\sin\theta  \frac{\partial}{\partial y},
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2x}{dt^2}+2,nu,frac{dx}{dt}+omega_0^{,2},x=omega_0^{,2},sum_{n=0}^inftyX_n,{{rm{e}^{-{{rm{i},n,omega,t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="61" BORDER="0"
 SRC="|."$dir".q|img411.png"
 ALT="\begin{displaymath}
\frac{d^2 x}{dt^2} + 2 \nu \frac{dx}{dt} + \omega_0^{ 2}\...
... 2} \sum_{n=0}^\infty X_n {\rm e}^{-{\rm i} n \omega t},
\end{displaymath}">|; 

$key = q/{displaymath}x(t)=sum_{n=0}^{infty}X_n,x_n,cos(n,omega,t-phi_n),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="61" BORDER="0"
 SRC="|."$dir".q|img419.png"
 ALT="\begin{displaymath}
x(t) = \sum_{n=0}^{\infty} X_n x_n \cos(n \omega t-\phi_n),
\end{displaymath}">|; 

$key = q/oint;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3474.png"
 ALT="$\oint$">|; 

$key = q/H=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img836.png"
 ALT="$H=0$">|; 

$key = q/displaystylex_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2922.png"
 ALT="$\displaystyle x_4$">|; 

$key = q/I_perp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1482.png"
 ALT="$I_\perp$">|; 

$key = q/dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1263.png"
 ALT="$dV$">|; 

$key = q/r,theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1867.png"
 ALT="$r,\theta$">|; 

$key = q/(r,,theta,,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2072.png"
 ALT="$(r,  \theta,  \phi)$">|; 

$key = q/{{bf{g}_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2166.png"
 ALT="${\bf g}_c$">|; 

$key = q/{displaymath}U=-rho,g,inty,ds=-rho,g,int_{-aslash2}^{aslash2}y,[1+y'^{,2}]^{1slash2},dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1761.png"
 ALT="\begin{displaymath}
U = - \rho g \int y ds = - \rho g \int_{-a/2}^{a/2} y [1+y'^{ 2}]^{1/2} dx,
\end{displaymath}">|; 

$key = q/displaystyleOmega_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2354.png"
 ALT="$\displaystyle \Omega_\phi$">|; 

$key = q/rho';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2263.png"
 ALT="$\rho'$">|; 

$key = q/{{bf{u}equiv(u,0,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="${\bf u} \equiv (u,0,0)$">|; 

$key = q/+infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img829.png"
 ALT="$+\infty$">|; 

$key = q/T_0=2pislashomega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img295.png"
 ALT="$T_0 = 2\pi/\omega_0$">|; 

$key = q/r_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img849.png"
 ALT="$r_c$">|; 

$key = q/displaystylev_perp,sin({mitOmega},t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img634.png"
 ALT="$\displaystyle v_\perp \sin({\mit\Omega} t),$">|; 

$key = q/{{bf{F}=sum_{i=1,N}{{bf{F}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1234.png"
 ALT="${\bf F} = \sum_{i=1,N} {\bf F}_i$">|; 

$key = q/Omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1221.png"
 ALT="$\Omega$">|; 

$key = q/{displaymath}T=sqrt{frac{4pi^2,a^3}{G,M}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img938.png"
 ALT="\begin{displaymath}
T = \sqrt{\frac{4\pi^2 a^3}{G M}}.
\end{displaymath}">|; 

$key = q/displaystyle-{{bf{f},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img916.png"
 ALT="$\displaystyle -{\bf f},$">|; 

$key = q/{displaymath}frac{l}{2,h}=sinhleft(frac{a}{2,h}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1780.png"
 ALT="\begin{displaymath}
\frac{l}{2 h} = \sinh\left(\frac{a}{2 h}\right).
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{AppendixAslashfigA.10.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="436" HEIGHT="228" BORDER="0"
 SRC="|."$dir".q|img3375.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{AppendixA/figA.10.eps}}
\end{figure}">|; 

$key = q/{displaymath}{{bf{R}'={{bf{R}-{{bf{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img2219.png"
 ALT="\begin{displaymath}
{\bf R}' = {\bf R} - {\bf r},
\end{displaymath}">|; 

$key = q/{calE};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img668.png"
 ALT="${\cal E}$">|; 

$key = q/{displaymath}Phi(r)=-frac{G,M}{r}left[1+frac{1}{4}left(frac{a}{r}right)^2+frac{9}{64}left(frac{a}{r}right)^4+cdotsright].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="384" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2386.png"
 ALT="\begin{displaymath}
\Phi(r) = - \frac{G M}{r}\left[1 + \frac{1}{4}\left(\frac{a...
...ght)^2 + \frac{9}{64}\left(\frac{a}{r}\right)^4+\cdots\right].
\end{displaymath}">|; 

$key = q/displaystyle+frac{1}{2},e,iota,sin[(2+c,m^2+g,m^2),T-alpha_0-gamma_0];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2963.png"
 ALT="$\displaystyle +\frac{1}{2} e \iota \sin [(2+c m^2+g m^2) T-\alpha_0-\gamma_0]$">|; 

$key = q/theta+2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3023.png"
 ALT="$\theta +2\pi$">|; 

$key = q/{displaymath}frac{1}{2},frac{partialdot{x}_j^{,2}}{partialdot{q}_i}=dot{x}_j,frac{partialdot{x}_j}{partialdot{q}_i},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1593.png"
 ALT="\begin{displaymath}
\frac{1}{2}  \frac{\partial\dot{x}_j^{ 2}}{\partial \dot{q}_i}
= \dot{x}_j \frac{\partial \dot{x}_j}{\partial \dot{q}_i},
\end{displaymath}">|; 

$key = q/b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img565.png"
 ALT="$b$">|; 

$key = q/dot{theta}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2291.png"
 ALT="$\dot{\theta}=0$">|; 

$key = q/x=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1733.png"
 ALT="$x=a$">|; 

$key = q/{displaymath}f(r)=-frac{k}{r^2}-frac{epsilon}{r^4},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img878.png"
 ALT="\begin{displaymath}
f(r) = - \frac{k}{r^2} - \frac{\epsilon}{r^4},
\end{displaymath}">|; 

$key = q/x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img204.png"
 ALT="$x$">|; 

$key = q/lambda_i=lambda_i^{,ast};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1296.png"
 ALT="$\lambda_i=\lambda_i^{ \ast}$">|; 

$key = q/displaystyleomega,sinalpha,sin({mitOmega},t'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1394.png"
 ALT="$\displaystyle \omega \sin\alpha \sin({\mit\Omega} t'),$">|; 

$key = q/d{mitOmega}'=2pi,sinpsi,dpsi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1075.png"
 ALT="$d{\mit\Omega}'=2\pi \sin\psi d\psi$">|; 

$key = q/displaystylex_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2906.png"
 ALT="$\displaystyle x_2$">|; 

$key = q/x=y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3377.png"
 ALT="$x=y$">|; 

$key = q/displaystyle(-sintheta,,costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img681.png"
 ALT="$\displaystyle (-\sin\theta,  \cos\theta),$">|; 

$key = q/(duslashdvartheta)_{u_{max}}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img969.png"
 ALT="$(du/d\vartheta)_{u_{max}}=0$">|; 

$key = q/{displaymath}({{bf{b}-{{bf{a})cdot({{bf{b}-{{bf{a})=|{{bf{a}|^2+|{{bf{b}|^2-2,{{bf{a}cdot{{bf{b}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3267.png"
 ALT="\begin{displaymath}
({\bf b} - {\bf a} ) \cdot ({\bf b} - {\bf a} ) = \vert{\bf a}\vert^2 + \vert{\bf b}\vert^2 - 2 {\bf a} \cdot
{\bf b}.
\end{displaymath}">|; 

$key = q/n,{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3205.png"
 ALT="$n {\bf a}$">|; 

$key = q/{displaymath}r=frac{r_c}{1-e,costheta},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img713.png"
 ALT="\begin{displaymath}
r = \frac{r_c}{1-e \cos\theta},
\end{displaymath}">|; 

$key = q/{{bf{r}_jtimes{{bf{f}_{ji};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.png"
 ALT="${\bf r}_j\times {\bf f}_{ji}$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{Chapter09slashfig9.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="490" HEIGHT="197" BORDER="0"
 SRC="|."$dir".q|img1643.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{Chapter09/fig9.03.eps}}
\end{figure}">|; 

$key = q/displaystyle+frac{15}{4},m,e,sin[(1-2,m-c,m^2),T+alpha_0],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="317" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2961.png"
 ALT="$\displaystyle +\frac{15}{4} m e \sin[(1-2 m-c m^2) T+\alpha_0],$">|; 

$key = q/y(x)rightarrowy(x)+deltay(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1721.png"
 ALT="$y(x)\rightarrow y(x)+\delta y(x)$">|; 

$key = q/displaystylecostheta,costheta',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2095.png"
 ALT="$\displaystyle \cos\theta \cos\theta',$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{Chapter15slashfig15.16.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="539" HEIGHT="402" BORDER="0"
 SRC="|."$dir".q|img3083.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{Chapter15/fig15.16.eps}}
\end{figure}">|; 

$key = q/I_parallelslashI_perp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1555.png"
 ALT="$I_\parallel/I_\perp$">|; 

$key = q/{calE}_1'=(1slash2),m_1,V_1'^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1042.png"
 ALT="${\cal E}_1'=(1/2) m_1 V_1'^{ 2}$">|; 

$key = q/displaystyle-frac{G,m}{a}-frac{G,m'}{R};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2233.png"
 ALT="$\displaystyle - \frac{G m}{a} - \frac{G m'}{R}$">|; 

$key = q/I_{xx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1255.png"
 ALT="$I_{xx}$">|; 

$key = q/{displaymath}m,frac{d{{bf{v}'}{dt}=q,{{bf{v}'times{{bf{B},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img625.png"
 ALT="\begin{displaymath}
m \frac{d{\bf v}'}{dt} = q {\bf v}'\times{\bf B},
\end{displaymath}">|; 

$key = q/{displaymath}(lambda_i-lambda_i^{,ast}),{{bf{x}_i^{ast,T}{{bf{x}_i=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1295.png"
 ALT="\begin{displaymath}
(\lambda_i - \lambda_i^{ \ast}) {\bf x}_i^{\ast T} {\bf x}_i = 0.
\end{displaymath}">|; 

$key = q/{displaymath}chi+Phi'simeq-frac{G,m'}{R}left[1+frac{r^2}{R^2},P_2(costheta)right]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2229.png"
 ALT="\begin{displaymath}
\chi+\Phi' \simeq - \frac{G m'}{R}\left[1 + \frac{r^2}{R^2} P_2(\cos\theta)\right]
\end{displaymath}">|; 

$key = q/{displaymath}({{bf{a}times{{bf{b})times{{bf{c}equiv({{bf{a}cdot{{bf{c}),{{bf{b}-({{bf{b}cdot{{bf{c}),{{bf{a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3353.png"
 ALT="\begin{displaymath}
({\bf a}\times{\bf b}) \times {\bf c}\equiv ({\bf a}\cdot {\bf c})  {\bf b} - ({\bf b}\cdot
{\bf c}) {\bf a}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{f}=-nablaU,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="\begin{displaymath}
{\bf f} = - \nabla U,
\end{displaymath}">|; 

$key = q/{displaymath}sigma=pi,a^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1087.png"
 ALT="\begin{displaymath}
\sigma = \pi a^2,
\end{displaymath}">|; 

$key = q/log(|deltav|);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3150.png"
 ALT="$\log(\vert\delta v\vert)$">|; 

$key = q/{displaymath}K=U+lambda,l=int_{-aslash2}^{aslash2}(-rho,g,y+lambda),[1+y'^{,2}]^{1slash2},dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1764.png"
 ALT="\begin{displaymath}
K = U + \lambda l = \int_{-a/2}^{a/2}(-\rho g y+\lambda) [1+y'^{ 2}]^{1/2} dx,
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{g}'=left(-{mitOmega}^2,R,coslambda,sinlambda,,0,,-g+{mitOmega}^2,R,cos^2lambdaright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img1140.png"
 ALT="\begin{displaymath}
{\bf g}' = \left(-{\mit\Omega}^2 R \cos\lambda \sin\lambda, 0, 
-g + {\mit\Omega}^2 R \cos^2\lambda\right).
\end{displaymath}">|; 

$key = q/displaystylechi+Phi'+Phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2232.png"
 ALT="$\displaystyle \chi +\Phi'+\Phi$">|; 

$key = q/Deltat=2pislashn';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2983.png"
 ALT="$\Delta t=2\pi/n'$">|; 

$key = q/T=2,pislashomega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img542.png"
 ALT="$T=2 \pi/\omega_0$">|; 

$key = q/0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2780.png"
 ALT="$0,$">|; 

$key = q/displaystyle-{mitrho},cos({mitOmega},t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img639.png"
 ALT="$\displaystyle -{\mit\rho} \cos({\mit\Omega} t),$">|; 

$key = q/{displaymath}int_{P}^Qf(x,y),dl={{rm{Area~under~the~curve},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img3373.png"
 ALT="\begin{displaymath}
\int_{P}^Q f(x,y) dl = {\rm Area&nbsp;under&nbsp;the&nbsp;curve},
\end{displaymath}">|; 

$key = q/{mitDelta}=pislash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img571.png"
 ALT="${\mit\Delta}=\pi/4$">|; 

$key = q/{displaymath}Phi'simeq-frac{G,m'}{R}left[1+frac{r}{R},P_1(costheta)+frac{r^2}{R^2},P_2(costheta)right]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2226.png"
 ALT="\begin{displaymath}
\Phi' \simeq - \frac{G m'}{R}\left[1+ \frac{r}{R} P_1(\cos\theta) + \frac{r^2}{R^2} P_2(\cos\theta)\right]
\end{displaymath}">|; 

$key = q/{displaymath}frac{d{{bf{L}}{dt}={{bf{T}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1235.png"
 ALT="\begin{displaymath}
\frac{d{\bf L}}{dt} = {\bf T}.
\end{displaymath}">|; 

$key = q/e'=0.016711;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2718.png"
 ALT="$e'=0.016711$">|; 

$key = q/displaystyle-C.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2666.png"
 ALT="$\displaystyle - C.$">|; 

$key = q/{displaymath}X_n=frac{2,sin(n,pislash5),A}{n,pi},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img431.png"
 ALT="\begin{displaymath}
X_n = \frac{2 \sin(n \pi/5) A}{n \pi},
\end{displaymath}">|; 

$key = q/displaystylev_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1016.png"
 ALT="$\displaystyle v_2.$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter08slashfig8.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="441" HEIGHT="271" BORDER="0"
 SRC="|."$dir".q|img1484.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter08/fig8.03.eps}}
\end{figure}">|; 

$key = q/{displaymath}theta=bar{theta}+lambda,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img2998.png"
 ALT="\begin{displaymath}
\theta = \bar{\theta}+\lambda,
\end{displaymath}">|; 

$key = q/displaystylefrac{dtheta}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3038.png"
 ALT="$\displaystyle \frac{d\theta}{dt}$">|; 

$key = q/{displaymath}E=frac{1}{2},mleft(frac{dx}{dt}right)^2+frac{1}{2},m,omega_0^{,2},x^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img340.png"
 ALT="\begin{displaymath}
E = \frac{1}{2} m\left(\frac{dx}{dt}\right)^2 + \frac{1}{2} m \omega_0^{ 2} x^2.
\end{displaymath}">|; 

$key = q/{displaymath}Q_1=-frac{partialU}{partialq_1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1812.png"
 ALT="\begin{displaymath}
Q_1 = -\frac{\partial U}{\partial q_1}.
\end{displaymath}">|; 

$key = q/{displaymath}a=x_1,{{rm{e}^{,{{rm{i},phi_1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img369.png"
 ALT="\begin{displaymath}
a = x_1 {\rm e}^{ {\rm i} \phi_1},
\end{displaymath}">|; 

$key = q/{displaymath}tan(thetaslash2)=left(frac{1+e}{1-e}right)^{1slash2}tan(Eslash2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img815.png"
 ALT="\begin{displaymath}
\tan (\theta/2) = \left(\frac{1+e}{1-e}\right)^{1/2}\tan (E/2).
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{g}'=frac{2,G,m'}{R^3}left(-frac{x}{2},{{bf{e}_x-frac{y}{2},{{bf{e}_y+z,{{bf{e}_zright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img2254.png"
 ALT="\begin{displaymath}
{\bf g}' = \frac{2 G m'}{R^3}\left(-\frac{x}{2} {\bf e}_x- \frac{y}{2} {\bf e}_y+ z {\bf e}_z\right).
\end{displaymath}">|; 

$key = q/{displaymath}frac{dvarpi_n}{dt}=-Omega_n,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2316.png"
 ALT="\begin{displaymath}
\frac{d\varpi_n}{dt}= -\Omega_n,
\end{displaymath}">|; 

$key = q/Phi(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2378.png"
 ALT="$\Phi(r)$">|; 

$key = q/displaystyle{{bf{G},{{bf{x}_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1936.png"
 ALT="$\displaystyle {\bf G} {\bf x}_k$">|; 

$key = q/e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img749.png"
 ALT="$e$">|; 

$key = q/deltagga;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3159.png"
 ALT="$\delta \gg a$">|; 

$key = q/cos(n,omega,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img408.png"
 ALT="$\cos(n \omega t)$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{AppendixAslashfigA.11.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="200" BORDER="0"
 SRC="|."$dir".q|img3383.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.11.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{1}{3},m,a^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1347.png"
 ALT="$\displaystyle \frac{1}{3} m a^2,$">|; 

$key = q/{displaymath}DeltaEsimeqfrac{2,m,nu,x_0^{,2}}{omega_r}int_0^{2pi}left(nu^2,cos^2ta,sintheta+omega_r^{,2},sin^2thetaright)dtheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="541" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img354.png"
 ALT="\begin{displaymath}
\Delta E \simeq \frac{2 m \nu x_0^{ 2}}{\omega_r}\int_0^...
...heta \sin\theta + \omega_r^{ 2} \sin^2\theta\right)d\theta.
\end{displaymath}">|; 

$key = q/n=13.17639646^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2702.png"
 ALT="$n=13.17639646^\circ$">|; 

$key = q/{{bf{e}_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1434.png"
 ALT="${\bf e}_{x'}$">|; 

$key = q/{displaymath}v_t=left(frac{rho,g,d}{rho_a,C}right)^{1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img284.png"
 ALT="\begin{displaymath}
v_t = \left(\frac{\rho g d}{\rho_a C}\right)^{1/2}.
\end{displaymath}">|; 

$key = q/displaystylex_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2839.png"
 ALT="$\displaystyle x_0$">|; 

$key = q/772;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3021.png"
 ALT="$772$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.08.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="287" BORDER="0"
 SRC="|."$dir".q|img3070.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.08.eps}}
\end{figure}">|; 

$key = q/u=(r_0,theta^2)^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img841.png"
 ALT="$u = (r_0 \theta^2)^{-1}$">|; 

$key = q/rho_2ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2576.png"
 ALT="$\rho_2\ll 1$">|; 

$key = q/l,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img454.png"
 ALT="$l \sin\theta$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.09.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="269" BORDER="0"
 SRC="|."$dir".q|img3075.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.09.eps}}
\end{figure}">|; 

$key = q/u_{max}=1slashr_{min};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img967.png"
 ALT="$u_{max}=1/r_{min}$">|; 

$key = q/{displaymath}alpha=frac{{{bf{r}cdot{{bf{b}times{{bf{c}}{{{bf{a}cdot{{bf{b}times{{bf{c}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img3348.png"
 ALT="\begin{displaymath}
\alpha = \frac{{\bf r}\cdot{\bf b}\times{\bf c}}{{\bf a}\cdot{\bf b}\times{\bf c}}.
\end{displaymath}">|; 

$key = q/(0,,0,,1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1334.png"
 ALT="$(0, 0, 1)$">|; 

$key = q/4.36times10^{-5};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2438.png"
 ALT="$4.36\times 10^{-5}$">|; 

$key = q/{displaymath}frac{partialU}{partialz}=left(frac{mu_1}{rho_1^{,3}}+frac{mu_2}{rho_2^{,3}}right)z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2562.png"
 ALT="\begin{displaymath}
\frac{\partial U}{\partial z} = \left(\frac{\mu_1}{\rho_1^{ 3}}+ \frac{\mu_2}{\rho_2^{ 3}}\right) z.
\end{displaymath}">|; 

$key = q/displaystylem_2,(-ddot{x}+ddot{x}')+m_3,(ddot{x}+ddot{x}')-g,(m_2-m_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="351" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1640.png"
 ALT="$\displaystyle m_2 (-\ddot{x} + \ddot{x}')+m_3 (\ddot{x}+\ddot{x}') - g (m_2-m_3)$">|; 

$key = q/dW;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3282.png"
 ALT="$dW$">|; 

$key = q/iota;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2822.png"
 ALT="$\iota$">|; 

$key = q/displaystyledeltaddot{Y}+2,deltadot{X};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2790.png"
 ALT="$\displaystyle \delta \ddot{Y}+2 \delta \dot{X}$">|; 

$key = q/{displaymath}E=frac{1}{2},left(I_perp,omega_{x'}^{,2}+I_perp,omega_{y'}^{,2}+I_parallel,omega_{z'}^{,2}right)+m,g,l,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="389" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1497.png"
 ALT="\begin{displaymath}
E = \frac{1}{2} \left(I_\perp \omega_{x'}^{ 2} + I_\perp\...
...+
I_\parallel \omega_{z'}^{ 2}\right) + m g l \cos\theta.
\end{displaymath}">|; 

$key = q/|y|slashall1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1228.png"
 ALT="$\vert y\vert/a\ll 1$">|; 

