On commuting Tonelli Hamiltonians: Time-periodic case

TL;DR

This paper demonstrates that for two commuting time-periodic Tonelli Hamiltonians, key dynamical sets and barrier functions coincide, revealing a fundamental connection in their structure.

Contribution

It establishes the equivalence of Aubry sets, Mather sets, and barrier functions for commuting time-periodic Tonelli Hamiltonians, a novel insight in Hamiltonian dynamics.

Findings

Aubry sets coincide for commuting Hamiltonians

Mather's barrier functions are identical in this case

Provides a unified view of dynamical structures in commuting systems

Abstract

We show that the Aubry sets, the Ma\~{n}\'{e} sets and Mather's barrier functions are the same for two commuting time-periodic Tonelli Hamiltonians.