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Symmetry and separation of variables for the Hamilton–Jacobi equation W2t −W2x −W2y =0

Symmetry and separation of variables for the Hamilton–Jacobi equation W2t −W2x −W2y =0

##### Abstract

We present a detailed group theoretical study of the problem of separation of variables for the characteristic equation of the wave equation in one time and two space dimensions. Using the well-known Lie algebra isomorphism between canonical vector fields under the Lie bracket operation and functions (modulo constants) under Poisson brackets, we associate, with each R-separable coordinate system of the equation, an orbit of commuting constants of the motion which are quadratic members of the universal enveloping algebra of the symmetry algebra o (3,2). In this, the first of two papers, we essentially restrict ourselves to those orbits where one of the constants of the motion can be split off, giving rise to a reduced equation with a nontrivial symmetry algebra. Our analysis includes several of the better known two-body problems, including the harmonic oscillator and Kepler problems, as special cases.

##### Type

Journal Article

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##### Citation

Boyer, C.P., Kalnins, E.G. & Miller, W., Jr. (1978). Symmetry and separation of variables for the Hamilton–Jacobi equation W2t −W2x −W2y =0. Journal of Mathematical Physics, 19, 200.

##### Date

1978-01

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##### Rights

Copyright 1978American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in the Journal of Mathematical Physics and may be found at http://jmp.aip.org/jmp/top.jsp