Need a Proof that Action-Angle Coordinates are Periodic

[jstrunk]

Can someone point me to a proof that Action-Angle coordinates in Hamilton-Jacobi Theory must be periodic.
I have looked all over and no one seems to prove it, they just assume it.
Thanks.

 

[gtoguy97]

A:Consider the Hamiltonian of an autonomous system $H(q,p)$ and Hamilton-Jacobi equation for it$$\frac{\partial S}{\partial t}+H(q,\frac{\partial S}{\partial q})=0.$$The action-angle coordinates are defined as $$J_i=\frac{\partial S}{\partial \theta_i},\quad q_i=q_i(\theta_1,...,\theta_n),$$where $\theta_i$ are the angles. To prove that $J_i$ is periodic one need to use Liouville theorem.Multiply Hamilton-Jacobi equation by $\frac{\partial q_i}{\partial \theta_j}$ and sum over $i$$$\frac{\partial S}{\partial \theta_j}+\sum_i \left(\frac{\partial q_i}{\partial \theta_j}\right) H(q,\frac{\partial S}{\partial q})=0.$$Using definition of $J_i$ and denoting $H_i=\left(\frac{\partial q_i}{\partial \theta_j}\right) H(q,\frac{\partial S}{\partial q})$ we have$$J_j+\sum_i H_i=0.$$This equation can be rewritten as$$dJ_j=-\sum_i dH_i.$$From Liouville theorem $H_i=H_i(J_1,..,J_n)$ and thus$$\oint J_j d\theta_j=-\oint \sum_i H_i d\theta_j=const.$$Since the period of $\theta_j$ is $2\pi$ we have$$\oint J_j d\theta_j=2\pi J_j=const.$$Therefore $J_j$ is periodic.