Weak KAM for commuting Hamiltonians

TL;DR

This paper proves that for two commuting Tonelli Hamiltonians, their weak KAM solutions coincide, leading to equal Aubry sets and Mather's $eta$ functions, using a geometric approach that generalizes previous results.

Contribution

It introduces a geometric proof of the commutation of Lax-Oleinik semi-groups and extends the equivalence of weak KAM solutions for commuting Hamiltonians.

Findings

Weak KAM solutions for G and H are identical on cotangent bundles.

Aubry sets, Peierls barriers, and flat parts of Mather's $eta$ functions are equal for commuting Hamiltonians.

The geometric method simplifies and generalizes previous algebraic proofs.

Abstract

For two commuting Tonelli Hamiltonians, we recover the commutation of the Lax-Oleinik semi-groups, a result of Barles and Tourin ([BT01]), using a direct geometrical method (Stoke's theorem). We also obtain a "generalization" of a theorem of Maderna ([Mad02]). More precisely, we prove that if the phase space is the cotangent of a compact manifold then the weak KAM solutions (or viscosity solutions of the critical stationary Hamilton-Jacobi equation) for G and for H are the same. As a corrolary we obtain the equality of the Aubry sets, of the Peierls barrier and of flat parts of Mather's $\alpha$ functions. This is also related to works of Sorrentino ([Sor09]) and Bernard ([Ber07b]).