Hamilton-Jacobi as model reduction, extension to Newtonian particle mechanics, and a wave mechanical curiosity

TL;DR

This paper reinterprets the Hamilton-Jacobi equation as a model reduction technique for classical mechanics, extending it to non-conservative systems and deriving a dissipative Schrödinger equation via geometric optics approximation.

Contribution

It introduces a novel perspective on Hamilton-Jacobi as a model reduction and extends its applicability to dissipative Newtonian systems, linking to wave mechanics.

Findings

Hamilton-Jacobi viewed as a model reduction of particle mechanics.

Extension to non-conservative, dissipative systems.

Derivation of a dissipative Schrödinger equation from geometric optics approximation.

Abstract

The Hamilton-Jacobi equation of classical mechanics is approached as a model reduction of conservative particle mechanics where the velocity degrees-of-freedom are eliminated. This viewpoint allows an extension of the association of the Hamilton-Jacobi equation from conservative systems to general Newtonian particle systems involving non-conservative forces, including dissipative ones. A geometric optics approximation leads to a dissipative Schr\"odinger equation, with the expected limiting form when the associated classical force system involves conservative forces.