Lagrangian submanifolds and Hamilton-Jacobi equation

Lagrangian submanifolds are becoming a very essential tool to generalize and geometrically understand results and procedures in the area of mathematical physics. Here we use general Lagrangian submanifolds to provide a geometric version of the Hamilton-Jacobi equation. This interpretation allows us to study some interesting applications of Hamilton-Jacobi equation in holonomic, nonholonomic and time-dependent dynamics from a geometrical point of view.