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

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.