$key = q/y_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2886.png"
 ALT="$y_j$">|; 

$key = q/1.37003;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3111.png"
 ALT="$1.37003$">|; 

$key = q/{displaymath}frac{dV}{du}=-h^2left(6,r_0,u^2+uright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img842.png"
 ALT="\begin{displaymath}
\frac{d V}{du} = - h^2\left(6 r_0 u^2 + u\right).
\end{displaymath}">|; 

$key = q/1slashsqrt{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img388.png"
 ALT="$1/\sqrt{2}$">|; 

$key = q/displaystylev_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$\displaystyle v_z.$">|; 

$key = q/displaystylefrac{r_c}{e^2-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img730.png"
 ALT="$\displaystyle \frac{r_c}{e^2-1},$">|; 

$key = q/{displaymath}U=-frac{G,M_m,M}{a_m}+frac{G,M_m,(I_parallel-I_perp)}{a_m^{,3}},P_2[cos(gamma_m)],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2302.png"
 ALT="\begin{displaymath}
U = - \frac{G M_m M}{a_m} + \frac{G M_m (I_\parallel-I_\perp)}{a_m^{ 3}} P_2[\cos(\gamma_m)],
\end{displaymath}">|; 

$key = q/{displaymath}n_m=frac{dlambda_m}{dt}simeqleft(frac{G,M}{a_m^{,3}}right)^{1slash2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img2314.png"
 ALT="\begin{displaymath}
n_m = \frac{d\lambda_m}{dt}\simeq \left(\frac{G M}{a_m^{ 3}}\right)^{1/2}
\end{displaymath}">|; 

$key = q/m^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2885.png"
 ALT="$m^2$">|; 

$key = q/(0,,0,,a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3422.png"
 ALT="$(0, 0, a)$">|; 

$key = q/{displaymath}frac{dsigma(theta)}{d{mitOmega}}=frac{a^2}{4},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1085.png"
 ALT="\begin{displaymath}
\frac{d\sigma(\theta)}{d{\mit\Omega}} = \frac{a^2}{4},
\end{displaymath}">|; 

$key = q/displaystylefrac{75}{T_i({{rm{yr})}left{sum_{j<i}left(frac{M_j}{M_0}right)left(f^2+frac{175}{64}left(frac{R_j}{R_i}right)^4+cdotsright]right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="515" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2416.png"
 ALT="$\displaystyle \frac{75}{T_i({\rm yr})}\left\{\sum_{j&lt;i}\left(\frac{M_j}{M_0}\ri...
...}\right)^2 + \frac{175}{64}\left(\frac{R_j}{R_i}
\right)^4+\cdots\right]\right.$">|; 

$key = q/{displaymath}U=-mu_1left(frac{1}{rho_1}+frac{rho_1^{,2}}{2}right)-mu_2left(frac{2}+frac{rho_2^{,2}}{2}right)+frac{mu_1,mu_2}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2565.png"
 ALT="\begin{displaymath}
U = - \mu_1\left(\frac{1}{\rho_1}+\frac{\rho_1^{ 2}}{2}\rig...
...rho_2}+\frac{\rho_2^{ 2}}{2}\right) + \frac{\mu_1 \mu_2}{2}.
\end{displaymath}">|; 

$key = q/displaystylealpha+frac{alpha^2}{3}-frac{alpha^3}{9}-frac{31,alpha^4}{81}+{calO}(alpha^5).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2587.png"
 ALT="$\displaystyle \alpha + \frac{\alpha^2}{3} - \frac{\alpha^3}{9} -\frac{31 \alpha^4}{81}+{\cal O}(\alpha^5).$">|; 

$key = q/deltaq_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1800.png"
 ALT="$\delta q_2$">|; 

$key = q/delta{{bf{q}^{(0)}=delta{{bf{q}(t=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1962.png"
 ALT="$\delta {\bf q}^{(0)}
= \delta {\bf q}(t=0)$">|; 

$key = q/{displaymath}s=x'+{{rm{i},y'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1196.png"
 ALT="\begin{displaymath}
s = x' + {\rm i} y'.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{Chapter15slashfig15.13.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="539" HEIGHT="402" BORDER="0"
 SRC="|."$dir".q|img3080.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{Chapter15/fig15.13.eps}}
\end{figure}">|; 

$key = q/displaystyle-dot{x},frac{partialU}{partialx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2550.png"
 ALT="$\displaystyle -\dot{x} \frac{\partial U}{\partial x},$">|; 

$key = q/lambda_k^ast=lambda_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1933.png"
 ALT="$\lambda_k^\ast = \lambda_k$">|; 

$key = q/displaystyle-frac{g}{l},y'-2,{mitOmega},sinlambda,dot{x}'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1194.png"
 ALT="$\displaystyle - \frac{g}{l} y' - 2 {\mit\Omega} \sin\lambda \dot{x}'.$">|; 

$key = q/{displaymath}{{bf{L}=(I_{x'x'},omega_{x'},,I_{y'y'},omega_{y'},,I_{z'z'},omega_{z'}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1373.png"
 ALT="\begin{displaymath}
{\bf L} = (I_{x'x'} \omega_{x'}, I_{y'y'} \omega_{y'}, 
I_{z'z'} \omega_{z'}).
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{r}=r,{{bf{e}_r.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img683.png"
 ALT="\begin{displaymath}
{\bf r} = r {\bf e}_r.
\end{displaymath}">|; 

$key = q/0.05488;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2825.png"
 ALT="$0.05488$">|; 

$key = q/lambda_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1922.png"
 ALT="$\lambda_k$">|; 

$key = q/epsilonll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2476.png"
 ALT="$\epsilon\ll 1$">|; 

$key = q/{{bf{e}_{x''};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1420.png"
 ALT="${\bf e}_{x''}$">|; 

$key = q/displaystyleP_1(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2107.png"
 ALT="$\displaystyle P_1(x)$">|; 

$key = q/{displaymath}deltambox{{boldmath{{theta{}=deltatheta_x,{{bf{e}_x+deltatheta_y,{{bf{e}_y+deltatheta_z,{{bf{e}_z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img3334.png"
 ALT="\begin{displaymath}
\delta\mbox{\boldmath$\theta$} = \delta\theta_x  {\bf e}_x + \delta\theta_y  {\bf e}_y +
\delta\theta_z  {\bf e}_z.
\end{displaymath}">|; 

$key = q/dtau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3421.png"
 ALT="$d\tau$">|; 

$key = q/I_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1261.png"
 ALT="$I_{ij}$">|; 

$key = q/{displaymath}delta{{bf{q}(t)=sum_{k=1,{calF}}eta_k(t),{{bf{x}_k.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1970.png"
 ALT="\begin{displaymath}
\delta {\bf q}(t) = \sum_{k=1,{\cal F}}\eta_k(t) {\bf x}_k.
\end{displaymath}">|; 

$key = q/{displaymath}R_c=1.41left(frac{rho'}{rho}right)^{1slash3}a'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img2264.png"
 ALT="\begin{displaymath}
R_c = 1.41\left(\frac{\rho'}{\rho}\right)^{1/3} a'.
\end{displaymath}">|; 

$key = q/displaystyley_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2916.png"
 ALT="$\displaystyle y_3$">|; 

$key = q/{{bf{M}^{ast,T}={{bf{M};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1927.png"
 ALT="${\bf M}^{\ast T}={\bf M}$">|; 

$key = q/10^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2694.png"
 ALT="$10^4$">|; 

$key = q/x=x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img218.png"
 ALT="$x=x_0$">|; 

$key = q/v(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$v(0)$">|; 

$key = q/-pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1343.png"
 ALT="$-\pi/2$">|; 

$key = q/displaystyleddot{x}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1148.png"
 ALT="$\displaystyle \ddot{x}'$">|; 

$key = q/{displaymath}frac{dsigma}{d{mitOmega}'}=frac{q^4}{(4,pi,epsilon_0)^2,{calE}^{,2}}frac{cospsi}{sin^4psi}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1089.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d{\mit\Omega}'} = \frac{q^4}{(4 \pi \epsilon_0)^2 {\cal E}^{ 2}}\frac{\cos\psi}{\sin^4\psi}.
\end{displaymath}">|; 

$key = q/|deltaY|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2777.png"
 ALT="$\vert\delta Y\vert$">|; 

$key = q/OQ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3320.png"
 ALT="$OQ$">|; 

$key = q/{displaymath}p_phi=frac{partialL}{partialdot{phi}}=m,l^2,sin^2theta,dot{phi}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1678.png"
 ALT="\begin{displaymath}
p_\phi = \frac{\partial L}{\partial \dot{\phi}} = m l^2 \sin^2\theta \dot{\phi}
\end{displaymath}">|; 

$key = q/L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1603.png"
 ALT="$L$">|; 

$key = q/{displaymath}m_i,{{bf{r}_itimesfrac{d^2{{bf{r}_i}{dt^2}=frac{d}{dt}!left(m_i,{{bfrac{d{{bf{r}_i}{dt}right)=frac{d{{bf{l}_i}{dt},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img153.png"
 ALT="\begin{displaymath}
m_i {\bf r}_i\times \frac{d^2{\bf r}_i }{dt^2}= \frac{d}{dt...
..._i\times
\frac{d{\bf r}_i}{dt}\right) = \frac{d{\bf l}_i}{dt},
\end{displaymath}">|; 

$key = q/displaystyledeltatheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2358.png"
 ALT="$\displaystyle \delta\theta$">|; 

$key = q/{mitOmega}=omega_{z'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1491.png"
 ALT="${\mit\Omega} = \omega_{z'}$">|; 

$key = q/9,Tslash10<t<T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img427.png"
 ALT="$9 T/10&lt;t&lt;T$">|; 

$key = q/displaystyle{{bf{a}cdot{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3256.png"
 ALT="$\displaystyle {\bf a} \cdot {\bf b}$">|; 

$key = q/d>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img704.png"
 ALT="$d&gt;0$">|; 

$key = q/alpha_0-gamma_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2857.png"
 ALT="$\alpha_0-\gamma_0$">|; 

$key = q/{displaymath}int_P^Qleft[f(x,y),dx+g(x,y),dyright].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img3387.png"
 ALT="\begin{displaymath}
\int_P^Q \left[ f(x,y) dx + g(x,y) dy\right].
\end{displaymath}">|; 

$key = q/displaystyleU;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2025.png"
 ALT="$\displaystyle U$">|; 

$key = q/displaystyle|{{bf{a}|^2,|{{bf{b}|^2-|{{bf{a}|^2,|{{bf{b}|^2,cos^2theta=|{{bf{a}|^2,|{{bf{b}|^2,sin^2theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="333" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3313.png"
 ALT="$\displaystyle \vert{\bf a}\vert^2  \vert{\bf b}\vert^2 - \vert{\bf a}\vert^2 \...
...ert^2  \cos^2\theta = \vert{\bf a}\vert^2 \vert{\bf b}\vert^2  \sin^2\theta.$">|; 

$key = q/n=n';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img403.png"
 ALT="$n=n'$">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}=omega,left(-sintheta,sinphi,,sintheta,cosphi,,costhetaright){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2307.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$}= \omega \left(-\sin\theta \sin\phi, \sin\theta \cos\phi, \cos\theta\right)
\end{displaymath}">|; 

$key = q/displaystylerho_2-frac{rho_2^{,2}}{3}+frac{rho_2^{,3}}{3}+frac{rho_2^{,4}}{81}+{calO}(rho_2^{,5}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2585.png"
 ALT="$\displaystyle \rho_2 -\frac{\rho_2^{ 2}}{3} + \frac{\rho_2^{ 3}}{3} + \frac{\rho_2^{ 4}}{81}+ {\cal O}(\rho_2^{ 5}),$">|; 

$key = q/displaystyle{{bf{a}cdot{{bf{b}+{{bf{a}cdot{{bf{c}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3262.png"
 ALT="$\displaystyle {\bf a} \cdot {\bf b} + {\bf a}\cdot {\bf c}.$">|; 

$key = q/x_1,x_2,x_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1564.png"
 ALT="$x_1, x_2, x_3$">|; 

$key = q/displaystylepsi_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2407.png"
 ALT="$\displaystyle \psi_i$">|; 

$key = q/displaystylez_1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2728.png"
 ALT="$\displaystyle z_1.$">|; 

$key = q/partial^2Uslashpartialq_i,partialq_jequivpartial^2Uslashpartialq_j,partialq_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1907.png"
 ALT="$\partial^2 U/\partial q_i \partial q_j\equiv
\partial^2 U/\partial q_j \partial q_i$">|; 

$key = q/{{bf{A};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1279.png"
 ALT="${\bf A}$">|; 

$key = q/3218;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3016.png"
 ALT="$3218$">|; 

$key = q/{displaymath}{{bf{T}=frac{d{{bf{L}}{dt'}+mbox{{boldmath{{omega{}times{{bf{L}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1355.png"
 ALT="\begin{displaymath}
{\bf T} = \frac{d{\bf L}}{dt'} + \mbox{\boldmath$\omega$}\times{\bf L}.
\end{displaymath}">|; 

$key = q/{displaymath}Q=2pi,frac{E}{DeltaE}=frac{omega_r}{2,nu}simeqfrac{omega_0}{2,nu}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img362.png"
 ALT="\begin{displaymath}
Q = 2\pi \frac{E}{\Delta E} = \frac{\omega_r}{2 \nu}\simeq \frac{\omega_0}{2 \nu}.
\end{displaymath}">|; 

$key = q/{{bf{f}_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img913.png"
 ALT="${\bf f}_{21}$">|; 

$key = q/C_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2609.png"
 ALT="$C_i$">|; 

$key = q/{mitOmega},sinlambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1212.png"
 ALT="${\mit\Omega} \sin\lambda$">|; 

$key = q/deltal=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1725.png"
 ALT="$\delta l =0$">|; 

$key = q/theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$\theta $">|; 

$key = q/omega_4simeq2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2921.png"
 ALT="$\omega_4\simeq 2$">|; 

$key = q/displaystyle3-mu_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2621.png"
 ALT="$\displaystyle 3 - \mu_2.$">|; 

$key = q/{displaymath}{{bf{S}={{bf{a}times{{bf{b},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3318.png"
 ALT="\begin{displaymath}
{\bf S} = {\bf a}\times {\bf b},
\end{displaymath}">|; 

$key = q/{displaymath}f_{ext}(t)=m,omega_0^{,2},X_1,cos(omega,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img363.png"
 ALT="\begin{displaymath}
f_{ext}(t) = m \omega_0^{ 2} X_1 \cos(\omega t).
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2{{bf{r}}{dt^2}=-frac{G,M}{r^3},{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img665.png"
 ALT="\begin{displaymath}
\frac{d^2{\bf r}}{dt^2} = -\frac{G M}{r^3} {\bf r}.
\end{displaymath}">|; 

$key = q/{displaymath}L_psi=I_parallel,(costheta,dot{phi}+dot{psi})=I_parallel,{mitOmega},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1490.png"
 ALT="\begin{displaymath}
L_\psi = I_\parallel (\cos\theta \dot{\phi}+\dot{\psi})= I_\parallel {\mit\Omega},
\end{displaymath}">|; 

$key = q/{displaymath}h=r^2,dot{theta}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img736.png"
 ALT="\begin{displaymath}
h = r^2 \dot{\theta}
\end{displaymath}">|; 

$key = q/3.34;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2458.png"
 ALT="$3.34$">|; 

$key = q/{displaymath}U=-frac{mu_1}{rho_1}-frac{mu_2}{rho_2}-frac{omega^2}{2},(x^2+y^2){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2548.png"
 ALT="\begin{displaymath}
U = - \frac{\mu_1}{\rho_1} - \frac{\mu_2}{\rho_2} - \frac{\omega^2}{2} (x^2+y^2)
\end{displaymath}">|; 

$key = q/{displaymath}tantheta=frac{I_perp}{I_parallel},tanalpha.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1404.png"
 ALT="\begin{displaymath}
\tan\theta = \frac{I_\perp}{I_\parallel} \tan\alpha.
\end{displaymath}">|; 

$key = q/{{bf{v}=(v_x,v_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img575.png"
 ALT="${\bf v} = (v_x, v_z)$">|; 

$key = q/displaystyleleft(frac{e+1}{e-1}right)^{1slash2}tanh(Hslash2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img834.png"
 ALT="$\displaystyle \left(\frac{e+1}{e-1}\right)^{1/2} \tanh (H/2).$">|; 

$key = q/|{{bf{b}|=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3273.png"
 ALT="$\vert{\bf b}\vert=0$">|; 

$key = q/y'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1418.png"
 ALT="$y''$">|; 

$key = q/gamma=pmsqrt{lambda}_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1957.png"
 ALT="$\gamma=\pm \sqrt{\lambda}_k$">|; 

$key = q/displaystylefrac{3}{2},frac{epsilon,iota_m,mu_m,n_m^{,2}}{omega,Omega_n},frac{cos(2,theta_0)}{sintheta_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2357.png"
 ALT="$\displaystyle \frac{3}{2} \frac{\epsilon \iota_m \mu_m n_m^{ 2}}{\omega \Omega_n} \frac{\cos(2 \theta_0)}{\sin\theta_0},$">|; 

$key = q/displaystyleI_perp,sin^2theta_0,ddot{phi}_1-I_{parallel},sintheta_0,omega,dot{theta}_1-beta_m,sin(2,theta_0),sin(Omega_n,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="451" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2346.png"
 ALT="$\displaystyle I_\perp \sin^2\theta_0 \ddot{\phi}_1 - I_{\parallel} \sin\theta_0 \omega \dot{\theta}_1 - \beta_m \sin(2 \theta_0) \sin(\Omega_n t),$">|; 

$key = q/x=y=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1884.png"
 ALT="$x=y=0$">|; 

$key = q/dslashdt';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1107.png"
 ALT="$d/dt'$">|; 

$key = q/E_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img212.png"
 ALT="$E_1$">|; 

$key = q/e>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img729.png"
 ALT="$e&gt;1$">|; 

$key = q/{displaymath}m,l,frac{d^2theta}{dt^2}+nu,frac{dtheta}{dt}+m,g,sintheta=A,cos(omega,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img3026.png"
 ALT="\begin{displaymath}
m l \frac{d^2\theta}{dt^2}+ \nu \frac{d\theta}{dt} + m g \sin\theta = A \cos(\omega  t),
\end{displaymath}">|; 

$key = q/displaystylelambda_i,{{bf{x}_i^T{{bf{x}_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1307.png"
 ALT="$\displaystyle \lambda_i {\bf x}_i^T{\bf x}_j,$">|; 

$key = q/{displaymath}tan^{-1}left[frac{(I_parallel-I_perp),tanalpha}{I_parallel+I_perp,tan^2alpha}right]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1554.png"
 ALT="\begin{displaymath}
\tan^{-1}\left[\frac{(I_\parallel-I_\perp) \tan\alpha}{I_\parallel+I_\perp \tan^2\alpha}\right]
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter13slashfig13.07.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="239" BORDER="0"
 SRC="|."$dir".q|img2626.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter13/fig13.07.eps}}
\end{figure}">|; 

$key = q/|n|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3207.png"
 ALT="$\vert n\vert$">|; 

$key = q/{{bf{a}={{bf{a}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3363.png"
 ALT="${\bf a} = {\bf a} (t)$">|; 

$key = q/displaystyle0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img477.png"
 ALT="$\displaystyle 0.$">|; 

$key = q/chi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2227.png"
 ALT="$\chi$">|; 

$key = q/m^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2785.png"
 ALT="$m^4$">|; 

$key = q/{displaymath}U(r)=frac{k}{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img990.png"
 ALT="\begin{displaymath}
U(r) = \frac{k}{r}.
\end{displaymath}">|; 

$key = q/{calO}(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img280.png"
 ALT="${\cal O}(1)$">|; 

$key = q/displaystylea_y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="$\displaystyle a_y ,$">|; 

$key = q/{displaymath}alpha_1=sqrt{1+e}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img785.png"
 ALT="\begin{displaymath}
\alpha_1 = \sqrt{1+e}.
\end{displaymath}">|; 

$key = q/lambda'=-cosh(aslash2,h);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1777.png"
 ALT="$\lambda' = - \cosh (a/2 h)$">|; 

$key = q/v(0)slashA=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$v(0)/A=0$">|; 

$key = q/{displaymath}a=frac{r_p+r_a}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img759.png"
 ALT="\begin{displaymath}
a = \frac{r_p+r_a}{2}.
\end{displaymath}">|; 

$key = q/{displaymath}e=frac{r_2-r_1}{r_2+r_1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img784.png"
 ALT="\begin{displaymath}
e = \frac{r_2-r_1}{r_2+r_1}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{r}_itimes{{bf{f}_{ij}+{{bf{r}_jtimes{{bf{f}_{ji}=({{bf{r}_i-{{bf{r}_j)times{{bf{f}_{ij}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img159.png"
 ALT="\begin{displaymath}
{\bf r}_i\times {\bf f}_{ij}+{\bf r}_j\times {\bf f}_{ji} =
({\bf r}_i-{\bf r}_j)\times {\bf f}_{ij}.
\end{displaymath}">|; 

$key = q/{displaymath}R_c=2.44left(frac{rho'}{rho}right)^{1slash3}a'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img2265.png"
 ALT="\begin{displaymath}
R_c = 2.44\left(\frac{\rho'}{\rho}\right)^{1/3} a'.
\end{displaymath}">|; 

$key = q/displaystyle8.2^{''},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2373.png"
 ALT="$\displaystyle 8.2^{''},$">|; 

$key = q/4%;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2692.png"
 ALT="$4\%$">|; 

