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Motion in a General Central Force-Field

Consider the motion of an object in a general (attractive) central force-field characterized by the potential energy per unit mass function $V(r)$. Since the force-field is central, it still remains true that
\begin{displaymath}
h = r^2\,\dot{\theta}
\end{displaymath} (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} (300)

where $u=r^{-1}$.

Suppose, for instance, that we wish to find the potential $V(r)$ which causes an object to execute the spiral orbit

\begin{displaymath}
r = r_0\,\theta^{\,2}.
\end{displaymath} (301)

Substitution of $u = (r_0\,\theta^2)^{-1}$ into Equation (300) yields
\begin{displaymath}
\frac{d V}{du} = - h^2\left(6\,r_0\,u^2 + u\right).
\end{displaymath} (302)

Integrating, we obtain
\begin{displaymath}
V(u) = -h^2\left(2\,r_0\,u^3 + \frac{u^2}{2}\right),
\end{displaymath} (303)

or
\begin{displaymath}
V(r) = - h^2\left(\frac{2\,r_0}{r^3} + \frac{1}{2\,r^2}\right).
\end{displaymath} (304)

In other words, the spiral pattern (301) is obtained from a mixture of an inverse-square and inverse-cube potential.


next up previous
Next: Motion in a Nearly Up: Planetary Motion Previous: Kepler Problem
Richard Fitzpatrick 2011-03-31