The Hamilton–Jacobi equation: An alternative approach
Bahram Houchmandzadeh
Abstract
The Hamilton–Jacobi equation (HJE) is one of the most elegant approaches to Lagrangian systems such as geometrical optics and classical mechanics, establishing the duality between trajectories and waves and paving the way naturally for 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.
Topics & Concepts
PhysicsFormalism (music)Duality (order theory)LagrangianClassical mechanicsTheoretical physicsRelation (database)QuantumQuantum opticsGeometrical opticsDiagrammatic reasoningAlgebra over a fieldQuantum mechanicsClassical physicsAnalytical dynamicsCanonical quantizationCalculus (dental)Analytical mechanicsCanonical coordinatesQuantum Mechanics and Non-Hermitian PhysicsControl and Stability of Dynamical SystemsQuantum chaos and dynamical systems