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

Authors

  • G.F. Torres del Castillo
  • G.S. Ana
  • a González.

Keywords:

Hamilton--Jacobi equation, canonical transformations, envelopes, eikonal equation

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.

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Published

2014-01-01

How to Cite

[1]
G. Torres del Castillo, G. Ana, and a González., “Complete solutions of the Hamilton--Jacobi equation and the envelope method”, Rev. Mex. Fís., vol. 60, no. 6 Nov-Dec, pp. 414–0, Jan. 2014.

Issue

Vol. 60 No. 6 Nov-Dec (2014): Revista Mexicana de Física.

Section

Articles

License

Authors retain copyright and grant the Revista Mexicana de Física right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.