# A conceptual approach to the problem of action-angle variables

## TL;DR

This paper introduces a unified, conceptual framework for understanding the existence of action-angle variables in various dynamical systems, providing new insights, simplified proofs, and extending results to complex and singular cases.

## Contribution

It offers the shortest, most conceptual proof of the classical theorem and unifies existing results while introducing new findings in diverse contexts.

## Key findings

- Unified approach to action-angle variables
- Simplified proof of Arnold--Liouville--Mineur theorem
- New results in contact, presymplectic, Dirac, and stochastic systems

## Abstract

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 conceptual easy to understand 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.