$key = q/{displaymath}theta_{dev}simeq-frac{{mitOmega}^2,R}{2,g},sin(2,lambda)simeq-0.1^circ,sin(2,lambda).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1143.png"
 ALT="\begin{displaymath}
\theta_{dev} \simeq - \frac{{\mit\Omega}^2 R}{2 g} \sin(2 \lambda)\simeq
-0.1^\circ \sin(2 \lambda).
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{c}={{bf{a}-{{bf{b}={{bf{a}+(-{{bf{b}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3203.png"
 ALT="\begin{displaymath}
{\bf c} = {\bf a} - {\bf b} = {\bf a} +(-{\bf b}).
\end{displaymath}">|; 

$key = q/{displaymath}omega_ast=sqrt{1-frac{1}{4Q^2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3051.png"
 ALT="\begin{displaymath}
\omega_\ast = \sqrt{1-\frac{1}{4Q^2}},
\end{displaymath}">|; 

$key = q/{displaymath}int_{v_0}^vfrac{v',dv'}{f(v')}=frac{x-x_0}{m},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img248.png"
 ALT="\begin{displaymath}
\int_{v_0}^v \frac{v' dv'}{f(v')} = \frac{x-x_0}{m},
\end{displaymath}">|; 

$key = q/{displaymath}E-e,sinE=2pileft(frac{t}{T}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img808.png"
 ALT="\begin{displaymath}
E - e \sin E = 2\pi\left(\frac{t}{T}\right),
\end{displaymath}">|; 

$key = q/n';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2972.png"
 ALT="$n'$">|; 

$key = q/1slashlambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3161.png"
 ALT="$1/\lambda$">|; 

$key = q/{{bf{g}=(0,-g);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img576.png"
 ALT="${\bf g} = (0, -g)$">|; 

$key = q/sin^2lambda_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2281.png"
 ALT="$\sin^2\lambda_s$">|; 

$key = q/{displaymath}{{bf{a}=a_x,{{bf{e}_x+a_y,{{bf{e}_y+a_z,{{bf{e}_z,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img3218.png"
 ALT="\begin{displaymath}
{\bf a} = a_x {\bf e}_x+ a_y {\bf e}_y+a_z {\bf e}_z,
\end{displaymath}">|; 

$key = q/dL^2slashdt'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1411.png"
 ALT="$d L^2/dt'=0$">|; 

$key = q/x(t=0)=x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img233.png"
 ALT="$x(t=0)=x_0$">|; 

$key = q/{displaymath}dot{C}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img2519.png"
 ALT="\begin{displaymath}
\dot{C} = 0.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{v}=frac{d{{bf{r}}{dt}=dot{r},{{bf{e}_r+r,dot{{bf{e}_r,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img684.png"
 ALT="\begin{displaymath}
{\bf v} = \frac{d{\bf r}}{dt} = \dot{r} {\bf e}_r + r \dot{\bf e}_r,
\end{displaymath}">|; 

$key = q/{{bf{u}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="${\bf u}(t)$">|; 

$key = q/n=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img874.png"
 ALT="$n=1$">|; 

$key = q/displaystyle-frac{Y}{R^3}-frac{3}{2},m^{2},sin[2,(1-m),T],X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2765.png"
 ALT="$\displaystyle -\frac{Y}{R^3}-\frac{3}{2} m^{2} \sin[2 (1-m) T] X$">|; 

$key = q/displaystyle{mitrho},sin({mitOmega},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img640.png"
 ALT="$\displaystyle {\mit\rho} \sin({\mit\Omega} t)$">|; 

$key = q/{displaymath}sum_{i=1,{calF}}dot{q}_i,frac{partialK}{partialdot{q}_i}=2,K.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1843.png"
 ALT="\begin{displaymath}
\sum_{i=1,{\cal F}} \dot{q}_i \frac{\partial K}{\partial \dot{q}_i} = 2 K.
\end{displaymath}">|; 

$key = q/{displaymath}J=int_a^bG(y,y',x),dx{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1749.png"
 ALT="\begin{displaymath}
J = \int_a^b G(y,y',x) dx
\end{displaymath}">|; 

$key = q/sin(omega_ast,t_0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3144.png"
 ALT="$\sin(\omega_\ast t_0)=0$">|; 

$key = q/displaystyleZ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2757.png"
 ALT="$\displaystyle Z$">|; 

$key = q/int{{bf{A}cdotd{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3472.png"
 ALT="$\int {\bf A}\cdot d{\bf r}$">|; 

$key = q/displaystylefrac{1}{3},m,b^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1346.png"
 ALT="$\displaystyle \frac{1}{3} m b^2,$">|; 

$key = q/{displaymath}phi=tan^{-1}left(frac{b}{a}right)=tan^{-1}(sqrt{e^2-1}){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img733.png"
 ALT="\begin{displaymath}
\phi = \tan^{-1}\left(\frac{b}{a}\right) = \tan^{-1}(\sqrt{e^2-1})
\end{displaymath}">|; 

$key = q/f<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img224.png"
 ALT="$f &lt;0$">|; 

$key = q/{displaymath}{{bf{a}cdot{{bf{b}=a_x^{~2}+a_y^{~2}+a_z^{~2}=|{{bf{a}|^2=a^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img3265.png"
 ALT="\begin{displaymath}
{\bf a} \cdot {\bf b} = a_x^{&nbsp;2}+a_y^{&nbsp;2} + a_z^{&nbsp;2} = \vert{\bf a}\vert^2=a^2.
\end{displaymath}">|; 

$key = q/{displaymath}M_{ij}=M_{ji}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1900.png"
 ALT="\begin{displaymath}
M_{ij} = M_{ji}.
\end{displaymath}">|; 

$key = q/displaystylefrac{partialH}{partialp}=frac{p}{m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1863.png"
 ALT="$\displaystyle \frac{\partial H}{\partial p} = \frac{p}{m},$">|; 

$key = q/Q^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img391.png"
 ALT="$Q^{-1}$">|; 

$key = q/displaystylei,sin(T-gamma_0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2812.png"
 ALT="$\displaystyle i \sin(T-\gamma_0),$">|; 

$key = q/{{bf{b}times{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3340.png"
 ALT="${\bf b}\times {\bf c}$">|; 

$key = q/f(r)=-c,exp(-rslasha)slashr^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img903.png"
 ALT="$f(r) = -c \exp(-r/a)/r^2$">|; 

$key = q/{displaymath}T=2pi,sqrt{frac{V}{g,A}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img508.png"
 ALT="\begin{displaymath}
T = 2\pi \sqrt{\frac{V}{g A}}.
\end{displaymath}">|; 

$key = q/f(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img205.png"
 ALT="$f(x)$">|; 

$key = q/displaystyle0+int_0^1y^2,dy=frac{1}{3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3382.png"
 ALT="$\displaystyle 0 + \int_0^1 y^2 dy = \frac{1}{3}.$">|; 

$key = q/displaystyleL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1634.png"
 ALT="$\displaystyle L$">|; 

$key = q/Qsimeq1.370;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3094.png"
 ALT="$Q\simeq 1.370$">|; 

$key = q/displaystylefrac{m_1}{m_1+m_2},{{bf{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img940.png"
 ALT="$\displaystyle \frac{m_1}{m_1+m_2} {\bf r},$">|; 

$key = q/{displaymath}chi(r,theta)=-frac{Omega^2,r^2}{2},sin^2theta=-frac{Omega^2,r^2}{3},left[1-P_2(costheta)right]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2170.png"
 ALT="\begin{displaymath}
\chi(r,\theta) = - \frac{\Omega^2 r^2}{2} \sin^2\theta= -\frac{\Omega^2 r^2}{3} \left[1-P_2(\cos\theta)\right]
\end{displaymath}">|; 

$key = q/{displaymath}{calL}=K-U=frac{1}{2}left(I_perp,dot{theta}^2+I_perp,sin^2theta,dotallel,omega^2right)+epsilon,alpha_s,cos(2,theta){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="498" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2288.png"
 ALT="\begin{displaymath}
{\cal L} = K - U = \frac{1}{2}\left(I_\perp \dot{\theta}^2 ...
...parallel \omega^2\right) +\epsilon \alpha_s \cos(2 \theta)
\end{displaymath}">|; 

$key = q/{displaymath}left(array{{ccc}I_{xx}-lambda&I_{xy}&I_{xz}I_{yx}&I_{yy}-lambda&I_{gammaarray{right)=left(array{{c}000array{right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="415" HEIGHT="76" BORDER="0"
 SRC="|."$dir".q|img1329.png"
 ALT="\begin{displaymath}
\left(\begin{array}{ccc}
I_{xx}-\lambda&amp;I_{xy}&amp;I_{xz}\\\\
I_{...
...y}\right) = \left(\begin{array}{c}0\ 0\ 0\end{array}\right),
\end{displaymath}">|; 

$key = q/r<a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2387.png"
 ALT="$r&lt;a$">|; 

$key = q/r_{min};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img968.png"
 ALT="$r_{min}$">|; 

$key = q/{displaymath}g_x=G,m',frac{partial}{partialx}!left(frac{1}{|{{bf{r}'-{{bf{r}|}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2056.png"
 ALT="\begin{displaymath}
g_x = G m' \frac{\partial}{\partial x}\!\left(\frac{1}{\vert{\bf r}'-{\bf r}\vert}\right),
\end{displaymath}">|; 

$key = q/D=180^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3006.png"
 ALT="$D=180^\circ$">|; 

$key = q/displaystyledeltapsi_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2410.png"
 ALT="$\displaystyle \delta\psi_i$">|; 

$key = q/{displaymath}K=frac{1}{2},deltadot{{bf{q}^T,{{bf{M},deltadot{{bf{q},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1975.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} \delta \dot{\bf q}^T {\bf M} \delta\dot{\bf q},
\end{displaymath}">|; 

$key = q/n=2pislashT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img769.png"
 ALT="$n=2\pi/T$">|; 

$key = q/v(0)=-2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$v(0)=-2$">|; 

$key = q/A_r={{bf{A}cdot{{bf{e}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3491.png"
 ALT="$A_r = {\bf A}\cdot {\bf e}_r$">|; 

$key = q/{displaymath}deltaH=sum_{i=1,{calF}}left(deltadot{q}_i,p_i+dot{q}_i,deltap_i-fra{q}_i-frac{partialL}{partialq_i},deltaq_iright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1847.png"
 ALT="\begin{displaymath}
\delta H = \sum_{i=1,{\cal F}} \left(\delta\dot{q}_i p_i + ...
...dot{q}_i - \frac{\partial L}{\partial q_i} \delta q_i\right).
\end{displaymath}">|; 

$key = q/i=1,5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2611.png"
 ALT="$i=1,5$">|; 

$key = q/m_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$m_3$">|; 

$key = q/q_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1664.png"
 ALT="$q_k$">|; 

$key = q/|deltaZ|llX_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2778.png"
 ALT="$\vert\delta Z\vert\ll X_0$">|; 

$key = q/k=l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1949.png"
 ALT="$k=l$">|; 

$key = q/0leqtleqTslash10;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img426.png"
 ALT="$0\leq t\leq T/10$">|; 

$key = q/{{bf{v}'=d{{bf{r}slashdt';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1113.png"
 ALT="${\bf v}' = d{\bf r}/dt'$">|; 

$key = q/v_0=dx(0)slashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img326.png"
 ALT="$v_0 = dx(0)/dt$">|; 

$key = q/mbox{{boldmath{{omega{}=(0,,0,,omega);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2522.png"
 ALT="$\mbox{\boldmath$\omega$}=(0, 0, \omega)$">|; 

$key = q/{displaymath}leftlangle|{{bf{r}'-{{bf{r}|^{-1}rightrangle=frac{1}{r}sum_{n=0,inffrac{r'}{r}right)^nP_n(costheta),P_n(costheta').{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="398" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img2116.png"
 ALT="\begin{displaymath}
\left\langle\vert{\bf r}'-{\bf r}\vert^{-1}\right\rangle = \...
...\left(\frac{r'}{r}\right)^n P_n(\cos\theta) P_n(\cos\theta').
\end{displaymath}">|; 

$key = q/displaystyle+frac{G}{R_i^{,2}},sum_{j>i}M_j,left(frac{R_i}{R_j}right)!left[frac{right)^2+frac{27}{16}left(frac{R_i}{R_j}right)^4+cdotsright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2403.png"
 ALT="$\displaystyle + \frac{G}{R_i^{ 2}} \sum_{j&gt; i}M_j \left(\frac{R_i}{R_j}\righ...
...{R_i}{R_j}\right)^2+ \frac{27}{16}\left(\frac{R_i}{R_j}\right)^4+\cdots\right],$">|; 

$key = q/{displaymath}h^2=a,(1-e^2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2526.png"
 ALT="\begin{displaymath}
h^2 = a (1-e^2).
\end{displaymath}">|; 

$key = q/m'=1.99times10^{30},{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2248.png"
 ALT="$m'=1.99\times 10^{30} {\rm kg}$">|; 

$key = q/{displaymath}Phi({{bf{r})=-Gintfrac{rho({{bf{r}')}{|{{bf{r}'-{{bf{r}|},d^3{{bf{r}',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2069.png"
 ALT="\begin{displaymath}
\Phi({\bf r}) = - G\int\frac{\rho({\bf r}')}{\vert{\bf r}'-{\bf r}\vert} d^3{\bf r}',
\end{displaymath}">|; 

$key = q/(deltadot{Psi})_{obs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2444.png"
 ALT="$(\delta\dot{\Psi})_{obs}$">|; 

$key = q/{displaymath}m_j,ddot{x}_j=f_j,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1581.png"
 ALT="\begin{displaymath}
m_j \ddot{x}_j = f_j,
\end{displaymath}">|; 

$key = q/0<mu_2leq0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2678.png"
 ALT="$0&lt;\mu_2\leq 0.5$">|; 

$key = q/displaystyler,sintheta,sinphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2074.png"
 ALT="$\displaystyle r \sin\theta \sin\phi,$">|; 

$key = q/Ox'y'z';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3233.png"
 ALT="$Ox'y'z'$">|; 

$key = q/displaystylem,g,l,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1486.png"
 ALT="$\displaystyle m g l \sin\theta$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter07slashfig7.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img1177.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter07/fig7.03.eps}}
\end{figure}">|; 

$key = q/displaystylea_x,(a_y,b_z-a_z,b_y)+a_y,(a_z,b_x-a_x,b_z)+a_z,(a_x,b_y-a_y,b_x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="486" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3306.png"
 ALT="$\displaystyle a_x (a_y  b_z-a_z  b_y) + a_y  (a_z  b_x- a_x  b_z)
+a_z (a_x  b_y - a_y  b_x)$">|; 

$key = q/displaystyle-frac{3}{2},m^2,sin(omega_4,T-alpha_4),deltaY-3,deltaX^{,2}+frac{3}{2},(deltaY^{,2}+deltaZ^{,2}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="426" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2879.png"
 ALT="$\displaystyle -\frac{3}{2} m^2 \sin(\omega_4 T-\alpha_4) \delta Y-3 \delta X^{ 2} + \frac{3}{2} (\delta Y^{ 2}+\delta Z^{ 2}),$">|; 

$key = q/{displaymath}DeltaEsimeqfrac{2pi,m,nu,x_0^{,2}}{omega_r},(nu^2+omega_r^{,2})=2piega_0^{,2},x_0^{,2}left(frac{nu}{omega_r}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="401" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img355.png"
 ALT="\begin{displaymath}
\Delta E \simeq \frac{2\pi m \nu x_0^{ 2}}{\omega_r} (\...
...m \omega_0^{ 2} x_0^{ 2}\left(\frac{\nu}{\omega_r}\right),
\end{displaymath}">|; 

$key = q/{{bf{r}={{bf{r}_2-{{bf{r}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img925.png"
 ALT="${\bf r}= {\bf r}_2-{\bf r}_1$">|; 

$key = q/{{bf{u};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="${\bf u}$">|; 

$key = q/{displaymath}m_i,frac{d^2{{bf{r}_i}{dt^2}={{bf{f}_i=sum_{j=1,N}^{jneqi}{{bf{f}_{ij}+{{bf{F}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="67" BORDER="0"
 SRC="|."$dir".q|img165.png"
 ALT="\begin{displaymath}
m_i \frac{d^2{\bf r}_i}{dt^2} = {\bf f}_i = \sum_{j=1,N}^{j\neq i} {\bf f}_{ij} + {\bf F}_i.
\end{displaymath}">|; 

$key = q/k=1,2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2009.png"
 ALT="$k=1,2$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{AppendixAslashfigA.08.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="342" BORDER="0"
 SRC="|."$dir".q|img3326.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{AppendixA/figA.08.eps}}
\end{figure}">|; 

$key = q/x_4,x_5,x_6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1565.png"
 ALT="$x_4, x_5, x_6$">|; 

$key = q/ErightarrowE+2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img822.png"
 ALT="$E\rightarrow E + 2\pi$">|; 

$key = q/displaystyley_5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2936.png"
 ALT="$\displaystyle y_5$">|; 

$key = q/{displaymath}{{bf{v}={{bf{v}'+mbox{{boldmath{{Omega{}times{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1112.png"
 ALT="\begin{displaymath}
{\bf v} = {\bf v}' + \mbox{\boldmath$\Omega$}\times{\bf r}.
\end{displaymath}">|; 

$key = q/y_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1784.png"
 ALT="$y_k$">|; 

$key = q/dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3385.png"
 ALT="$dy$">|; 

$key = q/displaystylepileft{1+frac{3}{4}sum_{j<i}left(frac{M_j}{M_0}right)left(frac{R_j}{^2+frac{175}{64}left(frac{R_j}{R_i}right)^4+cdotsright]right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="525" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2408.png"
 ALT="$\displaystyle \pi\left\{1 + \frac{3}{4}\sum_{j&lt;i}\left(\frac{M_j}{M_0}\right)
\...
...}\right)^2 + \frac{175}{64}\left(\frac{R_j}{R_i}
\right)^4+\cdots\right]\right.$">|; 

$key = q/displaystyle-n^{,2},a^{,3},frac{{{bf{r}}{|{{bf{r}|^{,3}}+n'^{,2},a'^{,3}left[fra{bf{r}'-{{bf{r}|^{,3}}-frac{{{bf{r}'}{|{{bf{r}'|^{,3}}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2713.png"
 ALT="$\displaystyle - n^{ 2} a^{ 3} \frac{{\bf r}}{\vert{\bf r}\vert^{ 3}}
+ n'...
...t{\bf r}'-{\bf r}\vert^{ 3}}-\frac{{\bf r}'}{\vert{\bf r}'\vert^{ 3}}\right],$">|; 

$key = q/displaystyleomega,cosalpha,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1396.png"
 ALT="$\displaystyle \omega \cos\alpha,$">|; 

$key = q/E=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img800.png"
 ALT="$E=0$">|; 

$key = q/14.8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3013.png"
 ALT="$14.8$">|; 

$key = q/displaystyle-frac{partialH}{partialr}=frac{p_theta^{,2}}{m,r^3}-frac{partialU}{partialr},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1879.png"
 ALT="$\displaystyle -\frac{\partial H}{\partial r} = \frac{p_\theta^{ 2}}{m r^3}-\frac{\partial U}{\partial r},$">|; 

$key = q/{displaymath}y',frac{d}{dx}!left(frac{partialF}{partialy'}right)-y',frac{partialF}{partialy}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1739.png"
 ALT="\begin{displaymath}
y' \frac{d}{dx}\!\left(\frac{\partial F}{\partial y'}\right)-y' \frac{\partial F}{\partial y} = 0.
\end{displaymath}">|; 

$key = q/11.45;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2450.png"
 ALT="$11.45$">|; 

$key = q/{displaymath}{mitOmega}_{x'}=left[frac{(I_{y'y'}-I_{x'x'}),(I_{z'z'}-I_{x'x'})}{I_{y'y'},I_{z'z'}}right]^{1slash2}!omega_{x'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1546.png"
 ALT="\begin{displaymath}
{\mit\Omega}_{x'} = \left[\frac{(I_{y'y'}-I_{x'x'}) (I_{z'z'}-I_{x'x'})}{I_{y'y'} I_{z'z'}}\right]^{1/2}\! \omega_{x'}.
\end{displaymath}">|; 

$key = q/displaystylev_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1014.png"
 ALT="$\displaystyle v_1,$">|; 

$key = q/50^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2482.png"
 ALT="$50^\circ$">|; 

$key = q/{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3185.png"
 ALT="${\bf a}$">|; 

$key = q/displaystylec_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3296.png"
 ALT="$\displaystyle c_{x'}$">|; 

$key = q/{{bf{e}_theta=nablathetaslash|nablatheta|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3490.png"
 ALT="${\bf e}_\theta = \nabla\theta/\vert\nabla\theta\vert$">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{AppendixAslashfigA.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="167" BORDER="0"
 SRC="|."$dir".q|img3277.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{AppendixA/figA.04.eps}}
\end{figure}">|; 

$key = q/{displaymath}Usimeq-frac{1}{2}sum_{i,j=1,{calF}}G_{ij},deltaq_i,deltaq_j.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1906.png"
 ALT="\begin{displaymath}
U \simeq - \frac{1}{2}\sum_{i,j=1,{\cal F}} G_{ij} \delta q_i  \delta q_j.
\end{displaymath}">|; 

$key = q/I_{x'x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1359.png"
 ALT="$I_{x'x'}$">|; 

$key = q/displaystyle+frac{3}{8},m,iota,sin[(1-2,m-g,m^2),T+gamma_0].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2964.png"
 ALT="$\displaystyle +\frac{3}{8} m \iota \sin[(1-2 m-g m^2) T+\gamma_0].$">|; 

$key = q/displaystyleE-e,sinE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img816.png"
 ALT="$\displaystyle E - e \sin E$">|; 

$key = q/18.6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2315.png"
 ALT="$18.6$">|; 

$key = q/A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$A$">|; 

