Discussion
For a particle moving in an inverse square force field, \(V(r) = -\frac{K}{r}\), the action angle variable analysis gives :
Answer: B
On the key, the action-angle analysis gives two similar frequencies of oscillation.
Motion under the inverse-square force is planar, with two degrees of freedom, r and \(\theta\). Their actions \(J_{r}\) and \(J_{\theta}\) give the energy
\[ E = -\frac{2\pi^{2}mK^{2}}{(J_{r} + J_{\theta})^{2}} \]
which depends only on the sum \(J_{r} + J_{\theta}\). The two frequencies \(\nu = \partial E/\partial J\) are therefore equal: the radial oscillation and the angular revolution have the same period, which is why bound Kepler orbits are closed ellipses. This degeneracy is the point of the question.
Two dissimilar frequencies would give non-closed, precessing orbits, as in a perturbed potential. 'Only one frequency' misses that there are two degrees of freedom whose frequencies coincide.
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