This paper explores the profound connection between the Eikonal principle, primarily known in optics for describing wavefront propagation, and the Hamilton-Jacobi formulation of classical mechanics. We explicitly demonstrate how the Hamilton-Jacobi equation, a powerful partial differential equation that unifies various aspects of classical dynamics, naturally gives rise to an Eikonal-like behavior when considering the surfaces of constant action. By drawing an explicit analogy between the optical path length and Hamilton's principal function, we show that the trajectories of particles in classical mechanics can be interpreted as "rays" that are orthogonal to these propagating "wavefronts" of constant action in configuration space. This framework provides an elegant geometric interpretation of classical mechanics, where the evolution of a system is described by the propagation of a scalar field. The paper elucidates how the principle of least action, a cornerstone of classical mechanics, finds its optical counterpart in Fermat's principle of least time, thereby achieving a deeper, unified understanding of these seemingly disparate physical phenomena. This unification not only offers a powerful method for solving complex mechanical problems but also provides conceptual insights bridging classical wave theory and particle dynamics, laying foundational groundwork for quantum mechanics.
This paper explores the profound connection between the Eikonal principle, primarily known in optics for describing wavefront propagation, and the Hamilton-Jacobi formulation of classical mechanics. We explicitly demonstrate how the Hamilton-Jacobi equation, a powerful partial differential equation that unifies various aspects of classical dynamics, naturally gives rise to an Eikonal-like behavior when considering the surfaces of constant action. By drawing an explicit analogy between the optical path length and Hamilton's principal function, we show that the trajectories of particles in classical mechanics can be interpreted as "rays" that are orthogonal to these propagating "wavefronts" of constant action in configuration space. This framework provides an elegant geometric interpretation of classical mechanics, where the evolution of a system is described by the propagation of a scalar field. The paper elucidates how the principle of least action, a cornerstone of classical mechanics, finds its optical counterpart in Fermat's principle of least time, thereby achieving a deeper, unified understanding of these seemingly disparate physical phenomena. This unification not only offers a powerful method for solving complex mechanical problems but also provides conceptual insights bridging classical wave theory and particle dynamics, laying foundational groundwork for quantum mechanics.
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.
In this paper, we show that the difficulties of interpretation of the principle of least action concerning "final causes" or "efficient causes" are due to the existence of two different actions, the "Euler-Lagrange action" (or classical action) and the "Hamilton-Jacobi action". These two actions, which are not clearly differentiated in the texbooks, are solutions to the same Hamilton-Jacobi equation, but with very different initial conditions: smooth conditions for the Hamilton-Jacobi action, singular conditions for the Euler-Lagrange action. There are related by the Minplus Path Integral which is the analog in classical mechanics of the Feynmann Path Integral in quantum mechanics. Finally, we propose a clear-cut interpretation of the principle of least action: the Hamilton-Jacobi action does not use "final causes" and seems to be the action used by Nature; the Euler-Lagrange action uses "final causes" and is the action used by an observer to retrospectively determine the trajectory of the particle.
Most authors of textbooks on quantum mechanics either postulate or sketch a short `ad hoc` derivation of Schrodinger's equation. In this work we give a detailed derivation of Schrodinger's equation from the Hamilton-Jacobi equation and the Eikonal equation in geometrical optics. We start from the historical debates on the nature of light -- whether it is a beam of particles, or waves in the aether. We derive the Eikonal equation and show the conditions when a wave can behave as a beam of particles. Then we discuss several experiments with an electron gun, that show clearly diffraction and interference of a single electron. Next, in order to explain these experiments, we derive Schrodinger's equation by comparing Hamilton-Jacobi equation in classical mechanics with the Eikonal equation in geometrical optics. To do that, we first show how to derive the wave equation from the Eikonal equation (not the other way around!). Second, we use this method to derive Schrodinger's equation from the Hamilton-Jacobi equation. Next, we derive Born's statistical rule using the early understanding of de Broglie that both particles and waves exist. Afterwards, we show that historically people preferred to remove the particles (as well as their trajectories) altogether from de Broglie's ideas but retained Born's rule (the so called Copenhagen interpretation). These derivations of the foundations of quantum mechanics do not follow precisely the history of the subject. Rather we select some early ideas and experiments in a judicious manner to present Schrodinger's equation in a logical and ordered way. We use the electron gun experiments instead of black body radiation and photoelectric effect. Our derivation may bring more light and satisfaction for the undergraduate students about the confusing and rather mysterious subject of quantum mechanics.
Abstract The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.
A well-known formulation of classical mechanics of system of particles is based on solving the Hamilton-Jacobi equation ∂S ∂t + H ( qi , ∂S ∂qi , t ) = 0, (19.1) where S(qi , αi , t) is Hamilton’s principal function, H is the Hamiltonian, qi are the configuration space variables, αi are constants of integration, and t denotes time. If the Hamiltonian is independent of t and is a constant of motion equal to the total energy E, then S and Hamilton’s characteristic function W are related by S = W − Et . (19.2) Since W is independent of time, the surfaces with W = constant have fixed locations in configuration space. However, the S = constant surfaces move in time and may be considered as wavefronts propagating in this space [1]. For a single particle one can show that the wave velocity at any point is given by u = E|∇W | = E p = E mv . (19.3) T&F Cat # K11224, Chapter 19, Page 302, 15-9-2010 This shows that the velocity of a point on this surface is inversely proportional to the spatial velocity of the particle. Also one can show that the trajectories of the particle must always be normal to the surfaces of constant S. The momentum of the particle is obtained as [1] p = ∇W . (19.4) It is not clear whether this wave picture was known to classical physicists. However, in hindsight, we may now observe that there exists a correspondence between classical mechanics and geometrical optics. That is, the particle trajectories orthogonal to surfaces of constant S in classical mechanics, as we have discussed above, are similar to light rays traveling orthogonal to Huygens’ wavefronts.
Classical mechanics admits multiple equivalent formulations, from Newton's equations to the variational Lagrange-Hamilton framework and the scalar Hamilton-Jacobi (HJ) theory. In the HJ formulation, classical ensembles evolve through the continuity equation for a real density ρ=R2 coupled to Hamilton's principal function S. Here we develop a complementary formulation, the Hamilton-Jacobi-Schrödinger (HJS) theory, by embedding the pair (R,S) into a single complex field. Starting from a completely general complex ansatz ψ=f(R,S)eig(R,S), and imposing two minimal structural requirements, we obtain a unique map $ψ= R\, e^{iS/κ}\, $ together with a linear HJS equation whose ∣κ∣→0 limit reproduces the HJ formulation exactly. Remarkably, when Re(κ)=0, essential features of quantum mechanics, superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule and unitary evolution, follow naturally as structural consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.