$key = q/displaystylel,sintheta,sinphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1673.png"
 ALT="$\displaystyle l \sin\theta \sin\phi.$">|; 

$key = q/{displaymath}{mitOmega}=frac{2pi}{24,,{{rm{hrs}}=7.27times10^{-5},{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1121.png"
 ALT="\begin{displaymath}
{\mit\Omega} = \frac{2\pi}{24  {\rm hrs}}=7.27\times 10^{-5} {\rm rad./s}.
\end{displaymath}">|; 

$key = q/displaystyleDeltaE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img348.png"
 ALT="$\displaystyle \Delta E$">|; 

$key = q/{displaymath}{{bf{a}times{{bf{b}equiv(a_y,b_z-a_z,b_y,,a_z,b_x-a_x,b_z,,a_x,b_y-a_y,b_x)={{bf{c}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img3295.png"
 ALT="\begin{displaymath}
{\bf a}\times{\bf b} \equiv (a_y   b_z-a_z  b_y,  a_z  b_x - a_x  b_z,  a_x  b_y - a_y  b_x)
={\bf c}.
\end{displaymath}">|; 

$key = q/tsimeqTslash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1215.png"
 ALT="$t\simeq T/4$">|; 

$key = q/theta(0)slashA=0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$\theta (0)/A=0.5$">|; 

$key = q/mbox{{boldmath{{Omega{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1104.png"
 ALT="$\mbox{\boldmath$\Omega$}$">|; 

$key = q/n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img510.png"
 ALT="$n$">|; 

$key = q/|{{bf{r}|^n,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3553.png"
 ALT="$\vert{\bf r}\vert^n,$">|; 

$key = q/{{bf{a}={{bf{e}_x+3,{{bf{e}_y-{{bf{e}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3513.png"
 ALT="${\bf a} = {\bf e}_x + 3 {\bf e}_y-{\bf e}_z$">|; 

$key = q/{displaymath}dsigma=2pi,b,db.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img977.png"
 ALT="\begin{displaymath}
d\sigma = 2\pi b db.
\end{displaymath}">|; 

$key = q/displaystylen^2,a^3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img768.png"
 ALT="$\displaystyle n^2 a^3,$">|; 

$key = q/r=sqrt{x^2+y^2+z^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3495.png"
 ALT="$r=\sqrt{x^2+y^2+z^2}$">|; 

$key = q/{displaymath}{{bf{A},{{bf{x}=lambda,{{bf{x}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1283.png"
 ALT="\begin{displaymath}
{\bf A}  {\bf x} = \lambda {\bf x}.
\end{displaymath}">|; 

$key = q/{displaymath}x=v_0,costheta,t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img592.png"
 ALT="\begin{displaymath}
x = v_0 \cos\theta t,
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{4},iota^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2915.png"
 ALT="$\displaystyle \frac{1}{4} \iota^2,$">|; 

$key = q/{displaymath}dtheta=frac{db}{|db(theta)slashdtheta|}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img974.png"
 ALT="\begin{displaymath}
d\theta = \frac{db}{\vert db(\theta)/d\theta\vert}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{AppendixAslashfigA.15.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="227" BORDER="0"
 SRC="|."$dir".q|img3461.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{AppendixA/figA.15.eps}}
\end{figure}">|; 

$key = q/{displaymath}mu=frac{m_1,m_2}{m_1+m_2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img927.png"
 ALT="\begin{displaymath}
\mu = \frac{m_1 m_2}{m_1+ m_2}
\end{displaymath}">|; 

$key = q/-frac{3}{2},iota,x_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2901.png"
 ALT="$-\frac{3}{2} \iota x_4$">|; 

$key = q/displaystylefrac{I_parallel,{mitOmega}}{I_perp,costheta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1529.png"
 ALT="$\displaystyle \frac{I_\parallel {\mit\Omega}}{I_\perp \cos\theta}.$">|; 

$key = q/phi_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img537.png"
 ALT="$\phi_2$">|; 

$key = q/{displaymath}U({{bf{r})=-frac{G,M,m}{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img666.png"
 ALT="\begin{displaymath}
U({\bf r}) = - \frac{G M m}{r}.
\end{displaymath}">|; 

$key = q/displaystyler,sintheta,cosphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2073.png"
 ALT="$\displaystyle r \sin\theta \cos\phi,$">|; 

$key = q/{displaymath}T_phi({{rm{yr})=frac{n_s}{Omega_phi}=frac{2,T_s({{rm{day})}{3,epsilon,costheta},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2299.png"
 ALT="\begin{displaymath}
T_\phi({\rm yr}) = \frac{n_s}{\Omega_\phi} = \frac{2 T_s({\rm day})}{3 \epsilon \cos\theta},
\end{displaymath}">|; 

$key = q/displaystyleB,cos(omega_0,t'-{mitDelta}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img540.png"
 ALT="$\displaystyle B \cos(\omega_0 t'-{\mit\Delta}),$">|; 

$key = q/{displaymath}left(array{{c}x'''y'''z'''array{right)=left(array{{ccc}1&0&00&costharray{right)left(array{{c}x''y''z''array{right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="76" BORDER="0"
 SRC="|."$dir".q|img1432.png"
 ALT="\begin{displaymath}
\left(\begin{array}{c}x'''\ y'''\ z'''\end{array}\right)=
...
...right)\left(\begin{array}{c}x''\ y''\ z''\end{array}\right).
\end{displaymath}">|; 

$key = q/x=r,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2795.png"
 ALT="$x=r \cos\theta$">|; 

$key = q/displaystylefrac{partialL}{partialdot{theta}}=m,r^2,dot{theta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1611.png"
 ALT="$\displaystyle \frac{\partial L}{\partial \dot{\theta}} = m r^2 \dot{\theta},$">|; 

$key = q/{displaymath}n,{{bf{a}equiv(n,a_x,,n,a_y,,n,a_z).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img3232.png"
 ALT="\begin{displaymath}
n {\bf a} \equiv (n a_x, n a_y, n a_z).
\end{displaymath}">|; 

$key = q/{{bf{e}_r=nablarslash|nablar|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3489.png"
 ALT="${\bf e}_r=\nabla r/\vert\nabla r\vert$">|; 

$key = q/displaystyle-n^2,a^3,frac{y_1}{r^3}-frac{3}{2},n'^{,2},sin[2,(n-n'),t],x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2747.png"
 ALT="$\displaystyle -n^2 a^3 \frac{y_1}{r^3}-\frac{3}{2} n'^{ 2} \sin[2 (n-n') t] x_1$">|; 

$key = q/-3,e^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2893.png"
 ALT="$-3 e^2$">|; 

$key = q/{displaymath}{{bf{a}times({{bf{b}+{{bf{c})={{bf{a}times{{bf{b}+{{bf{a}times{{bf{c},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3302.png"
 ALT="\begin{displaymath}
{\bf a}\times({\bf b} +{\bf c})= {\bf a} \times{\bf b}+{\bf a}\times{\bf c},
\end{displaymath}">|; 

$key = q/displaystylefrac{3}{8},I_parallel,mu_m,n_m^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2327.png"
 ALT="$\displaystyle \frac{3}{8} I_\parallel \mu_m n_m^{ 2},$">|; 

$key = q/displaystylev_1,(costheta,,sintheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1028.png"
 ALT="$\displaystyle v_1 (\cos\theta, \sin\theta),$">|; 

$key = q/{{bf{A}^T={{bf{A};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1281.png"
 ALT="${\bf A}^T = {\bf A}$">|; 

$key = q/displaystylefrac{r_p}{1-e},e^2,(1-cos^2E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img804.png"
 ALT="$\displaystyle \frac{r_p}{1-e} e^2 (1-\cos^2 E)$">|; 

$key = q/R';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2218.png"
 ALT="$R'$">|; 

$key = q/{displaymath}ddot{x}=frac{g,(m_1-m_2)}{m_1+m_2+Islasha^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1629.png"
 ALT="\begin{displaymath}
\ddot{x} = \frac{g (m_1-m_2)}{m_1+m_2 + I/a^2},
\end{displaymath}">|; 

$key = q/{displaymath}Phi(r,theta)=-Gint_0^inftyint_0^piint_0^{2pi}frac{r'^{,2},rho(r',thintheta'}{|{{bf{r}-{{bf{r}'|},dr',dtheta',dphi',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="438" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2078.png"
 ALT="\begin{displaymath}
\Phi(r,\theta) = - G\int_0^\infty\int_0^\pi\int_0^{2\pi}
\fr...
...in\theta'}{\vert{\bf r}-{\bf r}'\vert} dr' d\theta' d\phi',
\end{displaymath}">|; 

$key = q/|x|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img293.png"
 ALT="$\vert x\vert$">|; 

$key = q/{{bf{k}cdot{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3552.png"
 ALT="${\bf k}\cdot{\bf r}$">|; 

$key = q/displaystylefrac{12,beta}{7}+frac{144,beta^2}{49}+frac{1567,beta^3}{343}+{calO}(beta^4),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2593.png"
 ALT="$\displaystyle \frac{12 \beta}{7} + \frac{144 \beta^2}{49} + \frac{1567 \beta^3}{343} + {\cal O}(\beta^4),$">|; 

$key = q/{{bf{r}_itimes;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img169.png"
 ALT="${\bf r}_i\times$">|; 

$key = q/(r,,vartheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img955.png"
 ALT="$(r, \vartheta)$">|; 

$key = q/eta_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1972.png"
 ALT="$\eta_k$">|; 

$key = q/(dsigma(theta)slashd{mitOmega}),d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1078.png"
 ALT="$(d\sigma(\theta)/d{\mit\Omega}) d{\mit\Omega}$">|; 

$key = q/r_c+x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img852.png"
 ALT="$r_c+x$">|; 

$key = q/e=0.967;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img885.png"
 ALT="$e=0.967$">|; 

$key = q/displaystyleb;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img720.png"
 ALT="$\displaystyle b$">|; 

$key = q/{displaymath}tilde{Q}_phi=lambda,frac{partialf}{partialphi}=frac{m,g,a,sintheta}{1+m,a^2slashI}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1830.png"
 ALT="\begin{displaymath}
\tilde{Q}_\phi = \lambda \frac{\partial f}{\partial\phi} =
\frac{m g a \sin\theta}{1+ m a^2/I}.
\end{displaymath}">|; 

$key = q/dT=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3459.png"
 ALT="$dT=0$">|; 

$key = q/{displaymath}U=-m_1,g,x-m_2,g,(l-x+x')-m_3,g,(l-x+l'-x').{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="465" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1633.png"
 ALT="\begin{displaymath}
U = - m_1 g x - m_2 g (l-x+x')- m_3 g (l-x+l'-x').
\end{displaymath}">|; 

$key = q/{displaymath}deltaf=frac{partialf}{partialq_1},deltaq_1+frac{partialf}{partialq_2},deltaq_2=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1801.png"
 ALT="\begin{displaymath}
\delta f = \frac{\partial f}{\partial q_1} \delta q_1 + \frac{\partial f}{\partial
q_2} \delta q_2 = 0
\end{displaymath}">|; 

$key = q/stackrel{displaystylerightarrow}{QR};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3190.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{QR}$">|; 

$key = q/n,{{bf{a}=|n|,(-{{bf{a});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3206.png"
 ALT="$n {\bf a} = \vert n\vert (-{\bf a})$">|; 

$key = q/mbox{{boldmath{{omega{}=(omega_{x'},,omega_{y'},,omega_{z'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1357.png"
 ALT="$\mbox{\boldmath$\omega$}= (\omega_{x'}, \omega_{y'}, \omega_{z'})$">|; 

$key = q/{mitOmega}^2,rho={mitOmega}^2,R,coslambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1131.png"
 ALT="${\mit\Omega}^2 \rho = {\mit\Omega}^2 R \cos\lambda$">|; 

$key = q/U_{yy}=-9slash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2686.png"
 ALT="$U_{yy}=-9/4$">|; 

$key = q/displaystylefrac{3}{2},frac{epsilon,(n_s^{,2}+mu_m,n_m^{,2})}{omega},costheta_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2355.png"
 ALT="$\displaystyle \frac{3}{2} \frac{\epsilon (n_s^{ 2}+\mu_m n_m^{ 2})}{\omega} \cos\theta_0,$">|; 

$key = q/dot{theta}=hslashr^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img866.png"
 ALT="$\dot{\theta} = h/r^2$">|; 

$key = q/K=(1slash2),m,dot{x}^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1858.png"
 ALT="$K = (1/2) m \dot{x}^{ 2}$">|; 

$key = q/{calE}=2,E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1090.png"
 ALT="${\cal E} = 2 E$">|; 

$key = q/v_0(t_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3131.png"
 ALT="$v_0(t_0)$">|; 

$key = q/(partialhslashpartialx)_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3441.png"
 ALT="$(\partial h/\partial x)_y$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{AppendixAslashfigA.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="253" BORDER="0"
 SRC="|."$dir".q|img3189.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{AppendixA/figA.02.eps}}
\end{figure}">|; 

$key = q/{displaymath}{{bf{r}=x,{{bf{e}_z+y,{{bf{e}_y+z,{{bf{e}_z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img3217.png"
 ALT="\begin{displaymath}
{\bf r} = x {\bf e}_z + y {\bf e}_y+z {\bf e}_z.
\end{displaymath}">|; 

$key = q/epsilon=(2slash5),R,{mitDelta}R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img908.png"
 ALT="$\epsilon= (2/5) R {\mit\Delta}R$">|; 

$key = q/displaystyleleft.+3,iota_m,sin(2,theta),sin(lambda_m-phi),sin(lambda_m-varpi_n)-1right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="384" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2320.png"
 ALT="$\displaystyle \left.
+ 3 \iota_m \sin(2 \theta) \sin(\lambda_m-\phi) \sin(\lambda_m-\varpi_n)-1\right]$">|; 

$key = q/i=1,{calF};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1559.png"
 ALT="$i=1,{\cal F}$">|; 

$key = q/{displaymath}a_n=x_n,{{rm{e}^{,{{rm{i},phi_n},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img420.png"
 ALT="\begin{displaymath}
a_n = x_n  {\rm e}^{ {\rm i} \phi_n},
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.27.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="274" BORDER="0"
 SRC="|."$dir".q|img3145.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.27.eps}}
\end{figure}">|; 

$key = q/U_x=partialU(x_0,y_0,0)slashpartialx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2641.png"
 ALT="$U_x=\partial U(x_0,y_0,0)/\partial x$">|; 

$key = q/displaystyleR_X,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2827.png"
 ALT="$\displaystyle R_X,$">|; 

$key = q/L=K-U;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1659.png"
 ALT="$L=K-U$">|; 

$key = q/{displaymath}mu_1,rho_1^{,2}+mu_2,rho_2^{,2}=x^2+y^2+mu_1,mu_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2563.png"
 ALT="\begin{displaymath}
\mu_1 \rho_1^{ 2} + \mu_2 \rho_2^{ 2} = x^2+y^2+\mu_1 \mu_2,
\end{displaymath}">|; 

$key = q/Ox;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3214.png"
 ALT="$Ox$">|; 

$key = q/f(q_1,q_2,cdots,q_{calF},t)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1796.png"
 ALT="$f(q_1,q_2,\cdots,q_{\cal F},t)=0$">|; 

$key = q/{displaymath}cos^3uequiv(3slash4),cosu+(1slash4),cos(3,u),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img469.png"
 ALT="\begin{displaymath}
\cos^3 u \equiv (3/4) \cos u+ (1/4) \cos(3 u),
\end{displaymath}">|; 

$key = q/{displaymath}m,frac{d{{bf{v}}{dt}={{bf{f}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="\begin{displaymath}
m \frac{d{\bf v}}{dt} = {\bf f}.
\end{displaymath}">|; 

$key = q/d^3{{bf{r}'=r'^{,2},sintheta',dr',dtheta',dphi';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2077.png"
 ALT="$d^3{\bf r}' = r'^{ 2} \sin\theta' dr' d\theta' d\phi'$">|; 

$key = q/lambda_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3155.png"
 ALT="$\lambda_2$">|; 

$key = q/p_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1679.png"
 ALT="$p_\phi$">|; 

$key = q/x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img549.png"
 ALT="$x'$">|; 

$key = q/deltaY;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2783.png"
 ALT="$\delta Y$">|; 

$key = q/displaystylefrac{d}{dt}!left(frac{partialL}{partialdot{theta}}right)-frac{partialL}{partialtheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1614.png"
 ALT="$\displaystyle \frac{d}{dt}\!\left(\frac{\partial L}{\partial\dot{\theta}}\right)- \frac{\partial L}{\partial \theta}$">|; 

$key = q/deltax_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1568.png"
 ALT="$\delta x_j$">|; 

$key = q/Q_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3106.png"
 ALT="$Q_\infty$">|; 

$key = q/{displaymath}{{bf{a}=left(frac{d}{dt'}+mbox{{boldmath{{Omega{}timesright)left({{f{v}'+mbox{{boldmath{{Omega{}times{{bf{r}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1114.png"
 ALT="\begin{displaymath}
{\bf a} = \left(\frac{d}{dt'}+ \mbox{\boldmath$\Omega$}\time...
...\left({\bf v}' + \mbox{\boldmath$\Omega$}\times{\bf r}\right),
\end{displaymath}">|; 

$key = q/j=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2904.png"
 ALT="$j=2$">|; 

$key = q/AAAAAAcdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3086.png"
 ALT="$AAAAAA\cdots$">|; 

$key = q/I_perp<I_parallel;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1407.png"
 ALT="$I_\perp&lt; I_\parallel$">|; 

$key = q/displaystylev_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1644.png"
 ALT="$\displaystyle v_x$">|; 

$key = q/+x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img696.png"
 ALT="$+x$">|; 

$key = q/v_0(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3128.png"
 ALT="$v_0(t)$">|; 

$key = q/sinthetasimeqr'slashl;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1187.png"
 ALT="$\sin\theta \simeq r'/l$">|; 

$key = q/2370;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3017.png"
 ALT="$2370$">|; 

$key = q/{displaymath}R'^{,-1}=left(R^2-2,{{bf{R}cdot{{bf{r}+r^2right)^{-1slash2}=left(R^2-2,R,r,costheta+r^2right)^{-1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="487" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img2223.png"
 ALT="\begin{displaymath}
R'^{ -1} = \left(R^2 - 2 {\bf R}\cdot{\bf r}+ r^2\right)^{-1/2}
= \left(R^2 - 2 R r \cos\theta+ r^2\right)^{-1/2}.
\end{displaymath}">|; 

$key = q/z_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2731.png"
 ALT="$z_1'$">|; 

$key = q/C>C_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$C&gt; C_1$">|; 

$key = q/{displaymath}x_n=frac{omega_0^{,2}}{left[(omega_0^{,2}-n^2,omega^2)^2+4,nu^2,n^2,omega^2right]^{1slash2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img423.png"
 ALT="\begin{displaymath}
x_n = \frac{\omega_0^{ 2}}{\left[(\omega_0^{ 2}-n^2 \omega^2)^2
+ 4 \nu^2 n^2 \omega^2\right]^{1/2}},
\end{displaymath}">|; 

$key = q/{{bf{a}equiv(0,,1,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3248.png"
 ALT="${\bf a} \equiv (0, 1, 0)$">|; 

$key = q/f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img221.png"
 ALT="$f$">|; 

$key = q/displaystyleiota,sin(n,t-gamma_0).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2818.png"
 ALT="$\displaystyle \iota \sin(n t-\gamma_0).$">|; 

$key = q/e=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img726.png"
 ALT="$e=1$">|; 

$key = q/theta=23.44^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2270.png"
 ALT="$\theta = 23.44^\circ$">|; 

$key = q/{displaymath}{mitOmega}=omega,cosalpha.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1466.png"
 ALT="\begin{displaymath}
{\mit\Omega} = \omega \cos\alpha.
\end{displaymath}">|; 

$key = q/pi,bslashsqrt{a^2+b^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1093.png"
 ALT="$\pi b/\sqrt{a^2+b^2}$">|; 

$key = q/displaystylefrac{omega,Aleft[-(1-omega^2),sin(omega,t)+(omegaslashQ),cos(omega,t)right]}{left[(1-omega^2)^2+omega^2slashQ^2right]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3056.png"
 ALT="$\displaystyle \frac{\omega  A\left[-(1-\omega^2) \sin(\omega  t)+ (\omega/Q) \cos(\omega  t)\right]}
{\left[(1-\omega^2)^2+\omega^2/Q^2\right]}.$">|; 

$key = q/displaystyledeltaX+frac{1}{2},deltaY^{,2}+frac{1}{2},deltaZ^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2799.png"
 ALT="$\displaystyle \delta X +\frac{1}{2} \delta Y^{ 2}+\frac{1}{2} \delta Z^{ 2},$">|; 

$key = q/|epsilon|ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2147.png"
 ALT="$\vert\epsilon\vert\ll 1$">|; 

$key = q/v_t=m,gslashc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img581.png"
 ALT="$v_t = m g/c$">|; 

$key = q/{{bf{b}=3,{{bf{e}_x+2,{{bf{e}_y+{{bf{e}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3514.png"
 ALT="${\bf b} = 3 {\bf e}_x+ 2 {\bf e}_y+{\bf e}_z$">|; 

$key = q/{displaymath}(AB)^2=(OA)^2+(OB)^2-2,(OA),(OB),costheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3268.png"
 ALT="\begin{displaymath}
(AB)^2 = (OA)^2 + (OB)^2 - 2  (OA) (OB) \cos\theta,
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{grad},Tequivleft(frac{partialT}{partialx},,frac{partialT}{partialy},,frac{partialT}{partialz}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3455.png"
 ALT="\begin{displaymath}
{\bf grad} T \equiv \left(\frac{\partial T}{\partial x},  ...
...artial T}{\partial y}, 
\frac{\partial T}{\partial z}\right).
\end{displaymath}">|; 

