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

TL;DR

The paper addresses how to treat the Hamilton-Jacobi equation as a velocity-elimination reduction of Newtonian mechanics and extend it to general Newtonian systems with non-conservative forces. It develops a gradient-based ansatz to derive a Hamilton-Jacobi equation and shows how a geometric optics limit yields a (nonlinear) dissipative Schrödinger-type equation, while a gradient-free extension leads to a coupled S,R system for general forces. A self-contained derivation of the classical Hamilton-Jacobi equation is provided, revealing the necessity of multiple solution sheets for different initial conditions. The work suggests links to variational duals, wave mechanics analogies, and potential applications to continuum/dissipative systems and hydrodynamic quantum analogs.

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 Schrödinger equation, with the expected limiting form when the associated classical force system involves conservative forces.