# Generalized action-angle coordinates in toric contact spaces

## TL;DR

This paper extends the concept of action-angle coordinates to contact geometry, providing a framework for integrable systems in contact manifolds and illustrating it with examples from five-dimensional toric Sasaki-Einstein spaces.

## Contribution

It introduces generalized contact action-angle variables for integrable contact Hamiltonian systems, expanding the tools available beyond symplectic geometry.

## Key findings

- Defined generalized contact action-angle variables.
- Applied the framework to five-dimensional toric Sasaki-Einstein spaces.
- Demonstrated the commutativity of first integrals in contact systems.

## Abstract

In this paper we are concerned with completely integrable Hamiltonian systems in the setting of contact geometry. Unlike the symplectic case, contact structures are automatically Hamiltonian. Using the Jacobi brackets defined on contact manifolds, we discuss the commutativity of the first integrals for contact Hamiltonian systems and introduce the generalized contact action-angle variables. We exemplify the general scheme in the case of the five-dimensional toric Sasaki-Einstein spaces $T^{1,1}$ and $Y^{p,q}$.