A Conceptual Approach to the Problem of Action-Angle Variables

In this paper we develop a general conceptual approach to the problem of existence of action-angle variables for dynamical systems, which establishes and uses the fundamental conservation property of associated torus actions : anything which is preserved by the system is also preserved by the associated torus actions. This approach allows us to obtain, among other things: (a) the shortest and most easy-to-understand conceptual proof of the classical Arnold–Liouville–Mineur theorem; (b) basically all known results in the literature about the existence of action-angle variables in various contexts can be recovered in a unifying way, with simple proofs, using our approach; (c) new results on action-angle variables in many different contexts, including systems on contact manifolds, systems on presymplectic and Dirac manifolds, action-angle variables near singularities, stochastic systems, and so on. Even when there are no natural action variables, our approach still leads to useful normal forms for dynamical systems, which are not necessarily integrable.