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Open AccessArticle
A Discrete Hamilton–Jacobi Theory for Contact Hamiltonian Dynamics
by
Oğul Esen
Oğul Esen 1,2
,
Cristina Sardón
Cristina Sardón 3,4,* and
Marcin Zajac
Marcin Zajac 5
1
Department of Mathematics, Gebze Technical University, 41400 Gebze, Turkey
2
Center for Mathematics and Its Applications, Khazar University, Baku 1009, Azerbaijan
3
Department of Applied Mathematics, Universidad Politécnica de Madrid, C/José Gutiérrez Abascal, 2, 28006 Madrid, Spain
4
Departament of Quantitative Methods, ICADE, Universidad Pontificia de Comillas, C/de Alberto Aguilera, 23, 28015 Madrid, Spain
5
Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(15), 2342; https://doi.org/10.3390/math12152342 (registering DOI)
Submission received: 3 June 2024
/
Revised: 5 July 2024
/
Accepted: 9 July 2024
/
Published: 26 July 2024
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.
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MDPI and ACS Style
Esen, O.; Sardón, C.; Zajac, M.
A Discrete Hamilton–Jacobi Theory for Contact Hamiltonian Dynamics. Mathematics 2024, 12, 2342.
https://doi.org/10.3390/math12152342
AMA Style
Esen O, Sardón C, Zajac M.
A Discrete Hamilton–Jacobi Theory for Contact Hamiltonian Dynamics. Mathematics. 2024; 12(15):2342.
https://doi.org/10.3390/math12152342
Chicago/Turabian Style
Esen, Oğul, Cristina Sardón, and Marcin Zajac.
2024. "A Discrete Hamilton–Jacobi Theory for Contact Hamiltonian Dynamics" Mathematics 12, no. 15: 2342.
https://doi.org/10.3390/math12152342
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