A C1 Arnol'd-Liouville theorem

TL;DR

This paper extends the Arnol'd-Liouville theorem to C1 Hamiltonians, emphasizing the importance of Lipschitz regularity for invariant Lagrangian tori in understanding Hamiltonian dynamics.

Contribution

It proves a C1 version of the Arnol'd-Liouville theorem, highlighting the role of Lipschitz regularity in the structure of invariant tori and integrability.

Findings

Lipschitz regularity is crucial for the foliation by invariant Lagrangian tori.

C1 regularity ensures the continuity of Arnol'd-Liouville coordinates.

Various notions of C0 and Lipschitz integrability are explored.

Abstract

In this paper, we prove a version of Arnol'd-Liouville theorem for C 1 commuting Hamiltonians. We show that the Lipschitz regularity of the foliation by invariant Lagrangian tori is crucial to determine the Dynamics on each Lagrangian torus and that the C 1 regularity of the foliation by invariant Lagrangian tori is crucial to prove the continuity of Arnol'd-Liouville coordinates. We also explore various notions of C 0 and Lipschitz integrability.