Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic Mechanics

TL;DR

This paper develops a geometric framework for the Hamilton-Jacobi equation applicable to nonholonomic systems, introducing linear almost Poisson structures and morphisms to facilitate reduction and analysis.

Contribution

It introduces a unified geometric formalism for Hamilton-Jacobi theory in nonholonomic mechanics using linear almost Poisson structures and morphisms.

Findings

Formulated Hamilton-Jacobi equation for nonholonomic systems

Constructed linear almost Poisson structures on dual bundles

Applied orbit theorem and morphisms for system reduction

Abstract

In this paper, we study the underlying geometry in the classical Hamilton-Jacobi equation. The proposed formalism is also valid for nonholonomic systems. We first introduce the essential geometric ingredients: a vector bundle, a linear almost Poisson structure and a Hamiltonian function, both on the dual bundle (a Hamiltonian system). From them, it is possible to formulate the Hamilton-Jacobi equation, obtaining as a particular case, the classical theory. The main application in this paper is to nonholonomic mechanical systems. For it, we first construct the linear almost Poisson structure on the dual space of the vector bundle of admissible directions, and then, apply the Hamilton-Jacobi theorem. Another important fact in our paper is the use of the orbit theorem to symplify the Hamilton-Jacobi equation, the introduction of the notion of morphisms preserving the Hamiltonian system; indeed, this concept will be very useful to treat with reduction procedures for systems with symmetries. Several detailed examples are given to illustrate the utility of these new developments.