Aubry-Mather theory for contact Hamiltonian systems II

TL;DR

This paper extends Aubry-Mather and weak KAM theories to contact Hamiltonian systems with specific dependence on the contact variable, revealing new properties of invariant sets and phenomena in the contact setting.

Contribution

It introduces new invariant sets, compares properties of Aubry and Mañé sets, and explores differences in solutions for contact Hamiltonian systems, advancing the theoretical framework.

Findings

Properties of the Mañé set for Lipschitz dependence

Comparison and graph properties of Aubry sets in contact systems

Existence of a new flow-invariant set of strongly static orbits

Abstract

In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems $H(x,u,p)$ with certain dependence on the contact variable $u$. For the Lipschitz dependence case, we obtain some properties of the Ma\~{n}\'{e} set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set $\tilde{\mathcal{S}}_s$ consists of strongly static orbits, which coincides with the Aubry set $\tilde{\mathcal{A}}$ in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show $\tilde{\mathcal{S}}_s\subsetneqq\tilde{\mathcal{A}}$ in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of $H$ on $u$ fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the minimal viscosity solution and non-minimal ones.