The Hamilton-Jacobi Equation: an intuitive approach

Abstract

The Hamilton-Jacobi equation (HJE) is one of the most elegant approach to Lagrangian systems such as geometrical optics and classical mechanics, establishing the duality between trajectories and waves and paving the way naturally for the quantum mechanics. Usually, this formalism is taught at the end of a course on analytical mechanics through its technical aspects and its relation to canonical transformations. I propose that the teaching of this subject be centered on this duality along the lines proposed here, and the canonical transformations be taught only after some familiarity with the HJE has been gained by the students.

References in corpus (5)

• Making Sense of the Legendre Transform
• From Classical to Quantum Mechanics through Optics
• Dynamics of the Chemical Master Equation, a strip of chains of equations in d-dimensional space
• Integrability and chaos: the classical uncertainty
• A giant disparity and a dynamical phase transition in large deviations of the time-averaged size of stochastic populations

Cited by in corpus (4)

• An eikonal equation approach to thermodynamics and the gradient flows in information geometry
• Shadow Cast by the Kerr MOG Black Hole under the Influence of Plasma and Constraints from EHT Observations
• The Carleman convexification method for Hamilton-Jacobi equations on the whole space
• Gravitational time dilation, free fall, and matter waves