Variational principle for contact Hamiltonian systems and its applications

TL;DR

This paper explores a variational principle for contact Hamiltonian systems and applies it to derive a solution representation for evolutionary equations and analyze the ergodic problem of stationary equations.

Contribution

It introduces a new variational principle for contact Hamiltonian systems and applies it to solve evolutionary and ergodic problems in this context.

Findings

Provided a representation formula for the solution semigroup of the evolutionary PDE.

Studied the ergodic problem and identified solution pairs (u,c) satisfying the stationary PDE.

Extended the variational principle to applications in contact Hamiltonian systems.

Abstract

In \cite{WWY}, the authors provided an implicit variational principle for the contact Hamilton's equations \begin{align*} \left\{ \begin{array}{l} \dot{x}=\frac{\partial H}{\partial p}(x,u,p),\\ \dot{p}=-\frac{\partial H}{\partial x}(x,u,p)-\frac{\partial H}{\partial u}(x,u,p)p,\quad (x,p,u)\in T^*M\times\mathbf{R},\\ \dot{u}=\frac{\partial H}{\partial p}(x,u,p)\cdot p-H(x,u,p), \end{array} \right. \end{align*} where $M$ is a closed, connected and smooth manifold and $H=H(x,u,p)$ is strictly convex, superlinear in $p$ and Lipschitz in $u$. In the present paper, we focus on two applications of the variational principle: 1. We provide a representation formula for the solution semigroup of the evolutionary equation \[ w_t(x,t)+H(x,w(x,t),w_x(x,t))=0; \] 2. We study the ergodic problem of the stationary equation via the solution semigroup. More precisely, we find pairs $(u,c)$ with $u\in C(M,\mathbf{R})$ and $c\in\mathbf{R}$ which, in the viscosity sense, satisfy the stationary partial differential equation \[ H(x,u(x),u_x(x))=c. \]