$key = q/r=sqrt{x^2+y^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img524.png"
 ALT="$r=\sqrt{x^2+y^2}$">|; 

$key = q/displaystyle-frac{e,r_c}{e^2-1}=-e,a.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img732.png"
 ALT="$\displaystyle -\frac{e r_c}{e^2-1}=-e a.$">|; 

$key = q/displaystyle-2,omega,dot{x},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1226.png"
 ALT="$\displaystyle -2 \omega \dot{x},$">|; 

$key = q/{displaymath}epsilon=0.00429,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img2182.png"
 ALT="\begin{displaymath}
\epsilon = 0.00429,
\end{displaymath}">|; 

$key = q/rho_1^{,2}=(x+mu_2)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2667.png"
 ALT="$\rho_1^{ 2} = (x+\mu_2)^2$">|; 

$key = q/{displaymath}V(r)=-frac{G,M}{r}left(1+frac{epsilon}{r^2}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img907.png"
 ALT="\begin{displaymath}
V(r) = - \frac{G M}{r}\left(1+\frac{\epsilon}{r^2}\right),
\end{displaymath}">|; 

$key = q/(dhslashdr)_{{rm{max};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3438.png"
 ALT="$(dh/dr)_{\rm max}$">|; 

$key = q/textstylesimeq;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img476.png"
 ALT="$\textstyle \simeq$">|; 

$key = q/164.8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2442.png"
 ALT="$164.8$">|; 

$key = q/displaystyle-e,cos(T-alpha_0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2808.png"
 ALT="$\displaystyle -e \cos(T-\alpha_0),$">|; 

$key = q/(pmaslash2,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1758.png"
 ALT="$(\pm a/2, 0)$">|; 

$key = q/displaystylefrac{r_c}{1-e^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img719.png"
 ALT="$\displaystyle \frac{r_c}{1-e^2},$">|; 

$key = q/{displaymath}(lambda_k^ast-lambda_k),{{bf{x}_k^{ast,T},{{bf{M},{{bf{x}_k=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1931.png"
 ALT="\begin{displaymath}
(\lambda_k^\ast-\lambda_k) {\bf x}_k^{\ast T} {\bf M} {\bf x}_k = 0.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d{{bf{p}}{dt}={{bf{f},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="\begin{displaymath}
\frac{d{\bf p}}{dt} = {\bf f},
\end{displaymath}">|; 

$key = q/ds=sqrt{dx^2+dy^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1762.png"
 ALT="$ds = \sqrt{dx^2+dy^2}$">|; 

$key = q/displaystyleL_{y'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1400.png"
 ALT="$\displaystyle L_{y'}$">|; 

$key = q/x_c=-r_cslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img727.png"
 ALT="$x_c = -r_c/2$">|; 

$key = q/rho_n(r)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2129.png"
 ALT="$\rho_n(r)=0$">|; 

$key = q/displaystylem_i,frac{d^2{{bf{r}_i}{dt^2}=sum_{j=1,N}^{jneqi}{{bf{f}_{ij}+{{bf{F}_i.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1230.png"
 ALT="$\displaystyle m_i \frac{d^2{\bf r}_i}{dt^2} = \sum_{j=1,N}^{j\neq i}
{\bf f}_{ij} + {\bf F}_i.$">|; 

$key = q/{displaymath}ddot{{bf{r}+2,mbox{{boldmath{{omega{}timesdot{{bf{r}+mbox{{boldmathf{r}'),{{bf{r}'}{|{{bf{r}'|^{,2}}-{{bf{r}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="554" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2742.png"
 ALT="\begin{displaymath}
\ddot{\bf r} +2 \mbox{\boldmath$\omega$}\times\dot{\bf r}+...
...f r}') {\bf r}'}{\vert{\bf r}'\vert^{ 2}} - {\bf r}\right].
\end{displaymath}">|; 

$key = q/displaystylem_1,frac{d^2{{bf{r}_1}{dt^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img915.png"
 ALT="$\displaystyle m_1 \frac{d^2{\bf r}_1}{dt^2}$">|; 

$key = q/{displaymath}b^2=frac{k^2}{4,E^{,2}},cot^2(thetaslash2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img996.png"
 ALT="\begin{displaymath}
b^2 = \frac{k^2}{4 E^{ 2}} \cot^2(\theta/2).
\end{displaymath}">|; 

$key = q/r=r_0,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img899.png"
 ALT="$r=r_0 \cos\theta$">|; 

$key = q/T_s({{rm{day})=365.24;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2363.png"
 ALT="$T_s({\rm day})=365.24$">|; 

$key = q/dl;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3386.png"
 ALT="$dl$">|; 

$key = q/displaystylez.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$\displaystyle z.$">|; 

$key = q/displaystyleA,[1-cos(2,phi)],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img657.png"
 ALT="$\displaystyle A [1-\cos (2 \phi)],$">|; 

$key = q/y'={{rm{constant};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1748.png"
 ALT="$y' = {\rm constant}$">|; 

$key = q/t=T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img404.png"
 ALT="$t=T$">|; 

$key = q/{displaymath}sigma=intfrac{dsigma}{d{mitOmega}},d{mitOmega}=pi,a^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img989.png"
 ALT="\begin{displaymath}
\sigma = \int \frac{d\sigma}{d{\mit\Omega}} d{\mit\Omega} = \pi a^2.
\end{displaymath}">|; 

$key = q/displaystyleddot{y}+2,omega,dot{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2541.png"
 ALT="$\displaystyle \ddot{y} + 2 \omega \dot{x}$">|; 

$key = q/y=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3551.png"
 ALT="$y=a$">|; 

$key = q/{displaymath}ddot{r}=-h,frac{d^2u}{dtheta^2},dot{theta}=-u^2,h^2,frac{d^2u}{dtheta^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img746.png"
 ALT="\begin{displaymath}
\ddot{r} = - h  \frac{d^2 u}{d\theta^2} \dot{\theta} = - u^2 h^2 \frac{d^2 u}{d\theta^2}.
\end{displaymath}">|; 

$key = q/{{bf{r}_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2695.png"
 ALT="${\bf r}_E$">|; 

$key = q/|{{bf{F}|,|{{bf{r}|,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3321.png"
 ALT="$\vert{\bf F}\vert \vert{\bf r}\vert \sin\theta$">|; 

$key = q/{displaymath}M=bar{theta}-alpha,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img2995.png"
 ALT="\begin{displaymath}
M = \bar{\theta} - \alpha,
\end{displaymath}">|; 

$key = q/z''';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1428.png"
 ALT="$z'''$">|; 

$key = q/mu=m_1,m_2slash(m_1+m_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img930.png"
 ALT="$\mu= m_1 m_2/(m_1+m_2)$">|; 

$key = q/rho_2=1+rho_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2589.png"
 ALT="$\rho_2=1+\rho_1$">|; 

$key = q/dot{{bf{r}=ddot{{bf{r}={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2560.png"
 ALT="$\dot{\bf r} =\ddot{\bf r} = {\bf0}$">|; 

$key = q/{displaymath}{{bf{a}'simeq{{bf{a}+deltatheta_z,{{bf{e}_ztimes{{bf{a},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img3329.png"
 ALT="\begin{displaymath}
{\bf a}' \simeq {\bf a} +\delta\theta_z  {\bf e}_z\times {\bf a},
\end{displaymath}">|; 

$key = q/{displaymath}frac{dU}{dx}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img217.png"
 ALT="\begin{displaymath}
\frac{dU}{dx} = 0.
\end{displaymath}">|; 

$key = q/{{bf{A}^ast={{bf{A};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1280.png"
 ALT="${\bf A}^\ast = {\bf A}$">|; 

$key = q/5.75;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2446.png"
 ALT="$5.75$">|; 

$key = q/chi+Phi';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2230.png"
 ALT="$\chi+\Phi'$">|; 

$key = q/{displaymath}{{bf{l}={{bf{r}times{{bf{p}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img3325.png"
 ALT="\begin{displaymath}
{\bf l}={\bf r}\times {\bf p}.
\end{displaymath}">|; 

$key = q/Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$Q$">|; 

$key = q/{displaymath}left(array{{cc}gamma^2+U_{xx}&-2,gamma+U_{xy}[0.5ex]2,gamma+U_{xy}&rray{right)=left(array{{c}0[0.5ex]0array{right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2651.png"
 ALT="\begin{displaymath}
\left(
\begin{array}{cc}
\gamma^2 + U_{xx}&amp; -2 \gamma+U_{xy...
... = \left(
\begin{array}{c}
0\ [0.5ex]
0
\end{array}\right).
\end{displaymath}">|; 

$key = q/{displaymath}frac{d}{dt}!left(frac{partialK}{partialdot{q}_i}right)=sum_{j=1,{catialx_j}{partialq_i}+frac{partialK}{partialq_i},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="61" BORDER="0"
 SRC="|."$dir".q|img1600.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial K}{\partial \dot{q}_i}\ri...
...\partial x_j}{\partial q_i} + \frac{\partial K}{\partial q_i},
\end{displaymath}">|; 

$key = q/deltay'=d(deltay)slashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1728.png"
 ALT="$\delta y' = d(\delta y)/dx$">|; 

$key = q/hat{mbox{{boldmath{{omega{}}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1317.png"
 ALT="$\hat{\mbox{\boldmath$\omega$}}_i$">|; 

$key = q/{displaymath}psi+zeta=frac{pi}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1069.png"
 ALT="\begin{displaymath}
\psi + \zeta =\frac{\pi}{2}.
\end{displaymath}">|; 

$key = q/{{bf{r}_2-{{bf{r}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img951.png"
 ALT="${\bf r}_2-{\bf r}_1$">|; 

$key = q/{{bf{r}equiv(x,,y,,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="${\bf r}\equiv (x, y, z)$">|; 

$key = q/displaystylen,({{bf{a}+{{bf{b});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3212.png"
 ALT="$\displaystyle n ({\bf a} + {\bf b})$">|; 

$key = q/{{bf{grad},h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3437.png"
 ALT="${\bf grad} h$">|; 

$key = q/displaystyleint_0^az,dzint_{-(a-z)slash2}^{(a-z)slash2}dyint_{-(a-z)slash2}^{(a-z)slash2}dx=int_0^az,(a-z)^2,dz=int_0^a(z,a^2-2,a,z^2+z^3),dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="624" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3428.png"
 ALT="$\displaystyle \int_0^a z dz \int_{-(a-z)/2}^{(a-z)/2} dy\int_{-(a-z)/2}^{(a-z)/2} dx = \int_0^a z (a-z)^2 dz=\int_0^a (z a^2-2 a z^2+z^3) dz$">|; 

$key = q/x_j=x_j(q_1,q_2,cdots,q_{calF},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1585.png"
 ALT="$x_j=x_j(q_1,q_2,\cdots, q_{\cal F},t)$">|; 

$key = q/displaystyleT+2,e,sin[(1+c,m^2),T-alpha_0]+frac{5}{4},e^2,sin[2,(1+c,m^2),T-2,alpha_0];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="511" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2968.png"
 ALT="$\displaystyle T + 2 e \sin[(1+c m^2) T-\alpha_0] + \frac{5}{4} e^2 \sin[2 (1+c m^2) T-2 \alpha_0]$">|; 

$key = q/{displaymath}int_a^bleft[frac{partialF}{partialy}-frac{d}{dx}!left(frac{partialFft[frac{partialF}{partialy'},deltayright]_a^b=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1731.png"
 ALT="\begin{displaymath}
\int_a^b\left[\frac{\partial F}{\partial y}- \frac{d}{dx}\!\...
... +\left[\frac{\partial F}{\partial y'} \delta y\right]_a^b=0.
\end{displaymath}">|; 

$key = q/deltav;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3149.png"
 ALT="$\delta v$">|; 

$key = q/E_1'=(1slash2),m_1,v_1'^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1039.png"
 ALT="$E_1'=(1/2) m_1 v_1'^{ 2}$">|; 

$key = q/omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img311.png"
 ALT="$\omega$">|; 

$key = q/displaystyler;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img763.png"
 ALT="$\displaystyle r$">|; 

$key = q/{displaymath}p_i=frac{partialL}{partialdot{q}_i},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1661.png"
 ALT="\begin{displaymath}
p_i = \frac{\partial L}{\partial\dot{q}_i},
\end{displaymath}">|; 

$key = q/displaystylea_{z'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3244.png"
 ALT="$\displaystyle a_{z'}$">|; 

$key = q/displaystyle(m_1slashm_2),v_1,(-1,,0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1026.png"
 ALT="$\displaystyle (m_1/m_2) v_1 (-1, 0),$">|; 

$key = q/rho(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2127.png"
 ALT="$\rho(r,\theta)$">|; 

$key = q/displaystyleT_{z'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1366.png"
 ALT="$\displaystyle T_{z'}$">|; 

$key = q/displaystylem,ddot{x}-m,g,sintheta-lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1822.png"
 ALT="$\displaystyle m \ddot{x} - m g \sin\theta - \lambda$">|; 

$key = q/R=6.37times10^6,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1132.png"
 ALT="$R=6.37\times 10^6 {\rm m}$">|; 

$key = q/{displaymath}r=sqrt{x^2+y^2+z^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img3228.png"
 ALT="\begin{displaymath}
r = \sqrt{x^2 + y^2 + z^2}.
\end{displaymath}">|; 

$key = q/{displaymath}lambda=A,{{rm{e}^{,k,t'}+B,{{rm{e}^{-k,t'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1551.png"
 ALT="\begin{displaymath}
\lambda = A {\rm e}^{ k t'} + B {\rm e}^{-k t'}.
\end{displaymath}">|; 

$key = q/theta=cos^{-1}(zslashr);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3496.png"
 ALT="$\theta = \cos^{-1}(z/r)$">|; 

$key = q/{displaymath}H=K+U.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1846.png"
 ALT="\begin{displaymath}
H = K+ U.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}=frac{d{{bf{v}}{dt}=frac{d^2{{bf{r}}{dt^2}=ddot{r},{{bf{e}_r}),{{bf{e}_theta+r,dot{theta},,dot{{bf{e}_theta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="421" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img689.png"
 ALT="\begin{displaymath}
{\bf a} = \frac{d{\bf v}}{dt} = \frac{d^2{\bf r}}{dt^2}= \dd...
...eta}) {\bf e}_\theta + r \dot{\theta}  \dot{\bf e}_\theta.
\end{displaymath}">|; 

$key = q/displaystyleb_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2942.png"
 ALT="$\displaystyle b_1$">|; 

$key = q/xll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img263.png"
 ALT="$x\ll 1$">|; 

$key = q/{{bf{E}cdot{{bf{B}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img621.png"
 ALT="${\bf E}\cdot{\bf B} = 0$">|; 

$key = q/11.86;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2433.png"
 ALT="$11.86$">|; 

$key = q/17.60;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2453.png"
 ALT="$17.60$">|; 

$key = q/displaystylea',cos[(n-n'),t],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2738.png"
 ALT="$\displaystyle a' \cos[(n-n') t],$">|; 

$key = q/Q=1.348;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3093.png"
 ALT="$Q=1.348$">|; 

$key = q/{displaymath}frac{d}{dx}!left(y',frac{partialF}{partialy'}right)=y',frac{partialF}{partialy}+y'',frac{partialF}{partialy'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1741.png"
 ALT="\begin{displaymath}
\frac{d}{dx}\!\left(y' \frac{\partial F}{\partial y'}\right...
...\partial F}{\partial y} + y'' \frac{\partial F}{\partial y'}.
\end{displaymath}">|; 

$key = q/l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img449.png"
 ALT="$l$">|; 

$key = q/displaystylefrac{3}{4},I_parallel,iota_m,mu_m,n_m^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2329.png"
 ALT="$\displaystyle \frac{3}{4} I_\parallel \iota_m \mu_m n_m^{ 2},$">|; 

$key = q/{{bf{a}cdot{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3255.png"
 ALT="${\bf a} \cdot {\bf b}$">|; 

$key = q/thetall1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img459.png"
 ALT="$\theta\ll 1$">|; 

$key = q/{displaymath}cosgamma_m=frac{mbox{{boldmath{{omega{}cdot{{bf{r}_m}{|mbox{{boldmada_m-phi)+iota_m,costheta,sin(lambda_m-varpi_n),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="523" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2318.png"
 ALT="\begin{displaymath}
\cos\gamma_m = \frac{\mbox{\boldmath$\omega$}\cdot{\bf r}_m}...
...\lambda_m-\phi)+\iota_m \cos\theta \sin(\lambda_m-\varpi_n),
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter15slashfig15.21.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="345" BORDER="0"
 SRC="|."$dir".q|img3096.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter15/fig15.21.eps}}
\end{figure}">|; 

$key = q/costhetasimeq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3330.png"
 ALT="$\cos\theta\simeq 1$">|; 

$key = q/displaystyletheta(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2340.png"
 ALT="$\displaystyle \theta(t)$">|; 

$key = q/cos({{bf{k}cdot{{bf{r}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3555.png"
 ALT="$\cos({\bf k}\cdot {\bf r}).$">|; 

$key = q/displaystyle-frac{1}{omega_ast}left{theta(0)+frac{v(0)}{2Q}-frac{A,[(1-omega^2)-2+omega^2slashQ^2]}right}{{rm{e}^{-tslash2Q},sin(omega_ast,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="524" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3049.png"
 ALT="$\displaystyle -\frac{1}{\omega_\ast}\left\{ \theta(0) + \frac{v(0)}{2Q}
-\frac{...
...
{[(1-\omega^2)^2+ \omega^2/Q^2]}\right\}{\rm e}^{-t/2Q} \sin(\omega_\ast  t)$">|; 

$key = q/x'simeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1159.png"
 ALT="$x'\simeq 0$">|; 

$key = q/I_{xz}=I_{yz}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1345.png"
 ALT="$I_{xz}= I_{yz} = 0$">|; 

$key = q/1.3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2246.png"
 ALT="$1.3$">|; 

$key = q/3pislash2<omega,t<5pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img519.png"
 ALT="$3\pi/2&lt;\omega t&lt;5\pi/2$">|; 

$key = q/p_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1662.png"
 ALT="$p_i$">|; 

$key = q/phi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$\phi =0$">|; 

$key = q/7.42;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2455.png"
 ALT="$7.42$">|; 

$key = q/m_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$m_2$">|; 

$key = q/{displaymath}sum_{i=1,N}m_i,frac{d^2{{bf{r}_i}{dt^2}=sum_{i,j=1,N}^{jneqi}{{bf{f}_{ij}+sum_{i=1,N}{{bf{F}_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="67" BORDER="0"
 SRC="|."$dir".q|img166.png"
 ALT="\begin{displaymath}
\sum_{i=1,N} m_i \frac{d^2{\bf r}_i}{dt^2} = \sum_{i,j=1,N}^{j\neq i} {\bf f}_{ij} + \sum_{i=1,N}{\bf F}_i,
\end{displaymath}">|; 

$key = q/z_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2725.png"
 ALT="$z_1$">|; 

$key = q/phi=tan^{-1}(yslashx);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3497.png"
 ALT="$\phi=\tan^{-1}(y/x)$">|; 

$key = q/{displaymath}X_0=0.2,A,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img430.png"
 ALT="\begin{displaymath}
X_0 = 0.2 A,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter13slashfig13.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="249" BORDER="0"
 SRC="|."$dir".q|img2483.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter13/fig13.01.eps}}
\end{figure}">|; 

$key = q/{displaymath}tilde{A}=frac{A}{sqrt{(1-omega^2)^2+omega^2slashQ^2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img3058.png"
 ALT="\begin{displaymath}
\tilde{A} = \frac{A}{\sqrt{(1-\omega^2)^2+\omega^2/Q^2}}.
\end{displaymath}">|; 

$key = q/l=OP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1703.png"
 ALT="$l=OP$">|; 

$key = q/displaystyle-frac{3}{2},m^{2},cos[2,(1-m),T],Y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2766.png"
 ALT="$\displaystyle -\frac{3}{2} m^{2} 
\cos[2 (1-m) T] Y,$">|; 

$key = q/q_1=-q_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2017.png"
 ALT="$q_1=-q_2$">|; 

$key = q/E_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img213.png"
 ALT="$E_2$">|; 

$key = q/displaystylemu_2,(-cosomegat,,-sinomegat,,0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2500.png"
 ALT="$\displaystyle \mu_2 (-\cos \omega t, -\sin\omega t, 0),$">|; 

$key = q/{bf{F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3279.png"
 ALT="$\bf F$">|; 

$key = q/{displaymath}csimeq-frac{3}{4}-frac{225}{32},m+{calO}(m^2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2952.png"
 ALT="\begin{displaymath}
c \simeq- \frac{3}{4} - \frac{225}{32} m + {\cal O}(m^2).
\end{displaymath}">|; 

$key = q/{displaymath}rho_2=alpha-frac{alpha^2}{3}-frac{alpha^3}{9}-frac{23,alpha^4}{81}+{calO}(alpha^5),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2579.png"
 ALT="\begin{displaymath}
\rho_2 = \alpha - \frac{\alpha^2}{3} - \frac{\alpha^3}{9} -\frac{23 \alpha^4}{81}+{\cal O}(\alpha^5),
\end{displaymath}">|; 

$key = q/{calF};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1560.png"
 ALT="${\cal F}$">|; 

$key = q/J;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1753.png"
 ALT="$J$">|; 

$key = q/omega_perp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1389.png"
 ALT="$\omega_\perp$">|; 

$key = q/int_P^Q{{bf{grad},Tcdotd{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3464.png"
 ALT="$\int_P^Q {\bf grad } T\cdot d{\bf r}$">|; 

