Action-angle coordinates

Summary

Action-angle coordinates are canonical coordinates $(I, \theta)$ for an integrable system in which the Hamiltonian depends on the actions $I$ only. In these coordinates the actions are constant and the angles turn at constant speed, so the equations of motion are solved at once. They exist near compact regular level sets by the Liouville–Arnold theorem. They are the natural coordinates for perturbation theory, such as KAM theory.