A Discrete Hamilton–Jacobi Theory for Contact Hamiltonian Dynamics

Oğul Esen, Cristina Sardón () and Marcin Zajac
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Oğul Esen: Department of Mathematics, Gebze Technical University, 41400 Gebze, Turkey
Cristina Sardón: Department of Applied Mathematics, Universidad Politécnica de Madrid, C/José Gutiérrez Abascal, 2, 28006 Madrid, Spain
Marcin Zajac: Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland

Mathematics, 2024, vol. 12, issue 15, 1-24

Abstract: In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton–Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton–Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton–Jacobi equation is related to the continuous Hamilton–Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton–Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper, we provide a numerical example to test the theory.

Keywords: Hamilton–Jacobi theory; discrete dynamics; contact manifolds; discrete Hamilton–Jacobi (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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