Contact flows and integrable systems

TL;DR

This paper extends the Arnold-Liouville theorem to Hamiltonian systems on contact hypersurfaces, showing they can be foliated by invariant tori even without full integrability, and explores contact systems with constraints.

Contribution

It introduces a partial integrability theorem for contact Hamiltonian systems and analyzes Reeb flows on Brieskorn manifolds with constraints.

Findings

Invariant hypersurfaces are foliated by Lagrangian tori.

Reeb flows on Brieskorn manifolds exhibit integrable behavior.

Partial integrability extends classical results to contact systems.

Abstract

We consider Hamiltonian systems restricted to the hypersurfaces of contact type and obtain a partial version of the Arnold-Liouville theorem: the system not need to be integrable on the whole phase space, while the invariant hypersurface is foliated on an invariant Lagrangian tori. In the second part of the paper we consider contact systems with constraints. As an example, the Reeb flows on Brieskorn manifolds are considered.