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A Discrete Hamilton--Jacobi Theory for Contact Hamiltonian Dynamics

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.