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.

Topics
Hamilton-Jacobi equations, Hamiltonian mechanics, Lagrangian mechanics, Mechanical waves, Classical mechanics, Integral calculus, Geometrical optics, Educational aids, Theories in quantum mechanics, Schrodinger equations
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2011
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