Symmetry and separation of variables for the Hamilton–Jacobi equation W2t −W2x −W2y =0
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.
Permanent Research Commons link: https://hdl.handle.net/10289/1247
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.
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