A Geometric Framework for Discrete Hamilton-Jacobi Equation

In classical mechanics the Hamilton-Jacobi Equation is useful to integrate partially or completely Hamilton's equations [2]. Recent developments have provided this theory with an intrinsic formulation, see for instance [3]. Another branch in mechanics that has been studied from a geometric viewpoint is discrete lagrangian and hamiltonian mechanics [5, 6]. In this contribution we aim to mingle those two theories to describe the discrete Hamilton-Jacobi Equation. This has already started to be studied in the literature [7], but not intrinsically. We will show here that the use of Lagrangian submanifolds [8] creates the natural setting to describe geometrically the discrete Hamilton-Jacobi equation.