# Need a Proof that Action-Angle Coordinates are Periodic

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• jstrunk
In summary, the proof of periodicity of Action-Angle coordinates in Hamilton-Jacobi Theory relies on the use of Liouville theorem and shows that the action variables $J_i$ are periodic with period $2\pi$.
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.

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.

## 1. What are action-angle coordinates?

Action-angle coordinates are a set of mathematical tools used to describe the motion of a system in classical mechanics. They consist of two types of variables: the action variables, which represent the energy stored in the system, and the angle variables, which represent the position of the system. Together, they provide a complete description of the system's dynamics.

## 2. How do action-angle coordinates relate to periodic motion?

Action-angle coordinates are particularly useful for studying systems with periodic motion. This is because in these coordinates, the action variables are constant and the angle variables increase linearly with time. This allows us to easily identify and analyze the periodic behavior of a system.

## 3. Why is it important to have periodic action-angle coordinates?

Periodic action-angle coordinates are important because they allow us to simplify the study of a system's dynamics. By using these coordinates, we can reduce the complexity of the equations of motion and gain a better understanding of the behavior of the system. This is particularly useful in areas such as celestial mechanics, where many systems exhibit periodic behavior.

## 4. How can we prove that action-angle coordinates are periodic?

The proof that action-angle coordinates are periodic relies on mathematical analysis and the use of certain theorems, such as Liouville's theorem and Poincaré's theorem. These theorems demonstrate that for systems with periodic motion, the action variables are constant and the angle variables increase linearly with time, confirming the periodic nature of the coordinates.

## 5. Are action-angle coordinates applicable to all systems?

No, action-angle coordinates are not applicable to all systems. They are most commonly used for systems with periodic motion, such as oscillating systems or celestial bodies in orbit. For systems with non-periodic motion, other coordinate systems may be more appropriate for analysis.

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