What are action-angle variables and how do they relate to coordinates?

[LarryS]

I'm looking for one clear, simple mathematical definition of action-angle variables. I believe they are also called action-angle coordinates. As always, thanks in advance.

 

[atyy]

p191, http://books.google.com/books?id=olMpStYOlnoC&printsec=frontcover&source=gbs_navlinks_s

 

[charng]

Action-angle variables, also known as action-angle coordinates, are a set of canonical variables that describe the dynamics of a system with periodic motion. They are defined as the action variables, which represent the average amount of time a system spends in a particular region of its phase space, and the angle variables, which represent the phase of the system at a given time. In other words, action-angle variables describe both the energy and the phase of a system's motion.

Mathematically, action-angle variables can be defined as follows:

Let (q,p) be the canonical coordinates of a system, where q represents the position and p represents the momentum. The action variables, I_i, are given by:

I_i = \oint p_i dq_i

where the integral is taken over a closed path in phase space. The angle variables, \theta_i, are defined as:

\theta_i = \int \frac{dq_i}{\frac{\partial H}{\partial p_i}}

where H is the Hamiltonian of the system.

In summary, action-angle variables are a set of canonical variables that describe the dynamics of a system with periodic motion by representing the energy and phase of the system's motion. They are useful in studying and analyzing the behavior of systems with periodic motion, such as celestial bodies or oscillating systems.