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Consider the motion of an object in a general (attractive) central force-field characterized by the potential energy per unit mass function
. Since the force-field
is central, it still remains true that
![\begin{displaymath}
h = r^2\,\dot{\theta}
\end{displaymath}](img736.png) |
(299) |
is a constant of the motion. As is easily demonstrated, Equation (253)
generalizes to
![\begin{displaymath}
\frac{d^2 u}{d\theta^2} + u = - \frac{1}{h^2}\frac{dV}{du},
\end{displaymath}](img839.png) |
(300) |
where
.
Suppose, for instance, that we wish to find the potential
which causes
an object to execute the spiral orbit
![\begin{displaymath}
r = r_0\,\theta^{\,2}.
\end{displaymath}](img840.png) |
(301) |
Substitution of
into Equation (300) yields
![\begin{displaymath}
\frac{d V}{du} = - h^2\left(6\,r_0\,u^2 + u\right).
\end{displaymath}](img842.png) |
(302) |
Integrating, we obtain
![\begin{displaymath}
V(u) = -h^2\left(2\,r_0\,u^3 + \frac{u^2}{2}\right),
\end{displaymath}](img843.png) |
(303) |
or
![\begin{displaymath}
V(r) = - h^2\left(\frac{2\,r_0}{r^3} + \frac{1}{2\,r^2}\right).
\end{displaymath}](img844.png) |
(304) |
In other words, the spiral pattern (301) is obtained from a mixture
of an inverse-square and inverse-cube potential.
Next: Motion in a Nearly
Up: Planetary Motion
Previous: Kepler Problem
Richard Fitzpatrick
2011-03-31