The Liouville theorem as a problem of common eigenfunctions

TL;DR

This paper reformulates the Liouville theorem in Hamiltonian mechanics as a problem of finding common eigenfunctions of constants of motion, linking phase space invariance to eigenfunction theory.

Contribution

It introduces a novel eigenfunction-based formulation of the Liouville theorem for Hamiltonian systems with multiple degrees of freedom.

Findings

Reformulation of Liouville theorem using eigenfunctions

Connection between phase space invariance and eigenfunctions

Framework for solving Hamilton-Jacobi equations via eigenfunctions

Abstract

It is shown that, by appropriately defining the eigenfunctions of a function defined on the extended phase space, the Liouville theorem on solutions of the Hamilton--Jacobi equation can be formulated as the problem of finding common eigenfunctions of $n$ constants of motion in involution, where $n$ is the number of degrees of freedom of the system.