Variational attraction of the KAM torus for the conformally symplectic system

TL;DR

This paper investigates how the KAM torus influences the convergence speed of the Lax-Oleinik semigroup in conformally symplectic systems, highlighting the variational role of invariant Lagrangian graphs.

Contribution

It reveals the variational significance of invariant Lagrangian graphs and explains the impact of the KAM torus on the convergence rate of the Lax-Oleinik semigroup in such systems.

Findings

KAM torus affects the $W^{1,inity}$-convergence speed.

Invariant Lagrangian graphs have variational significance.

Analysis applies to conformally symplectic systems with positive definite Hamiltonian.

Abstract

For the conformally symplectic system \[ \left\{ \begin{aligned} \dot{q}&=H_p(q,p),\quad(q,p)\in T^*\mathbb{T}^n\\ \dot p&=-H_q(q,p)-\lambda p, \quad \lambda>0 \end{aligned} \right. \] with a positive definite Hamiltonian, we discuss the variational significance of invariant Lagrangian graphs and explain how the KAM torus impacts the $W^{1,\infty}-$convergence speed of the Lax-Oleinik semigroup.