On the Geometry of the Hamilton-Jacobi Equation and Generating Functions

Sebastián Ferraro, Manuel de León, Juan Carlos Marrero, David Martín de Diego, Miguel Vaquero

Published 2016-06-02Version 1

In this paper we develop a geometric version of the Hamilton-Jacobi equation in the Poisson setting. Specifically, we "geometrize" what is usually called a complete solution of the Hamilton-Jacobi equation. We use some well-known results about symplectic groupoids, in particular cotangent groupoids, as a keystone for the construction of our framework. Our methodology follows the ambitious program proposed by A. Weinstein, [55], in order to develop geometric formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids and Lie groupoids. This procedure allows us to take symmetries into account, and as a by-product, we recover results from [12, 25, 27], but even in those situations our approach is new. A theory of generating functions for the Poisson structures considered here is also developed following the same pattern. As an application we develop numerical methods based on generating functions, solving a longstanding problem of the area: how to obtain a generating function for the identity in Poisson manifolds. Some conclusions, current and future directions of research are shown at the end of the paper.