Extended Hamilton-Jacobi theory, contact manifolds and integrability by quadratures
arXiv:1909.11393 · doi:10.1063/1.5133153
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.
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- On the role of geometry in statistical mechanics and thermodynamics I: Geometric perspective
- Hamilton-Jacobi Formalism on Locally Conformally Symplectic Manifolds
- Reviewing the Geometric Hamilton-Jacobi Theory concerning Jacobi and Leibniz identities
- Hamilton-Jacobi theory and integrability for autonomous and non-autonomous contact systems
- Application of Herglotz's Variational Principle to Electromagnetic Systems with Dissipation
- An overview of the Hamilton--Jacobi theory: the classical and geometrical approaches and some extensions and applications