Complete solutions of the Hamilton-Jacobi equation and the envelope method

TL;DR

This paper explores the relationship between complete solutions of the Hamilton-Jacobi equation, canonical transformations, and the envelope method, with applications to geometrical optics, revealing new ways to generate solutions.

Contribution

It establishes a connection between pairs of complete solutions via time-independent canonical transformations and introduces the envelope method for generating solutions.

Findings

Parameters of two complete solutions are related by a canonical transformation.

Envelope of solutions can generate new solutions via the difference or sum of solutions.

Applications to geometrical optics demonstrate practical relevance.

Abstract

It is shown that the parameters contained in any two complete solutions of the Hamilton-Jacobi equation, corresponding to a given Hamiltonian, are related by means of a time-independent canonical transformation and that, in some cases, a generating function of this transformation is given by the envelope of a family of surfaces defined by the difference of the two complete solutions. Conversely, in those cases, one of the complete solutions is given by the envelope of a family of surfaces defined by the sum of the other complete solution and the generating function of the canonical transformation. Some applications of these results to geometrical optics are also given.