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A Discrete Hamilton–Jacobi Theory for Contact Hamiltonian Dynamics

Author

Listed:
  • Oğul Esen

    (Department of Mathematics, Gebze Technical University, 41400 Gebze, Turkey
    Center for Mathematics and Its Applications, Khazar University, Baku AZ1096, Azerbaijan)

  • Cristina Sardón

    (Department of Applied Mathematics, Universidad Politécnica de Madrid, C/José Gutiérrez Abascal, 2, 28006 Madrid, Spain
    Departament of Quantitative Methods, ICADE, Universidad Pontificia de Comillas, C/de Alberto Aguilera, 23, 28015 Madrid, Spain)

  • Marcin Zajac

    (Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland)

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.

Suggested Citation

  • Oğul Esen & Cristina Sardón & Marcin Zajac, 2024. "A Discrete Hamilton–Jacobi Theory for Contact Hamiltonian Dynamics," Mathematics, MDPI, vol. 12(15), pages 1-24, July.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:15:p:2342-:d:1443816
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    References listed on IDEAS

    as
    1. Manuel de León & Manuel Lainz & Álvaro Muñiz-Brea, 2021. "The Hamilton–Jacobi Theory for Contact Hamiltonian Systems," Mathematics, MDPI, vol. 9(16), pages 1-24, August.
    2. Borin, Daniel & Livorati, André Luís Prando & Leonel, Edson Denis, 2023. "An investigation of the survival probability for chaotic diffusion in a family of discrete Hamiltonian mappings," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    3. Yuan, Cuilian & Yang, Hujiang & Meng, Xiankui & Tian, Ye & Zhou, Qin & Liu, Wenjun, 2023. "Modulational instability and discrete rogue waves with adjustable positions for a two-component higher-order Ablowitz–Ladik system associated with 4 × 4 Lax pair," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
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