Structural aspects of Hamilton-Jacobi theory
Abstract
In our previous papers [11,13] we showed that the Hamilton-Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton-Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (`slicing vector fields') on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton-Jacobi theory, by considering special cases like fibred manifolds and Hamiltonian dynamics, in the symplectic framework and the Poisson one. We also show how a set of functions on a tangent bundle can determine a second-order dynamics for which they are constants of the motion.
Cite
@article{arxiv.1511.00288,
title = {Structural aspects of Hamilton-Jacobi theory},
author = {José F. Cariñena and Xavier Gràcia and Giuseppe Marmo and Eduardo Martínez and Miguel C. Muñoz-Lecanda and Narciso Román-Roy},
journal= {arXiv preprint arXiv:1511.00288},
year = {2016}
}
Comments
26 pages. Minor changes (some minor mistakes are corrected)