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

Abstract

In this paper, we study the underlying geometry in the classical Hamilton-Jacobi theory. 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 theory, obtaining as a particular case, the classical theory. The main application in this paper arises in 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 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 theory.