Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems

A generalization of the Hamilton-Jacobi theory to arbitrary dynamical systems, including non-Hamiltonian ones, is considered. The generalized Hamilton-Jacobi theory is constructed as a theory of ensemble of identical systems moving in the configuration space and described by the continual equation of motion and the continuity equation. For Hamiltonian systems, the usual Hamilton-Jacobi equations naturally follow from this theory. The proposed formulation of the Hamilton-Jacobi theory, as the theory of ensemble, allows interpreting in a natural way the transition from quantum mechanics in the Schrodinger's form to classical mechanics.