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

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 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.