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.
The Hamilton–Jacobi equation in classical mechanics and the related eikonal equation in geometrical optics are often described as the “point of closest approach” between classical and quantum mechanics. Most textbook treatments of Hamilton–Jacobi theory are aimed at graduate students and derive the equation only after a long introduction to canonical transformations. Most treatments of the eikonal equation only emphasize its use in geometrical optics. We show that both the Hamilton–Jacobi equation and the eikonal equation can be derived by a common procedure using only elementary aspects of the Lagrangian and Hamiltonian formalisms introduced in undergraduate classical mechanics courses. Through this common approach, we hope to highlight to undergraduates the deep connections between classical mechanics, classical wave theory, and Schrödinger’s wave mechanics.