Effects of transformed Hamiltonians on Hamilton-Jacobi theory in view of a stronger connection to wave mechanics

TL;DR

This paper explores generalized Hamiltonian transformations in Hamilton-Jacobi theory, revealing connections to wave mechanics and providing a pedagogical bridge to Schrödinger's equation without quantum mechanics tools.

Contribution

It introduces a broader class of Hamiltonian representations and clarifies their relation to wave mechanics and Schrödinger's equation, enhancing understanding of quantum-classical connections.

Findings

Generalized Hamiltonian forms relate to wave functions.

Hamilton-Jacobi theory can lead to Schrödinger's equation.

Distinction between K=0 and cyclic K regimes clarified.

Abstract

Hamilton-Jacobi theory is a fundamental subject of classical mechanics and has also an important role in the development of quantum mechanics. Its conceptual framework results from the advantages of transformation theory and, for this reason, relates to the features of the generating function of canonical transformations connecting a specific Hamiltonian H to a new Hamiltonian K chosen to simplify Hamilton's equations of motion. Usually, the choice is between K=0 and a cyclic K depending on the new conjugate momenta only. Here, we investigate more general representations of the new Hamiltonian. Furthermore, it is pointed out that the two common alternatives of K=0 and a cyclic K should be distinguished in more detail. An attempt is made to clearly discern the two regimes for the harmonic oscillator and, not surprisingly, some correspondences to the quantum harmonic oscillator appear. Thanks to this preparatory work, generalized coordinates and momenta associated with the Hamilton's principal function S (i.e., K=0) are used to determine dependences in the Hamilton's characteristic function W (i.e., cyclic K). The procedure leads to the Schrodinger's equation where the Hamilton's characteristic function plays the role of the Schrodinger's wave function, whereas the first integral invariant of Poincare' takes the place of the reduced Planck constant. This finding advances the pedagogical value of the Hamilton-Jacobi theory in view of an introduction to Schrodinger's wave mechanics without the tools of quantum mechanics.