$key = q/displaystyle{{bf{r}_{cm}-frac{m_2}{m_1+m_2},{{bf{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img922.png"
 ALT="$\displaystyle {\bf r}_{cm} - \frac{m_2}{m_1+m_2} {\bf r},$">|; 

$key = q/2:1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1229.png"
 ALT="$2:1$">|; 

$key = q/D=0^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3004.png"
 ALT="$D=0^\circ$">|; 

$key = q/theta_1(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3134.png"
 ALT="$\theta_1(t)$">|; 

$key = q/displaystyle3,omega^2,x+2,omega,dot{y},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1224.png"
 ALT="$\displaystyle 3 \omega^2 x + 2 \omega \dot{y},$">|; 

$key = q/Q>1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3052.png"
 ALT="$Q&gt;1/2$">|; 

$key = q/{displaymath}costheta=-1+frac{2,p}{a},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img894.png"
 ALT="\begin{displaymath}
\cos\theta = -1 + \frac{2 p}{a},
\end{displaymath}">|; 

$key = q/phi=tan^{-1}(bslasha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img701.png"
 ALT="$\phi=\tan^{-1}(b/a)$">|; 

$key = q/dAslashdt=hslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img753.png"
 ALT="$dA/dt=h/2$">|; 

$key = q/displaystyledeltaddot{Z}+(1+3,m^2slash2),deltaZ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2793.png"
 ALT="$\displaystyle \delta\ddot{Z} + (1+3 m^2/2) \delta Z$">|; 

$key = q/{displaymath}ddot{x}+left[-frac{3,f(r_c)}{r_c}-f'(r_c)right]x=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img856.png"
 ALT="\begin{displaymath}
\ddot{x} + \left[-\frac{3 f(r_c)}{r_c} - f'(r_c)\right] x = 0.
\end{displaymath}">|; 

$key = q/R_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img982.png"
 ALT="$R_1$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.11.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="274" BORDER="0"
 SRC="|."$dir".q|img3077.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.11.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{AppendixAslashfigA.02a.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="198" BORDER="0"
 SRC="|."$dir".q|img3204.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.02a.eps}}
\end{figure}">|; 

$key = q/displaystyle-frac{1}{4},iota^2,sin[2,(1+g,m^2),T-2,gamma_0]+frac{11}{8},m^2,cos[2,(1-m),T];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="469" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2960.png"
 ALT="$\displaystyle - \frac{1}{4} \iota^2 \sin[2 (1+g m^2) T-2 \gamma_0] + \frac{11}{8} m^2 \cos[2 (1-m) T]$">|; 

$key = q/{figure}{epsfysize{=5in{centerline{epsffile{Chapter13slashfig13.09.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="434" HEIGHT="400" BORDER="0"
 SRC="|."$dir".q|img2629.png"
 ALT="\begin{figure}
\epsfysize =5in
\centerline{\epsffile{Chapter13/fig13.09.eps}}
\end{figure}">|; 

$key = q/2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img344.png"
 ALT="$2\pi$">|; 

$key = q/a=r_pslash(e-1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img837.png"
 ALT="$a=r_p/(e-1)$">|; 

$key = q/M_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2393.png"
 ALT="$M_0$">|; 

$key = q/D=3slash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2684.png"
 ALT="$D=3/4$">|; 

$key = q/displaystylelambda_k,{{bf{M},{{bf{x}_k,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1937.png"
 ALT="$\displaystyle \lambda_k {\bf M} {\bf x}_k,$">|; 

$key = q/Omega_5=omega_4-Omega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2876.png"
 ALT="$\Omega_5=\omega_4-\Omega_1$">|; 

$key = q/x_b(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img413.png"
 ALT="$x_b(t)$">|; 

$key = q/y_i(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1782.png"
 ALT="$y_i(x)$">|; 

$key = q/{{bf{F}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img164.png"
 ALT="${\bf F}_i$">|; 

$key = q/displaystylefrac{3}{2},frac{epsilon,iota_m,mu_m,n_m^{,2}}{omega,Omega_n},costheta_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2359.png"
 ALT="$\displaystyle \frac{3}{2} \frac{\epsilon \iota_m \mu_m n_m^{ 2}}{\omega \Omega_n} \cos\theta_0.$">|; 

$key = q/theta=tan^{-1}(yslashx);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img673.png"
 ALT="$\theta=\tan^{-1}(y/x)$">|; 

$key = q/{displaymath}{mitOmega}=dot{phi}_0,sqrt{1+3,cos^2theta_0}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img1695.png"
 ALT="\begin{displaymath}
{\mit\Omega} = \dot{\phi}_0  \sqrt{1+ 3 \cos^2\theta_0}.
\end{displaymath}">|; 

$key = q/U=U(q_1,q_2,cdots,q_{calF});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1889.png"
 ALT="$U=U(q_1,q_2,\cdots, q_{\cal F})$">|; 

$key = q/thetasimeqalpha=0.2'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1478.png"
 ALT="$\theta\simeq \alpha = 0.2''$">|; 

$key = q/c=C,rho_a,A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img281.png"
 ALT="$c=C \rho_a A$">|; 

$key = q/-{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3201.png"
 ALT="$-{\bf a}$">|; 

$key = q/n=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img876.png"
 ALT="$n=2$">|; 

$key = q/frac{3}{2},m^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2897.png"
 ALT="$\frac{3}{2} m^2$">|; 

$key = q/{displaymath}vartheta_0left[omega_0^{,2}-omega^2-(1slash8),omega_0^{,2},varthetaega_0^{,2},vartheta_0^{,3},cos(3,omega,t)simeq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="554" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img470.png"
 ALT="\begin{displaymath}
\vartheta_0\left[\omega_0^{ 2}-\omega^2 - (1/8) \omega_0^{...
...omega_0^{ 2} \vartheta_0^{ 3}  \cos(3 \omega t)\simeq 0.
\end{displaymath}">|; 

$key = q/v_parallel;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img638.png"
 ALT="$v_\parallel$">|; 

$key = q/displaystyle(dot{phi})_{{rm{fast};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1528.png"
 ALT="$\displaystyle (\dot{\phi})_{\rm fast}$">|; 

$key = q/{displaymath}x=|a|,cos[{{rm{arg}(a)-{{rm{Re}(omega),t],{{rm{e}^{,{{rm{Im}(omega),t}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img312.png"
 ALT="\begin{displaymath}
x = \vert a\vert \cos[{\rm arg}(a)-{\rm Re}(\omega) t] {\rm e}^{ {\rm Im}(\omega) t}.
\end{displaymath}">|; 

$key = q/S_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2721.png"
 ALT="$S_1$">|; 

$key = q/18.36;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2457.png"
 ALT="$18.36$">|; 

$key = q/K(q_1,q_2,cdots,q_{calF},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1787.png"
 ALT="$K(q_1,q_2,\cdots,q_{\cal F},$">|; 

$key = q/{displaymath}e=frac{r_a-r_p}{r_a+r_p}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img760.png"
 ALT="\begin{displaymath}
e = \frac{r_a-r_p}{r_a+r_p}
\end{displaymath}">|; 

$key = q/5.32+0.43=5.75;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2474.png"
 ALT="$5.32+0.43= 5.75$">|; 

$key = q/{displaymath}frac{dotphi}{omega}simeq-frac{1}{2},epsilonleft(frac{R}{L}right)^2,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2480.png"
 ALT="\begin{displaymath}
\frac{\dot\phi}{\omega} \simeq - \frac{1}{2} \epsilon\left(\frac{R}{L}\right)^2 \cos\theta.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}times{{bf{b}=-{{bf{b}times{{bf{a},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3301.png"
 ALT="\begin{displaymath}
{\bf a}\times{\bf b} = - {\bf b} \times{\bf a},
\end{displaymath}">|; 

$key = q/displaystyleE_2'=left(frac{m_1}{m_1+m_2}right)E.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1049.png"
 ALT="$\displaystyle E_2'=\left(\frac{m_1}{m_1+m_2}\right) E.$">|; 

$key = q/V(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img838.png"
 ALT="$V(r)$">|; 

$key = q/m_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$m_1$">|; 

$key = q/n=m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2113.png"
 ALT="$n=m$">|; 

$key = q/{{bf{T}=sum_{i=1,N}{{bf{r}_itimes{{bf{F}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1237.png"
 ALT="${\bf T} = \sum_{i=1, N} {\bf r}_i\times{\bf F}_i$">|; 

$key = q/{displaymath}E_{n+1}=2pileft(frac{t}{T}right)+e,sinE_n.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img811.png"
 ALT="\begin{displaymath}
E_{n+1} = 2\pi\left(\frac{t}{T}\right) + e \sin E_n.
\end{displaymath}">|; 

$key = q/displaystyle{{bf{R};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1136.png"
 ALT="$\displaystyle {\bf R}$">|; 

$key = q/displaystylev_x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$\displaystyle v_x'$">|; 

$key = q/displaystylerho_2^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2511.png"
 ALT="$\displaystyle \rho_2^{ 2}$">|; 

$key = q/v_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img963.png"
 ALT="$v_\infty$">|; 

$key = q/displaystyleR_Y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2832.png"
 ALT="$\displaystyle R_Y$">|; 

$key = q/displaystyle-frac{1}{omega_ast}left{deltatheta_0+frac{deltav_0}{2Q}right}sin(omega_ast,{mitDelta}t),,{{rm{e}^{-{mitDelta}tslash2Q},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3143.png"
 ALT="$\displaystyle - \frac{1}{\omega_\ast}\left\{\delta \theta_0 + \frac{\delta v_0}{2Q}\right\}
\sin(\omega_\ast  {\mit\Delta} t)  {\rm e}^{- {\mit\Delta} t/2Q},$">|; 

$key = q/{displaymath}p_i=frac{partialL}{partialdot{q}_i}=frac{partialK}{partialdot{q}_i},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1844.png"
 ALT="\begin{displaymath}
p_i = \frac{\partial L}{\partial \dot{q}_i} = \frac{\partial K}{\partial \dot{q}_i},
\end{displaymath}">|; 

$key = q/displaystyle-frac{g}{l},x'+2,{mitOmega},sinlambda,dot{y}',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1193.png"
 ALT="$\displaystyle - \frac{g}{l} x' + 2 {\mit\Omega} \sin\lambda \dot{y}',$">|; 

$key = q/1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img358.png"
 ALT="$1/2$">|; 

$key = q/iota_m=5.16^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2308.png"
 ALT="$\iota_m=5.16^\circ$">|; 

$key = q/sinthetasimeqtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img460.png"
 ALT="$\sin\theta\simeq \theta$">|; 

$key = q/tanpsi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1058.png"
 ALT="$\tan\psi$">|; 

$key = q/{displaymath}tilde{Q}_x=lambda,frac{partialf}{partialx}=-frac{m,g,sintheta}{1+m,a^2slashI}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1828.png"
 ALT="\begin{displaymath}
\tilde{Q}_x = \lambda \frac{\partial f}{\partial x} = - \frac{m g \sin\theta}{1+ m a^2/I}.
\end{displaymath}">|; 

$key = q/{displaymath}deltaI=int_a^bleft(frac{partialF}{partialy},deltay+frac{partialF}{partialy'},deltay'right)dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1727.png"
 ALT="\begin{displaymath}
\delta I = \int_a^b\left(\frac{\partial F}{\partial y} \delta y+ \frac{\partial F}{\partial y'} \delta y'\right)dx,
\end{displaymath}">|; 

$key = q/(m,l^2omega^2slash12),sinalpha,cosalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1558.png"
 ALT="$(m l^2\omega^2/12) \sin\alpha \cos\alpha$">|; 

$key = q/v_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img339.png"
 ALT="$v_0=0$">|; 

$key = q/a=6.37times10^6,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2179.png"
 ALT="$a=6.37\times 10^6 {\rm m}$">|; 

$key = q/displaystyleddot{z}_1+n'^{,2},z_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2749.png"
 ALT="$\displaystyle \ddot{z}_1 + n'^{ 2} z_1$">|; 

$key = q/{displaymath}U=M_s,Phi=-frac{G,M_s,M}{a_s}+frac{G,M_s,(I_parallel-I_perp)}{a_s^{,3}},P_2[cos(gamma_s)],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2271.png"
 ALT="\begin{displaymath}
U = M_s \Phi = - \frac{G M_s M}{a_s} + \frac{G M_s (I_\parallel-I_\perp)}{a_s^{ 3}} P_2[\cos(\gamma_s)],
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}phisimeqdot{phi}_0,frac{pi}{{mitOmega}}=frac{pi}{sqrt{1+3,cos^2theta_0}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1696.png"
 ALT="\begin{displaymath}
{\mit\Delta}\phi \simeq \dot{\phi}_0 \frac{\pi}{{\mit\Omega}}= \frac{\pi}{\sqrt{1+3 \cos^2\theta_0}}
\end{displaymath}">|; 

$key = q/{displaymath}G_{ij}=-frac{partial^2U(q_{1,0},q_{2,0},cdots,q_{{calF},0})}{partialq_i,partialq_j}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1905.png"
 ALT="\begin{displaymath}
G_{ij} = -\frac{\partial^2 U(q_{1 0},q_{2 0},\cdots, q_{{\cal F} 0})}{\partial q_i \partial q_j}.
\end{displaymath}">|; 

$key = q/k_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img502.png"
 ALT="$k_2$">|; 

$key = q/{displaymath}frac{dE}{dt}=m,frac{dx}{dt},frac{d^2x}{dt^2}+m,omega_0^{,2},x,frac{}{dt}left(frac{d^2x}{dt^2}+omega_0^{,2},xright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img341.png"
 ALT="\begin{displaymath}
\frac{dE}{dt} = m \frac{dx}{dt} \frac{d^2 x}{dt^2} + m \o...
...ac{dx}{dt}\left(\frac{d^2 x}{dt^2} + \omega_0^{ 2} x\right).
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter12slashfig12.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="257" BORDER="0"
 SRC="|."$dir".q|img2178.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter12/fig12.02.eps}}
\end{figure}">|; 

$key = q/bar{theta}=gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2989.png"
 ALT="$\bar{\theta}=\gamma$">|; 

$key = q/{{bf{a}cdot{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3266.png"
 ALT="${\bf a} \cdot {\bf a}$">|; 

$key = q/displaystyleI_{xz}=I_{zx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1253.png"
 ALT="$\displaystyle I_{xz}=I_{zx}$">|; 

$key = q/dh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3435.png"
 ALT="$dh$">|; 

$key = q/displaystylealpha_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2326.png"
 ALT="$\displaystyle \alpha_m$">|; 

$key = q/displaystylefrac{m,g}{omega_0,nu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3034.png"
 ALT="$\displaystyle \frac{m g}{\omega_0 \nu},$">|; 

$key = q/{{bf{G}^{ast,T}={{bf{G};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1926.png"
 ALT="${\bf G}^{\ast T} = {\bf G}$">|; 

$key = q/{displaymath}tilde{Q}_1=lambda(t),frac{partialf}{partialq_1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1813.png"
 ALT="\begin{displaymath}
\tilde{Q}_1 = \lambda(t) \frac{\partial f}{\partial q_1},
\end{displaymath}">|; 

$key = q/{displaymath}frac{d{{bf{a}}{dt}=frac{dphi}{dt},{{bf{b}+phi,frac{d{{bf{b}}{dt}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img3369.png"
 ALT="\begin{displaymath}
\frac{d {\bf a}}{dt} = \frac{d\phi}{dt}  {\bf b} + \phi  \frac{d {\bf b}}{dt}.
\end{displaymath}">|; 

$key = q/r>r';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2087.png"
 ALT="$r &gt; r'$">|; 

$key = q/{displaymath}costheta=frac{{{bf{a}cdot{{bf{b}}{|{{bf{a}|,|{{bf{b}|}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img3276.png"
 ALT="\begin{displaymath}
\cos\theta = \frac{{\bf a} \cdot {\bf b}}{\vert{\bf a}\vert  \vert{\bf b}\vert }.
\end{displaymath}">|; 

$key = q/{{bf{x}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1290.png"
 ALT="${\bf x}_i$">|; 

$key = q/Q=0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$Q=0.5$">|; 

$key = q/{displaymath}{{bf{h}=alpha,{{bf{A}+beta,{{bf{B}+gamma,{{bf{C}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img3547.png"
 ALT="\begin{displaymath}
{\bf h} = \alpha {\bf A} + \beta {\bf B} + \gamma {\bf C}.
\end{displaymath}">|; 

$key = q/dot{mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1542.png"
 ALT="$\dot{\mu}$">|; 

$key = q/theta(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$\theta (0)=0$">|; 

$key = q/displaystylebeta_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2328.png"
 ALT="$\displaystyle \beta_m$">|; 

$key = q/displaystyleomega^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2487.png"
 ALT="$\displaystyle \omega^2$">|; 

$key = q/{displaymath}K=frac{1}{2}left(I_{xx},omega_x^{,2}+I_{yy},omega_y^{,2}+I_{zz},ome,omega_y,omega_z+2,I_{xz},omega_x,omega_zright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="623" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1278.png"
 ALT="\begin{displaymath}
K = \frac{1}{2}\left(I_{xx} \omega_x^{ 2}+ I_{yy} \omega_...
...z} \omega_y \omega_z + 2 I_{xz} \omega_x \omega_z\right).
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{h}cdot{{bf{r}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img670.png"
 ALT="\begin{displaymath}
{\bf h}\cdot{\bf r} = 0.
\end{displaymath}">|; 

$key = q/{displaymath}1-b^2,u_{max}^2-U(u_{max})slashE=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img970.png"
 ALT="\begin{displaymath}
1 - b^2 u_{max}^2 - U(u_{max})/E = 0.
\end{displaymath}">|; 

$key = q/{{bf{c}cdot{{bf{a}-{{bf{c}cdot{{bf{b}=c,|{{bf{a}-{{bf{b}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3523.png"
 ALT="${\bf c}\cdot{\bf a} - {\bf c}\cdot{\bf b} = c \vert{\bf a}-{\bf b}\vert$">|; 

$key = q/1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img615.png"
 ALT="$1$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter15slashfig15.36.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="345" BORDER="0"
 SRC="|."$dir".q|img3174.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter15/fig15.36.eps}}
\end{figure}">|; 

$key = q/nu=omega_0slash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img336.png"
 ALT="$\nu=\omega_0/4$">|; 

$key = q/{displaymath}R=frac{v_0^{,2},sin(2,theta)}{g}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img603.png"
 ALT="\begin{displaymath}
R = \frac{v_0^{ 2} \sin(2 \theta)}{g}
\end{displaymath}">|; 

$key = q/nuslashomega_0=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img377.png"
 ALT="$\nu/\omega_0 = 1$">|; 

$key = q/displaystyle{mitOmega},v_y',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img629.png"
 ALT="$\displaystyle {\mit\Omega} v_y',$">|; 

$key = q/displaystyle+frac{omega,Aleft[-(1-omega^2),sin(omega,t)+(omegaslashQ),cos(omega,t)right]}{left[(1-omega^2)^2+omega^2slashQ^2right]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3050.png"
 ALT="$\displaystyle +\frac{\omega A\left[-(1-\omega^2) \sin(\omega  t)+ (\omega/Q) \cos(\omega  t)\right]}
{\left[(1-\omega^2)^2+\omega^2/Q^2\right]}.$">|; 

$key = q/displaystyle{{bf{g}_c=r,Omega^2,sintheta,(sintheta,,costheta,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2168.png"
 ALT="$\displaystyle {\bf g}_c = r \Omega^2 \sin\theta (\sin\theta, \cos\theta, 0)$">|; 

$key = q/displaystyledeltav({mitDelta}t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3138.png"
 ALT="$\displaystyle \delta v({\mit\Delta} t)$">|; 

$key = q/6,5,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3172.png"
 ALT="$6,5,3$">|; 

$key = q/{mitOmega}=q_i,Bslashm_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img200.png"
 ALT="${\mit\Omega} = q_i B/m_i$">|; 

$key = q/{displaymath}|deltav({mitDelta}t)|simeqdeltav_0,{{rm{e}^{,lambda,{mitDeltat}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3152.png"
 ALT="\begin{displaymath}
\vert\delta v({\mit\Delta} t)\vert \simeq \delta v_0 {\rm e}^{ \lambda {\mit\Delta t}},
\end{displaymath}">|; 

$key = q/displaystylev_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1646.png"
 ALT="$\displaystyle v_y$">|; 

$key = q/dot{phi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1424.png"
 ALT="$\dot{\phi}$">|; 

$key = q/{displaymath}left(frac{partial^2U}{partialz^2}right)_{z=0}=frac{mu_1}{rho_1^{,3}}+frac{mu_2}{rho_2^{,3}}>0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2632.png"
 ALT="\begin{displaymath}
\left(\frac{\partial^2 U}{\partial z^2}\right)_{z=0} = \frac{\mu_1}{\rho_1^{ 3}} + \frac{\mu_2}{\rho_2^{ 3}}&gt;0.
\end{displaymath}">|; 

$key = q/displaystyley';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$\displaystyle y'$">|; 

$key = q/theta+dtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img973.png"
 ALT="$\theta+d\theta$">|; 

$key = q/displaystyle(a_y,c_y,b_x,,-a_x,b_x,c_y,,0)={{rm{LHS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3362.png"
 ALT="$\displaystyle (a_y  c_y  b_x,  -a_x  b_x  c_y,  0 ) = {\rm LHS},$">|; 

$key = q/displaystylea,sin(omega,t)+b,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img646.png"
 ALT="$\displaystyle a \sin(\omega t) + b t,$">|; 

$key = q/displaystylesum_{j>0}y_j,sin(omega_j,T-alpha_j),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2837.png"
 ALT="$\displaystyle \sum_{j&gt;0} y_j \sin(\omega_j T-\alpha_j),$">|; 

