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

In this paper, we study the underlying geometry in the classicalHamilton-Jacobi equation. The proposed formalism is also valid for nonholonomicsystems. We first introduce the essential geometric ingredients: a vectorbundle, a linear almost Poisson structure and a Hamiltonian function, both onthe dual bundle (a Hamiltonian system). From them, it is possible to formulatethe Hamilton-Jacobi equation, obtaining as a particular case, the classicaltheory. The main application in this paper is to nonholonomic mechanicalsystems. For it, we first construct the linear almost Poisson structure on thedual space of the vector bundle of admissible directions, and then, apply theHamilton-Jacobi theorem. Another important fact in our paper is the use of theorbit theorem to symplify the Hamilton-Jacobi equation, the introduction of thenotion of morphisms preserving the Hamiltonian system; indeed, this conceptwill be very useful to treat with reduction procedures for systems withsymmetries. Several detailed examples are given to illustrate the utility ofthese new developments.