A Discrete Hamilton--Jacobi Theory for Contact Hamiltonian Dynamics

TL;DR

This paper develops a novel discrete Hamilton--Jacobi theory for contact Hamiltonian dynamics, extending continuous geometric methods to discrete settings and providing numerical validation.

Contribution

It introduces a new discrete Hamilton--Jacobi framework for contact Hamiltonian systems based on geometric and variational principles.

Findings

Derived discrete Hamilton--Jacobi equations for contact systems.

Established connection between discrete and continuous Hamilton--Jacobi theories.

Validated the theory with a numerical example.

Abstract

In this paper, we propose a discrete Hamilton--Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton--Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton--Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton--Jacobi equation is related to the continuous Hamilton--Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton--Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper we provide a numerical example to test the theory.