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

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...

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Bibliographic Details
Published in:Journal of mathematical physics 2020-01, Vol.61 (1)
Main Authors: Grillo, Sergio, Padrón, Edith
Format: Article
Language:English
Subjects:
Chaos theory
Equations of motion
Fluid dynamics
Fluid flow
Fluid mechanics
Hamiltonian functions
Manifolds (mathematics)
Physics
Quadratures
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Summary: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.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.5133153