Implicit Contact Dynamics and Hamilton-Jacobi Theory

TL;DR

This paper develops a Hamilton-Jacobi theory for implicit contact Hamiltonian systems by characterizing their dynamics as Legendrian and Lagrangian submanifolds, utilizing Herglotz Lagrangian dynamics with non-regular Lagrangians.

Contribution

It introduces a novel Hamilton-Jacobi framework for implicit contact Hamiltonian systems using geometric submanifold characterizations and non-regular Lagrangian functions.

Findings

Provides a geometric interpretation of implicit contact Hamiltonian dynamics.

Derives a Hamilton-Jacobi theory using Legendrian and Lagrangian submanifolds.

Utilizes Herglotz Lagrangian dynamics for non-regular Lagrangians.

Abstract

In this paper we propose a Hamilton-Jacobi theory for implicit contact Hamiltonian systems in two different ways. One is the understanding of implicit contact Hamiltonian dynamics as a Legendrian submanifold of the tangent contact space, and another is as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. In these two scenarios, we propose a Hamilton-Jacobi theory specifically derived with the aid of Herglotz Lagrangian dynamics generated by non-regular Lagrangian functions.