%0 Journal Article
%T Structural aspects of Hamilton-Jacobi theory
%A Jos¨¦ F. Cari£¿ena
%A Xavier Gr¨¤cia
%A Giuseppe Marmo
%A Eduardo Mart¨ªnez
%A Miguel C. Mu£¿oz-Lecanda
%A Narciso Rom¨¢n-Roy
%J Physics
%D 2015
%I arXiv
%R 10.1142/S0219887816500171
%X 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.
%U http://arxiv.org/abs/1511.00288v1