Maximal superintegrability on N-dimensional curved spaces

Details Maximal superintegrability on N-dimensional curved spaces Maximal superintegrability on N-dimensional curved spaces

2002-11-08

arxiv.org/abs/math-ph/0211012v1

J. Phys. A36 (2003) L93-L99

Physics

Mathematical Physics

8 pages, LaTeX

Scientific paper

10.1088/0305-4470/36/7/101

A unified algebraic construction of the classical Smorodinsky-Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie groups SO(N+1), ISO(N), and SO(N,1) is presented. Firstly, general expressions for the Hamiltonian and its integrals of motion are given in a linear ambient space $R^{N+1}$, and secondly they are expressed in terms of two geodesic coordinate systems on the ND spaces themselves, with an explicit dependence on the curvature as a parameter. On the sphere, the potential is interpreted as a superposition of N+1 oscillators. Furthermore each Lie algebra generator provides an integral of motion and a set of 2N-1 functionally independent ones are explicitly given. In this way the maximal superintegrability of the ND Euclidean Smorodinsky-Winternitz system is shown for any value of the curvature.

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