$key = q/displaystyle(a_y,b_z-a_z,b_y),costheta+(a_z,b_x-a_x,b_z),sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3298.png"
 ALT="$\displaystyle (a_y  b_z - a_z  b_y)  \cos\theta + (a_z  b_x-a_x  b_z) \sin\theta$">|; 

$key = q/h=sqrt{G,M,r_p,(1+e)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img792.png"
 ALT="$h = \sqrt{G M r_p (1+e)}$">|; 

$key = q/{{bf{a}ast{{bf{b}=(1slash2,,-1slash2,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3294.png"
 ALT="${\bf a}\ast {\bf b} = (1/2,  -1/2, 0)$">|; 

$key = q/displaystyleintr^2,gamma,left(frac{1}{2},sin^2theta+cos^2thetaright),d^3{{bf{r}=frac{1}{2}intr^2,gamma,(1+cos^2theta),d^3{{bf{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="470" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2197.png"
 ALT="$\displaystyle \int r^2 \gamma \left(\frac{1}{2} \sin^2\theta+ \cos^2\theta\right) d^3{\bf r} = \frac{1}{2}\int r^2 \gamma (1+ \cos^2\theta) d^3{\bf r},$">|; 

$key = q/tilde{{bf{I};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1266.png"
 ALT="$\tilde{\bf I}$">|; 

$key = q/{{bf{b}equiv(1,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3249.png"
 ALT="${\bf b} \equiv (1, 0, 0)$">|; 

$key = q/tanhxsimeqx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img262.png"
 ALT="$\tanh x\simeq x$">|; 

$key = q/1slashalpha_1=10.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2986.png"
 ALT="$1/\alpha_1 = 10.5$">|; 

$key = q/1slashQ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3062.png"
 ALT="$1/Q$">|; 

$key = q/|{{bf{r}-{{bf{k}|^{-n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3554.png"
 ALT="$\vert{\bf r}-{\bf k}\vert^{-n}$">|; 

$key = q/displaystylea_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="$\displaystyle a_z.$">|; 

$key = q/displaystyle3left[frac{mu_1,(x+mu_2)}{rho_1^{,5}}+frac{mu_2,(x-mu_1)}{rho_2^{,5}}right]y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2658.png"
 ALT="$\displaystyle 3\left[\frac{\mu_1 (x+\mu_2)}{\rho_1^{ 5}}+\frac{\mu_2 (x-\mu_1)}{\rho_2^{ 5}}\right]y,$">|; 

$key = q/{displaymath}m_1,{{bf{v}_1+m_2,{{bf{v}_2=m_1,{{bf{v}_1'+m_2,{{bf{v}_2'={{bf{0}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1012.png"
 ALT="\begin{displaymath}
m_1 {\bf v}_1 + m_2 {\bf v}_2 = m_1 {\bf v}_1'+m_2 {\bf v}_2' = {\bf0}
\end{displaymath}">|; 

$key = q/e';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2528.png"
 ALT="$e'$">|; 

$key = q/j=4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2920.png"
 ALT="$j=4$">|; 

$key = q/a_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3220.png"
 ALT="$a_y$">|; 

$key = q/{displaymath}{{bf{a}cdot{{bf{b}times{{bf{c}={{bf{b}cdot{{bf{c}times{{bf{a}={{bf{c}cdot{{bf{a}times{{bf{b},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3343.png"
 ALT="\begin{displaymath}
{\bf a} \cdot {\bf b} \times{\bf c} = {\bf b} \cdot {\bf c} \times{\bf a} =
{\bf c} \cdot {\bf a} \times{\bf b},
\end{displaymath}">|; 

$key = q/{displaymath}K=frac{1}{2}left[I_perp,dot{theta}^{,2}+(I_perp,sin^2theta+I_parallot{phi},dot{psi}+I_parallel,dot{psi}^{,2}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="543" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1711.png"
 ALT="\begin{displaymath}
K = \frac{1}{2}\left[I_\perp \dot{\theta}^{ 2} + (I_\perp\...
...\dot{\phi} \dot{\psi} + I_\parallel \dot{\psi}^{ 2}\right],
\end{displaymath}">|; 

$key = q/displaystyle3+3^{4slash3},mu_2^{,2slash3}-14,mu_2slash3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2616.png"
 ALT="$\displaystyle 3 + 3^{4/3} \mu_2^{ 2/3}-14 \mu_2/3,$">|; 

$key = q/vsimeqv_t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img265.png"
 ALT="$v\simeq v_t$">|; 

$key = q/{displaymath}y^2-b,x=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img697.png"
 ALT="\begin{displaymath}
y^2 - b x = 0,
\end{displaymath}">|; 

$key = q/{displaymath}aleft[-omega^2-{{rm{i},2,nu,omega+omega_0^{,2}right]{{rm{e}^{-{{rm{i},omega,t}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img316.png"
 ALT="\begin{displaymath}
a \left[-\omega^2-{\rm i} 2 \nu \omega + \omega_0^{ 2}\right]
{\rm e}^{-{\rm i} \omega t} = 0,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashfig15.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="284" BORDER="0"
 SRC="|."$dir".q|img3027.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/fig15.01.eps}}
\end{figure}">|; 

$key = q/{displaymath}t_f=frac{v_0,sintheta}{g}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img600.png"
 ALT="\begin{displaymath}
t_f = \frac{v_0 \sin\theta}{g}
\end{displaymath}">|; 

$key = q/{{bf{v}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="${\bf v}'$">|; 

$key = q/{{bf{f}={{bf{f}_1+{{bf{f}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="${\bf f} = {\bf f}_1+{\bf f}_2$">|; 

$key = q/mu_2ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2612.png"
 ALT="$\mu_2\ll 1$">|; 

$key = q/-frac{3}{2},e,iota;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2896.png"
 ALT="$-\frac{3}{2} e \iota$">|; 

$key = q/displaystyle-frac{partialU}{partialy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2546.png"
 ALT="$\displaystyle -\frac{\partial U}{\partial y},$">|; 

$key = q/{displaymath}theta=2,sin^{-1}left(frac{alpha}{sqrt{4+alpha^2}}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img995.png"
 ALT="\begin{displaymath}
\theta = 2 \sin^{-1}\left(\frac{\alpha}{\sqrt{4+\alpha^2}}\right).
\end{displaymath}">|; 

$key = q/alpha<3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2477.png"
 ALT="$\alpha&lt;3$">|; 

$key = q/textstyle~~~~~~;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1609.png"
 ALT="$\textstyle&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$">|; 

$key = q/mbox{{boldmath{{omega{}^T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1273.png"
 ALT="$\mbox{\boldmath$\omega$}^T$">|; 

$key = q/xgg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img267.png"
 ALT="$x\gg 1$">|; 

$key = q/displaystylen',t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2976.png"
 ALT="$\displaystyle n' t,$">|; 

$key = q/{displaymath}E=frac{1}{2}left(I_perp,dot{theta}^{,2}+I_perp,sin^2theta,dot{phi}^+L_psi^{,2}slashI_parallelright)+m,g,l,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="419" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1498.png"
 ALT="\begin{displaymath}
E=\frac{1}{2}\left(I_\perp \dot{\theta}^{ 2} + I_\perp \s...
...{ 2} + L_\psi^{ 2}/I_\parallel\right) + m g l \cos\theta.
\end{displaymath}">|; 

$key = q/displaystyletheta_0+epsilon,theta_1(t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2341.png"
 ALT="$\displaystyle \theta_0+ \epsilon \theta_1(t),$">|; 

$key = q/r^{-alpha};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img191.png"
 ALT="$r^{-\alpha}$">|; 

$key = q/F_0=m,omega_0^{,2},X_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img516.png"
 ALT="$F_0=m \omega_0^{ 2} X_0$">|; 

$key = q/';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img855.png"
 ALT="$'$">|; 

$key = q/displaystyle-Gsum_{j>i}frac{M_j}{R_j}left[1+frac{1}{4}left(frac{R_i}{R_j}right)^2+frac{9}{64}left(frac{R_i}{R_j}right)^4+cdotsright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2396.png"
 ALT="$\displaystyle -G \sum_{j&gt; i}\frac{M_j}{R_j}\left[1+\frac{1}{4}\left(\frac{R_i}{R_j}\right)^2 + \frac{9}{64}\left(\frac{R_i}{R_j}\right)^4+\cdots\right].$">|; 

$key = q/Q=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3067.png"
 ALT="$Q=1/2$">|; 

$key = q/Q=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$Q=1$">|; 

$key = q/displaystylefrac{1}{b^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img563.png"
 ALT="$\displaystyle \frac{1}{b^2}$">|; 

$key = q/|omega-omega_0|llomega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img386.png"
 ALT="$\vert\omega-\omega_0\vert\ll \omega_0$">|; 

$key = q/2^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3123.png"
 ALT="$2^n$">|; 

$key = q/rho(r)=rho_0(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2138.png"
 ALT="$\rho(r)=\rho_0(r)$">|; 

$key = q/m_1,m_2,m_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1582.png"
 ALT="$m_1, m_2, m_3$">|; 

$key = q/-epsilon=9.6times10^{-8};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2249.png"
 ALT="$-\epsilon=9.6\times 10^{-8}$">|; 

$key = q/{{bf{r}equiv,stackrel{displaystylerightarrow}{OP};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3237.png"
 ALT="${\bf r} \equiv  \stackrel{\displaystyle \rightarrow}{OP}$">|; 

$key = q/Q=1.376;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$Q=1.376$">|; 

$key = q/U_{xx}=-3slash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2685.png"
 ALT="$U_{xx} = -3/4$">|; 

$key = q/displaystyle-frac{2pi,G}{r^{n+1}}int_0^rint_0^pir'^{,n+2},rho(r',theta'),P_n(costheta'),sintheta',dr',dtheta';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="414" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2122.png"
 ALT="$\displaystyle -\frac{2\pi G}{r^{n+1}}\int_0^r \int_0^\pi r'^{ n+2} 
\rho(r',\theta') P_n(\cos\theta') \sin\theta' dr' d\theta'$">|; 

$key = q/displaystyledot{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1864.png"
 ALT="$\displaystyle \dot{p}$">|; 

$key = q/U=kslashr^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1102.png"
 ALT="$U=k/r^2$">|; 

$key = q/displaystyley_1slasha,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2756.png"
 ALT="$\displaystyle y_1/a,$">|; 

$key = q/displaystylec_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2944.png"
 ALT="$\displaystyle c_1$">|; 

$key = q/{displaymath}x(t)=left[x_0,(1+omega_0,t)+v_0,tright]{{rm{e}^{-omega_0,t}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img329.png"
 ALT="\begin{displaymath}
x(t) = \left[x_0  (1+\omega_0 t) + v_0 t\right] {\rm e}^{-\omega_0 t}.
\end{displaymath}">|; 

$key = q/displaystyle+e,iota,sin[(2+c,m^2+g,m^2),T-alpha_0-gamma_0];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2971.png"
 ALT="$\displaystyle +e \iota \sin [(2+c m^2+g m^2) T-\alpha_0-\gamma_0]$">|; 

$key = q/{displaymath}rho_0(r)=left{array{{ccc}gamma&mbox{hspace{0.5cm}}&mbox{for{rleqa{}0&&mbox{for{r>a{}array{right.,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2128.png"
 ALT="\begin{displaymath}
\rho_0(r) = \left\{\begin{array}{ccc}
\gamma&amp;\mbox{\hspace{0...
...\mbox{for $r\leq a$}\\\\
0&amp;&amp;\mbox{for $r&gt;a$}\end{array}\right.,
\end{displaymath}">|; 

$key = q/displaystyle{{bf{x}_j-left(frac{{{bf{x}_i^T{{bf{x}_j}{{{bf{x}_i^T{{bf{x}_i}right){{bf{x}_i.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1316.png"
 ALT="$\displaystyle {\bf x}_j - \left(\frac{{\bf x}_i^T{\bf x}_j}{{\bf x}_i^T{\bf x}_i}\right) {\bf x}_i.$">|; 

$key = q/{displaymath}r=frac{r_p}{1-e},(1-e,cosE),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img797.png"
 ALT="\begin{displaymath}
r = \frac{r_p}{1-e} (1-e \cos E),
\end{displaymath}">|; 

$key = q/2,g,l^2slashv^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1710.png"
 ALT="$2 g l^2/v^2$">|; 

$key = q/{displaymath}{{bf{a}cdot{{bf{b}times{{bf{c}={{bf{a}times{{bf{b}cdot{{bf{c}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3342.png"
 ALT="\begin{displaymath}
{\bf a} \cdot {\bf b}\times{\bf c} = {\bf a} \times{\bf b} \cdot{\bf c}.
\end{displaymath}">|; 

$key = q/displaystylefrac{2,G,M_0}{R_i^{,2}}+frac{G}{R_i^{,2}}sum_{j<i}M_jleft[2+3left(frright)^2+frac{135}{32}left(frac{R_j}{R_i}right)^4+cdotsright];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="456" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2402.png"
 ALT="$\displaystyle \frac{2 G M_0}{R_i^{ 2}}+\frac{G}{R_i^{ 2}} \sum_{j&lt; i}M_j\le...
...R_j}{R_i}\right)^2
+ \frac{135}{32}\left(\frac{R_j}{R_i}\right)^4+\cdots\right]$">|; 

$key = q/{displaymath}bleft|frac{db}{dtheta}right|=frac{1}{2}left|frac{db^2}{dtheta}rightaslash2),cos(thetaslash2)=frac{a^2}{4},sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="435" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img987.png"
 ALT="\begin{displaymath}
b\left\vert\frac{db}{d\theta}\right\vert = \frac{1}{2}\left\...
...} \sin(\theta/2) \cos(\theta/2) = \frac{a^2}{4} \sin\theta.
\end{displaymath}">|; 

$key = q/displaystylePhi(R_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2394.png"
 ALT="$\displaystyle \Phi(R_i)$">|; 

$key = q/D=3,A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2670.png"
 ALT="$D=3 A$">|; 

$key = q/{displaymath}h_T(x,,y)=h(x_0,,y_0)+alpha,(x-x_0)+beta,(y-y_0).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="384" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img3445.png"
 ALT="\begin{displaymath}
h_T(x, y)= h(x_0, y_0) + \alpha (x-x_0)+\beta (y-y_0).
\end{displaymath}">|; 

$key = q/{displaymath}frac{d}{dt}!left(frac{partialL}{partialdot{q}_i}right)-frac{partialL}{partialq_i}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1605.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot{q}_i}\right) -\frac{\partial L}{\partial q_i} =0,
\end{displaymath}">|; 

$key = q/{displaymath}frac{d}{dt}!left(dot{x}_j,frac{partialdot{x}_j}{partialdot{q}_i}rig{d}{dt}!left(frac{partialx_j}{partialq_i}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="426" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1591.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\dot{x}_j \frac{\partial \dot{x}_j}{\pa...
...\frac{d}{dt}\!\left(
\frac{\partial x_j}{\partial q_i}\right).
\end{displaymath}">|; 

$key = q/v_0slashv_t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img611.png"
 ALT="$v_0/v_t$">|; 

$key = q/displaystyledeltatheta_0,cos(omega_ast,{mitDelta}t),,{{rm{e}^{-{mitDelta}tslash2Q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3140.png"
 ALT="$\displaystyle \delta\theta_0 \cos(\omega_\ast  {\mit\Delta} t)  {\rm e}^{-{\mit\Delta} t/2Q}$">|; 

$key = q/displaystyleI_parallel,omega,cosalpha.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1403.png"
 ALT="$\displaystyle I_\parallel \omega \cos\alpha.$">|; 

$key = q/displaystylefrac{3pi}{2}sum_{j<i}left(frac{M_j}{M_0}right)left(frac{R_j}{R_i}rigright)^2+frac{175}{64}left(frac{R_j}{R_i}right)^4+cdotsright];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="472" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2411.png"
 ALT="$\displaystyle \frac{3\pi}{2}\sum_{j&lt;i}\left(\frac{M_j}{M_0}\right)
\left(\frac{...
..._j}{R_i}\right)^2 + \frac{175}{64}\left(\frac{R_j}{R_i}
\right)^4+\cdots\right]$">|; 

$key = q/displaystylel,sintheta,cosphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1672.png"
 ALT="$\displaystyle l \sin\theta \cos\phi,$">|; 

$key = q/{displaymath}{{bf{x}_i^{ast,T}{{bf{A},{{bf{x}_i=lambda_i,{{bf{x}_i^{ast,T}{{bf{x}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1293.png"
 ALT="\begin{displaymath}
{\bf x}_i^{\ast T} {\bf A} {\bf x}_i = \lambda_i {\bf x}_i^{\ast T}{\bf x}_i.
\end{displaymath}">|; 

$key = q/{{bf{R}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2220.png"
 ALT="${\bf R}'$">|; 

$key = q/{displaymath}({{bf{G}-lambda,{{bf{M}),{bf{{x}={{bf{0}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1920.png"
 ALT="\begin{displaymath}
({\bf G} - \lambda {\bf M}) \bf {x} = {\bf0}.
\end{displaymath}">|; 

$key = q/{{bf{a}times({{bf{b}times{{bf{c})neq({{bf{a}times{{bf{b})times{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3351.png"
 ALT="${\bf a}\times({\bf b} \times {\bf c}) \neq ({\bf a}\times{\bf b} )\times {\bf c}$">|; 

$key = q/nu=4,omega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img337.png"
 ALT="$\nu=4 \omega_0$">|; 

$key = q/T=n,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2760.png"
 ALT="$T = n t$">|; 

$key = q/Qgg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img390.png"
 ALT="$Q\gg 1$">|; 

$key = q/r'slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2089.png"
 ALT="$r'/r$">|; 

$key = q/displaystyle(2,m)^{-1slash2},(1,,,,,1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2011.png"
 ALT="$\displaystyle (2 m)^{-1/2} (1,     1),$">|; 

$key = q/E<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img214.png"
 ALT="$E&lt;0$">|; 

$key = q/{displaymath}frac{1}{T}int_0^Tcos(n,omega,t),dt=delta_{n,0},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img401.png"
 ALT="\begin{displaymath}
\frac{1}{T} \int_0^T \cos (n \omega t) dt = \delta_{n 0},
\end{displaymath}">|; 

$key = q/{displaymath}frac{|{{bf{a}times{{bf{b}+{{bf{b}times{{bf{c}+{{bf{c}times{{bf{a}|}{|{{bf{b}-{{bf{c}|}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img3525.png"
 ALT="\begin{displaymath}
\frac{\vert{\bf a}\times {\bf b} + {\bf b}\times {\bf c} + {\bf c}\times {\bf a}\vert}{\vert{\bf b}-{\bf c}\vert}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d}{dt}!left(frac{partialK}{partialdot{q}_i}right)=Q_i+frac{partialK}{partialq_i}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1601.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial K}{\partial \dot{q}_i}\right) = Q_i + \frac{\partial K}{\partial q_i}.
\end{displaymath}">|; 

$key = q/{{bf{b}equiv(1slashsqrt{2},,-1slashsqrt{2},,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3252.png"
 ALT="${\bf b} \equiv (1/\sqrt{2}, -
1/\sqrt{2},  0)$">|; 

$key = q/v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$v$">|; 

$key = q/displaystylefrac{domega_{x'}}{dt'}+{mitOmega},omega_{y'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1382.png"
 ALT="$\displaystyle \frac{d\omega_{x'}}{dt'} + {\mit\Omega} \omega_{y'}$">|; 

$key = q/|{{bf{r}|=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3526.png"
 ALT="$\vert{\bf r}\vert = a$">|; 

$key = q/cos^2alpha+cos^2beta+cos^2gamma=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1331.png"
 ALT="$\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$">|; 

$key = q/x_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img421.png"
 ALT="$x_n$">|; 

$key = q/displaystylefrac{dv_z}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img579.png"
 ALT="$\displaystyle \frac{d v_z}{dt}$">|; 

$key = q/X(-t)=X(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img399.png"
 ALT="$X(-t)=X(t)$">|; 

$key = q/nullomega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img343.png"
 ALT="$\nu\ll\omega_0$">|; 

$key = q/{displaymath}dot{{bf{e}_r=dot{theta},(-sintheta,,costheta)=dot{theta},,{{bf{e}_theta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img687.png"
 ALT="\begin{displaymath}
\dot{\bf e}_r = \dot{\theta} (-\sin\theta, \cos\theta) = \dot{\theta}  {\bf e}_\theta.
\end{displaymath}">|; 

$key = q/x_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2729.png"
 ALT="$x_1'$">|; 

$key = q/displaystyle0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1338.png"
 ALT="$\displaystyle 0,$">|; 

$key = q/I;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img190.png"
 ALT="$I$">|; 

$key = q/sin^2phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2198.png"
 ALT="$\sin^2\phi$">|; 

$key = q/displaystylefrac{dv_x'}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img628.png"
 ALT="$\displaystyle \frac{d v_x'}{dt}$">|; 

$key = q/omega_5simeq1-2,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2933.png"
 ALT="$\omega_5\simeq 1-2 m$">|; 

$key = q/L_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$L_2$">|; 

$key = q/sin^2(lambda_m-phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2322.png"
 ALT="$\sin^2(\lambda_m-\phi)$">|; 

$key = q/displaystyle{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1468.png"
 ALT="$\displaystyle {\mit\Omega}$">|; 

$key = q/{displaymath}h=b,v_infty=bleft(frac{2,E}{mu}right)^{1slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img962.png"
 ALT="\begin{displaymath}
h = b v_\infty = b\left(\frac{2 E}{\mu}\right)^{1/2},
\end{displaymath}">|; 

$key = q/displaystyler,costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2075.png"
 ALT="$\displaystyle r \cos\theta.$">|; 

