Extended Hamilton-Jacobi theory, contact manifolds and integrability by quadratures

TL;DR

This paper extends Hamilton-Jacobi theory to contact manifolds, demonstrating how complete pseudo-isotropic solutions enable integrability by quadratures for contact Hamiltonian systems, with applications in thermodynamics and fluid mechanics.

Contribution

It introduces a Hamilton-Jacobi framework for contact systems and shows how such solutions lead to integrability, extending recent symplectic results to contact geometry.

Findings

Complete pseudo-isotropic solutions ensure integrability by quadratures.

Application of theory to thermodynamics and fluid mechanics systems.

Extension of integrability results from symplectic to contact manifolds.

Abstract

A Hamilton-Jacobi theory for general dynamical systems, defined on fibered phase spaces, has been recently developed. In this paper we shall apply such a theory to contact Hamiltonian systems, as those appearing in thermodynamics and on geodesic flows in fluid mechanics. We first study the partial and complete solutions of the Hamilton-Jacobi Equation (HJE) related to these systems. Then we show that, for a given contact system, the knowledge of what we have called a complete pseudo-isotropic solution ensures the integrability by quadratures of its equations of motion. This extends to contact manifolds a recent result obtained in the context of general symplectic and Poisson manifolds.