Action-angle coordinates for integrable systems on Poisson manifolds

TL;DR

This paper extends the classical action-angle theorem to integrable systems on Poisson manifolds, utilizing polyvector fields and a Poisson version of the Caratheodory-Jacobi-Lie theorem, and further generalizes to non-commutative systems.

Contribution

It provides the first proof of the action-angle theorem in the broader context of Poisson manifolds, including non-commutative integrable systems.

Findings

Proves the action-angle theorem on Poisson manifolds.

Develops a Poisson version of the Caratheodory-Jacobi-Lie theorem.

Generalizes the theorem to non-commutative integrable systems.

Abstract

We prove the action-angle theorem in the general, and most natural, context of integrable systems on Poisson manifolds, thereby generalizing the classical proof, which is given in the context of symplectic manifolds. The topological part of the proof parallels the proof of the symplectic case, but the rest of the proof is quite different, since we are naturally led to using the calculus of polyvector fields, rather than differential forms; in particular, we use in the end a Poisson version of the classical Caratheodory-Jacobi-Lie theorem, which we also prove. At the end of the article, we generalize the action-angle theorem to the setting of non-commutative integrable systems on Poisson manifolds.