$key = q/dy=y-y_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3448.png"
 ALT="$dy=y-y_0$">|; 

$key = q/alpha_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1954.png"
 ALT="$\alpha_k$">|; 

$key = q/{displaymath}v_z=v_0,sintheta,{{rm{e}^{-g,tslashv_t}-v_tleft(1-e^{-g,tslashv_t}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img590.png"
 ALT="\begin{displaymath}
v_z = v_0 \sin\theta {\rm e}^{-g t/v_t} - v_t\left(1-e^{-g t/v_t}\right).
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter05slashfig5.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="251" BORDER="0"
 SRC="|."$dir".q|img672.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter05/fig5.01.eps}}
\end{figure}">|; 

$key = q/{{bf{c}={{bf{p}+nu,{{bf{q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3539.png"
 ALT="${\bf c} = {\bf p} + \nu {\bf q}$">|; 

$key = q/nu=0.2,omega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img447.png"
 ALT="$\nu=0.2 \omega_0$">|; 

$key = q/displaystyle(0,,0,,-g),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1139.png"
 ALT="$\displaystyle (0, 0, -g),$">|; 

$key = q/{displaymath}{calE}=frac{v^2}{2}-frac{G,M}{r}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img667.png"
 ALT="\begin{displaymath}
{\cal E} = \frac{v^2}{2} - \frac{G M}{r}
\end{displaymath}">|; 

$key = q/displaystylealpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2584.png"
 ALT="$\displaystyle \alpha$">|; 

$key = q/displaystyley_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2924.png"
 ALT="$\displaystyle y_4$">|; 

$key = q/{displaymath}{{bf{a}ast{{bf{b}equiv(a_x,b_x,,a_y,b_y,,a_z,b_z).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img3287.png"
 ALT="\begin{displaymath}
{\bf a} \ast{\bf b} \equiv (a_x  b_x,  a_y  b_y,  a_z  b_z).
\end{displaymath}">|; 

$key = q/t=T_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img353.png"
 ALT="$t=T_r$">|; 

$key = q/E>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img215.png"
 ALT="$E&gt;0$">|; 

$key = q/{displaymath}deltaq_i(t)=deltaq_i,{{rm{e}^{,gamma,t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1911.png"
 ALT="\begin{displaymath}
\delta q_i(t) = \delta q_i {\rm e}^{ \gamma t},
\end{displaymath}">|; 

$key = q/epsilon<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2148.png"
 ALT="$\epsilon&lt;0$">|; 

$key = q/M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img195.png"
 ALT="$M$">|; 

$key = q/{displaymath}delta{{bf{q}_k(t)=left(alpha_k,{{rm{e}^{+sqrt{lambda_k},t}+beta_k,{{rm{e}^{-sqrt{lambda_k},t}right){{bf{x}_k.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img1953.png"
 ALT="\begin{displaymath}
\delta {\bf q}_k(t) = \left(\alpha_k  {\rm e}^{+\sqrt{\lamb...
...,t}+\beta_k {\rm e}^{-\sqrt{\lambda_k} t}\right)
{\bf x}_k.
\end{displaymath}">|; 

$key = q/I_{ji}=I_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1268.png"
 ALT="$I_{ji} = I_{ij}$">|; 

$key = q/{displaymath}m,frac{d{{bf{v}}{dt}=q,left({{bf{E}+{{bf{v}times{{bf{B}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img623.png"
 ALT="\begin{displaymath}
m \frac{d{\bf v}}{dt} = q \left({\bf E} + {\bf v}\times{\bf B}\right).
\end{displaymath}">|; 

$key = q/displaystylefrac{partialL}{partialtheta}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1612.png"
 ALT="$\displaystyle \frac{\partial L}{\partial\theta} = 0.$">|; 

$key = q/displaystyleP.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img827.png"
 ALT="$\displaystyle P.$">|; 

$key = q/pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img863.png"
 ALT="$\pi$">|; 

$key = q/{displaymath}dot{psi}=-{mitOmega}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1461.png"
 ALT="\begin{displaymath}
\dot{\psi}=-{\mit\Omega}.
\end{displaymath}">|; 

$key = q/Omega_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2877.png"
 ALT="$\Omega_4$">|; 

$key = q/{displaymath}t=left(frac{m}{k}right)^{1slash2}int_0^{(kslash2,E)^{1slash2},x}fra1}left(left[frac{k}{2,E}right]^{1slash2}xright),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="516" HEIGHT="67" BORDER="0"
 SRC="|."$dir".q|img236.png"
 ALT="\begin{displaymath}
t = \left(\frac{m}{k}\right)^{1/2}\int_0^{(k/2 E)^{1/2} x}...
.../2} \sin^{-1}\left(\left[\frac{k}{2 E}\right]^{1/2} x\right),
\end{displaymath}">|; 

$key = q/{displaymath}frac{df(0)}{dx}<0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img287.png"
 ALT="\begin{displaymath}
\frac{d f(0)}{dx} &lt; 0.
\end{displaymath}">|; 

$key = q/{displaymath}{calE}=frac{h^2}{2}left[left(frac{du}{dtheta}right)^2+u^2-2,u,u_cright],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img771.png"
 ALT="\begin{displaymath}
{\cal E} = \frac{h^2}{2}\left[\left(\frac{du}{d\theta}\right)^2
+ u^2 - 2 u u_c\right],
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{Chapter15slashfig15.35.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="517" HEIGHT="219" BORDER="0"
 SRC="|."$dir".q|img3169.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{Chapter15/fig15.35.eps}}
\end{figure}">|; 

$key = q/f_1,f_2,f_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1571.png"
 ALT="$f_1, f_2, f_3$">|; 

$key = q/rho_a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img277.png"
 ALT="$\rho_a$">|; 

$key = q/displaystyle-frac{partialH}{partialtheta}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1881.png"
 ALT="$\displaystyle -\frac{\partial H}{\partial \theta} = 0.$">|; 

$key = q/{displaymath}E_1=E_1'=E_2=E_2'=frac{E}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1071.png"
 ALT="\begin{displaymath}
E_1 = E_1' = E_2=E_2' = \frac{E}{2}.
\end{displaymath}">|; 

$key = q/x_0=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img338.png"
 ALT="$x_0=1$">|; 

$key = q/ddot{~}equivd^2slashdt^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2700.png"
 ALT="$\ddot{&nbsp;}\equiv d^2/dt^2$">|; 

$key = q/{displaymath}int_0^{(vslashv_t)^2}frac{dy}{1-y}=frac{x}{x_t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img271.png"
 ALT="\begin{displaymath}
\int_0^{(v/v_t)^2} \frac{dy}{1-y} = \frac{x}{x_t},
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{sin^2{mitDelta}}left[frac{sin^2theta}{A^2}+frac{2,costheta,sintheta,cos{mitDelta}}{A,B}+frac{cos^2theta}{B^2}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img564.png"
 ALT="$\displaystyle \frac{1}{\sin^2{\mit\Delta}}\left[\frac{\sin^2\theta}{A^2} +\frac...
...s\theta \sin\theta \cos{\mit\Delta}}{A B} + \frac{\cos^2\theta}{B^2}\right].$">|; 

$key = q/U_{yy}=A-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2672.png"
 ALT="$U_{yy} = A-1$">|; 

$key = q/u_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1511.png"
 ALT="$u_1$">|; 

$key = q/lambda_ineqlambda_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1303.png"
 ALT="$\lambda_i\neq \lambda_j$">|; 

$key = q/t=t_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3129.png"
 ALT="$t=t_0$">|; 

$key = q/{displaymath}L=frac{1}{2},m,l^2,(dot{theta}^{,2}+sin^2theta,dot{phi}^{,2})+m,g,l,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1677.png"
 ALT="\begin{displaymath}
L = \frac{1}{2} m l^2 (\dot{\theta}^{ 2}+ \sin^2\theta \dot{\phi}^{ 2})
+ m g l \cos\theta.
\end{displaymath}">|; 

$key = q/a_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2882.png"
 ALT="$a_j$">|; 

$key = q/2,e,sinM;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3009.png"
 ALT="$2 e \sin M$">|; 

$key = q/displaystyle-frac{3}{2},m^2,cos(omega_4,T-alpha_4),deltaY+3,deltaX,deltaY,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2881.png"
 ALT="$\displaystyle -\frac{3}{2} m^2 \cos(\omega_4 T-\alpha_4) \delta Y +3 \delta X \delta Y,$">|; 

$key = q/{displaymath}frac{x^2}{a^2}-frac{y^2}{b^2}=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img700.png"
 ALT="\begin{displaymath}
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.
\end{displaymath}">|; 

$key = q/T,costhetasimeqT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1185.png"
 ALT="$T \cos\theta\simeq T$">|; 

$key = q/{{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="${\bf v}$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter15slashfig15.31.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="345" BORDER="0"
 SRC="|."$dir".q|img3162.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter15/fig15.31.eps}}
\end{figure}">|; 

$key = q/langlecos^2phi'rangle=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2093.png"
 ALT="$\langle \cos^2\phi'\rangle = 1/2$">|; 

$key = q/{displaymath}frac{d{{bf{a}}{dt}=mbox{{boldmath{{omega{}times{{bf{a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3338.png"
 ALT="\begin{displaymath}
\frac{d {\bf a}}{dt} = \mbox{\boldmath$\omega$}\times {\bf a}.
\end{displaymath}">|; 

$key = q/f(u);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$f(u)$">|; 

$key = q/45^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img606.png"
 ALT="$45^\circ$">|; 

$key = q/z';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1134.png"
 ALT="$z'$">|; 

$key = q/displaystylefrac{d}{dt}!left(m,r^2,dot{theta}right)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1616.png"
 ALT="$\displaystyle \frac{d}{dt}\!\left(m r^2 \dot{\theta}\right) = 0,$">|; 

$key = q/{displaymath}{{bf{f}=-nablaU=-k,(x,,y)=-k,{{bf{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img526.png"
 ALT="\begin{displaymath}
{\bf f} = - \nabla U = -k (x,  y) = - k {\bf r},
\end{displaymath}">|; 

$key = q/r,deltatheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img741.png"
 ALT="$r \delta\theta$">|; 

$key = q/{displaymath}P_n(x)=frac{1}{2^n,n!},frac{d^n}{dx^n}!left[(x^2-1)^nright],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2103.png"
 ALT="\begin{displaymath}
P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}\!\left[(x^2-1)^n\right],
\end{displaymath}">|; 

$key = q/2,(1+c,m^2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2854.png"
 ALT="$2 (1+c m^2)$">|; 

$key = q/displaystyle-frac{4pi,G,gamma,a^3}{3}=-G,M,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2158.png"
 ALT="$\displaystyle - \frac{4\pi G \gamma a^3}{3} = - G M,$">|; 

$key = q/displaystylesqrt{-lambda_1}=sqrt{frac{k'}{m}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2004.png"
 ALT="$\displaystyle \sqrt{-\lambda_1} = \sqrt{\frac{k'}{m}},$">|; 

$key = q/{displaymath}U=frac{k}{2}left[(1+2,alpha+2,alpha^2),q_1^{,2}+4,alpha,(1+alpha),q_1,q_2+(1+2,alpha+2,alpha^2),q_2^{,2}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="564" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2034.png"
 ALT="\begin{displaymath}
U = \frac{k}{2}\left[(1+2 \alpha+2 \alpha^2) q_1^{ 2} + ...
...alpha) q_1 q_2 +(1+2 \alpha+2 \alpha^2) q_2^{ 2}\right],
\end{displaymath}">|; 

$key = q/{{bf{e}_{z'''}={{bf{e}_{z'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1439.png"
 ALT="${\bf e}_{z'''}={\bf e}_{z'}$">|; 

$key = q/q^2,d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img887.png"
 ALT="$q^2 d$">|; 

$key = q/{displaymath}epsilon=0.101.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img2189.png"
 ALT="\begin{displaymath}
\epsilon= 0.101.
\end{displaymath}">|; 

$key = q/P=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img830.png"
 ALT="$P=0$">|; 

$key = q/j=5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2932.png"
 ALT="$j=5$">|; 

$key = q/{displaymath}deltax_j=sum_{i=1,{calF}}frac{partialx_j}{partialq_i},deltaq_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1567.png"
 ALT="\begin{displaymath}
\delta x_j = \sum_{i=1,{\cal F}} \frac{\partial x_j}{\partial q_i} \delta q_i,
\end{displaymath}">|; 

$key = q/omega=sqrt{kslashm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img239.png"
 ALT="$\omega = \sqrt{k/m}$">|; 

$key = q/0.241;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2421.png"
 ALT="$0.241$">|; 

$key = q/{displaymath}frac{d}{dt}=frac{d}{dt'}+mbox{{boldmath{{Omega{}times,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1111.png"
 ALT="\begin{displaymath}
\frac{d}{dt} =\frac{d}{dt'}+ \mbox{\boldmath$\Omega$}\times,
\end{displaymath}">|; 

$key = q/AAAAAAcdotsrightarrowABABABcdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3088.png"
 ALT="$AAAAAA\cdots\rightarrow ABABAB\cdots$">|; 

$key = q/n,l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3176.png"
 ALT="$n,l$">|; 

$key = q/R=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2491.png"
 ALT="$R=1$">|; 

$key = q/displaystyle-frac{partialU}{partialx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2545.png"
 ALT="$\displaystyle -\frac{\partial U}{\partial x},$">|; 

$key = q/displaystylev_1(t_0+{mitDelta}t)-v_0(t_0+{mitDelta}t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3139.png"
 ALT="$\displaystyle v_1(t_0+{\mit\Delta} t)-v_0(t_0+{\mit\Delta} t),$">|; 

$key = q/f(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img846.png"
 ALT="$f(r)$">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}_theta=dot{theta},{{bf{e}_{x''}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1433.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$}_\theta = \dot{\theta} {\bf e}_{x''}.
\end{displaymath}">|; 

$key = q/Oxyz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2267.png"
 ALT="$Oxyz$">|; 

$key = q/displaystyle2,r+(e-1),frac{r^2}{r_p}-(e+1),r_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img803.png"
 ALT="$\displaystyle 2 r + (e-1) \frac{r^2}{r_p}- (e+1) r_p$">|; 

$key = q/{displaymath}t>t_hsimfrac{1}{lambda}ln!left(frac{a}{delta_0}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img3160.png"
 ALT="\begin{displaymath}
t&gt; t_h \sim \frac{1}{\lambda}\ln\!\left(\frac{a}{\delta_0}\right).
\end{displaymath}">|; 

$key = q/deltax(t)=deltax_0,exp(gamma,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2649.png"
 ALT="$\delta x(t) = \delta x_0 \exp(\gamma t)$">|; 

$key = q/{displaymath}T=frac{2pi}{left[-3,f(r_c)slashr_c-f'(r_c)right]^{1slash2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img864.png"
 ALT="\begin{displaymath}
T = \frac{2\pi}{\left[-3 f(r_c)/r_c - f'(r_c)\right]^{1/2}}.
\end{displaymath}">|; 

$key = q/{displaymath}left(array{{c}L_xL_yL_zarray{right)=left(array{{ccc}I_{xx}&I_{xy}&Ileft(array{{c}omega_xomega_yomega_zarray{right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="76" BORDER="0"
 SRC="|."$dir".q|img1242.png"
 ALT="\begin{displaymath}
\left(\begin{array}{c}L_x\ L_y\ L_z\end{array}\right)=
\le...
...gin{array}{c}\omega_x\ \omega_y\ \omega_z\end{array}\right),
\end{displaymath}">|; 

$key = q/lambda_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2311.png"
 ALT="$\lambda_m$">|; 

$key = q/displaystyleP+frac{P^3}{3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img824.png"
 ALT="$\displaystyle P + \frac{P^3}{3}$">|; 

$key = q/{displaymath}Phi({{bf{r})=-Gsum_{i=1,N}frac{m_i}{|{{bf{r}_i-{{bf{r}|}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2064.png"
 ALT="\begin{displaymath}
\Phi({\bf r}) = - G\sum_{i=1,N} \frac{m_i}{\vert{\bf r}_i-{\bf r}\vert}.
\end{displaymath}">|; 

$key = q/displaystyleddot{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1223.png"
 ALT="$\displaystyle \ddot{x}$">|; 

$key = q/{displaymath}t=pmleft(frac{m}{2,E}right)^{1slash2}int_{x_0}^xfrac{dx'}{sqrt{1-U(x')slashE}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="62" BORDER="0"
 SRC="|."$dir".q|img232.png"
 ALT="\begin{displaymath}
t=\pm\left(\frac{m}{2 E}\right)^{1/2} \int_{x_0}^x\frac{dx'}{\sqrt{1-U(x')/E}},
\end{displaymath}">|; 

$key = q/displaystyle-a',sin[(n-n'),t],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2739.png"
 ALT="$\displaystyle -a' \sin[(n-n') t],$">|; 

$key = q/BA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2208.png"
 ALT="$BA$">|; 

$key = q/{{bf{r}equiv(x',,y',,z');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3238.png"
 ALT="${\bf r}\equiv (x', y', z')$">|; 

$key = q/displaystyletheta-n,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2800.png"
 ALT="$\displaystyle \theta - n t$">|; 

$key = q/{displaymath}d{mitOmega}=2pi,sintheta,dtheta=frac{2pi,sintheta,db}{|dbslashdtheta|}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img976.png"
 ALT="\begin{displaymath}
d{\mit\Omega} = 2\pi \sin\theta d\theta = \frac{2\pi \sin\theta db}{\vert db/d\theta\vert}
\end{displaymath}">|; 

$key = q/displaystyleP_0(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2105.png"
 ALT="$\displaystyle P_0(x)$">|; 

$key = q/4586;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3018.png"
 ALT="$4586$">|; 

$key = q/displaystylefrac{d^2y}{dt^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img530.png"
 ALT="$\displaystyle \frac{d^2 y}{d t^2}$">|; 

$key = q/{displaymath}Deltaa=a_+-a_-=-epsilon,a=0.61,{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2250.png"
 ALT="\begin{displaymath}
\Delta a = a_+-a_- = -\epsilon a = 0.61 {\rm m}.
\end{displaymath}">|; 

$key = q/displaystylex_5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2934.png"
 ALT="$\displaystyle x_5$">|; 

$key = q/k>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img525.png"
 ALT="$k&gt;0$">|; 

$key = q/b+db;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img972.png"
 ALT="$b+db$">|; 

$key = q/{displaymath}frac{1}{2,a}+sqrt{a,(1-e^2)},cosI=frac{1}{2,a'}+sqrt{a',(1-e'^{,2})},cosI'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2530.png"
 ALT="\begin{displaymath}
\frac{1}{2 a} + \sqrt{a (1-e^2)} \cos I = \frac{1}{2 a'} + \sqrt{a' (1-e'^{ 2})} \cos I'.
\end{displaymath}">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}cdot{{bf{L}=I_{x'x'},omega_{x'}^{,2}+I_{y'y''}^{,2}+I_{z'z'},omega_{z'}^{,2}={{rm{constant}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="414" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img1414.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$}\cdot {\bf L} = I_{x'x'} \omega_{x'...
...ega_{y'}^{ 2}
+ I_{z'z'} \omega_{z'}^{ 2} ={\rm constant}.
\end{displaymath}">|; 

$key = q/{displaymath}left(frac{y}{B}-frac{x}{A},cos{mitDelta}right)^2=sin^2(omega_0,t'),lta}=left(1-frac{x^2}{A^2}right)sin^2{mitDelta},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="467" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img546.png"
 ALT="\begin{displaymath}
\left(\frac{y}{B} - \frac{x}{A} \cos{\mit\Delta}\right)^2=\...
...mit\Delta} = \left(1-\frac{x^2}{A^2}\right)\sin^2{\mit\Delta},
\end{displaymath}">|; 

$key = q/e<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img717.png"
 ALT="$e&lt;1$">|; 

$key = q/{displaymath}{{bf{g}_c=-nablachi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2211.png"
 ALT="\begin{displaymath}
{\bf g}_c = - \nabla\chi,
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}+{{bf{0}={{bf{a},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3200.png"
 ALT="\begin{displaymath}
{\bf a} + {\bf0} = {\bf a},
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{A}={{bf{grad},V.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img3471.png"
 ALT="\begin{displaymath}
{\bf A} = {\bf grad}  V.
\end{displaymath}">|; 

$key = q/a_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img415.png"
 ALT="$a_n$">|; 

$key = q/Omega_2simeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2910.png"
 ALT="$\Omega_2\simeq 0$">|; 

$key = q/{displaymath}frac{d^2theta}{dhat{t}^2}+frac{1}{Q},frac{dtheta}{dhat{t}}+sintheta=hat{A},cos(hat{omega},hat{t}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img3037.png"
 ALT="\begin{displaymath}
\frac{d^2\theta}{d\hat{t}^2} + \frac{1}{Q} \frac{d\theta}{d\hat{t}}
+ \sin\theta = \hat{A} \cos(\hat{\omega} \hat{t}).
\end{displaymath}">|; 

$key = q/D=90^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3007.png"
 ALT="$D=90^\circ$">|; 

$key = q/deltatheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img738.png"
 ALT="$\delta\theta$">|; 

$key = q/{{bf{b}times{{bf{c}equiv(0,,0,,b_x,c_y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3356.png"
 ALT="${\bf b}\times{\bf c} \equiv (0, 0, b_x  c_y)$">|; 

$key = q/{{bf{a}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="${\bf a}'$">|; 

$key = q/displaystyle-mu_1,frac{zeta}{rho_1^{,3}}-mu_2,frac{zeta}{rho_2^{,3}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2508.png"
 ALT="$\displaystyle -\mu_1 \frac{\zeta}{\rho_1^{ 3}} - \mu_2 \frac{\zeta}{\rho_2^{ 3}},$">|; 

1;

