Abstract

TORRES DEL CASTILLO, G.F.  and  ANAYA GONZALEZ, G.S.. Complete solutions of the Hamilton-Jacobi equation and the envelope method. Rev. mex. fis. [online]. 2014, vol.60, n.6, pp.414-418. ISSN 0035-001X.

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.

Keywords : Hamilton-Jacobi equation; canonical transformations; envelopes; eikonal equation.