A certain minimization property implies a certain integrability

TL;DR

This paper proves that for a compact connected manifold with a Tonelli Hamiltonian, the cotangent bundle can be partitioned into invariant Lagrangian graphs with specific regularity and dynamical properties, revealing a structured decomposition of the phase space.

Contribution

It establishes a partition of the cotangent bundle into invariant C0 Lagrangian graphs and shows that a residual subset of these are C1, with non-wandering dynamics, under Tonelli Hamiltonian conditions.

Findings

Partition of cotangent bundle into invariant Lagrangian graphs

Residual subset of graphs are C1 regular

Dynamics on these graphs are non-wandering

Abstract

The manifold M being compact and connected and H being a Tonelli Hamiltonian such that the cotangent bundle of M is equal to the dual tiered Mane set, we prove that there is a partition of the cotangent bundle of M into invariant C0 Lagrangian graphs. Moreover, among these graphs, those that are C1 cover a residual subset of this cotangent bundle The dynamic restricted to each of these sets is non wandering.