Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems

TL;DR

This paper extends the Hamilton-Jacobi theory to all dynamical systems, including non-Hamiltonian ones, by framing it as a theory of ensembles, which also provides insights into the quantum-classical transition.

Contribution

It introduces a generalized Hamilton-Jacobi framework applicable to arbitrary systems, unifying Hamiltonian and non-Hamiltonian dynamics through ensemble interpretation.

Findings

Derives Hamilton-Jacobi equations for Hamiltonian systems

Provides a natural interpretation of quantum to classical transition

Establishes a unified ensemble-based formulation

Abstract

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.