Hamilton-Jacobi theory and integrability for autonomous and non-autonomous contact systems

TL;DR

This paper develops a Hamilton-Jacobi framework for contact Hamiltonian systems, both autonomous and non-autonomous, providing two approaches and illustrating with three physical examples.

Contribution

It introduces two Hamilton-Jacobi equations tailored for contact systems, extending classical theory to include dissipative and time-dependent dynamics.

Findings

Derived two Hamilton-Jacobi equations for contact systems.

Applied the theory to three physical examples.

Enhanced understanding of integrability in dissipative systems.

Abstract

In this paper, we study the integrability of contact Hamiltonian systems, both time-dependent and independent. In order to do so, we construct a Hamilton--Jacobi theory for these systems following two approaches, obtaining two different Hamilton--Jacobi equations. Compared to conservative Hamiltonian systems, contact Hamiltonian systems depend of one additional parameter. The fact of obtaining two equations reflects whether we are looking for solutions depending on this additional parameter or not. In order to illustrate the theory developed in this paper, we study three examples: the free particle with a linear external force, the freely falling particle with linear dissipation and the damped and forced harmonic oscillator.