Structural aspects of Hamilton-Jacobi theory

TL;DR

This paper explores the geometric structures underlying Hamilton-Jacobi theory, generalizing previous work by analyzing dynamics decompositions on manifolds and their relation to classical Hamilton-Jacobi equations in symplectic and Poisson frameworks.

Contribution

It introduces a comprehensive geometric interpretation of Hamilton-Jacobi theory using manifold decompositions and extends the framework to include tangent bundle functions and constants of motion.

Findings

Identified geometric structures linking dynamics decomposition to Hamilton-Jacobi theory.

Extended Hamilton-Jacobi interpretation to symplectic and Poisson manifolds.

Showed how functions on tangent bundles determine second-order dynamics.

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.