Superintegrability on N-dimensional spaces of constant curvature from so(N+1) and its contractions

Details Superintegrability on N-dimensional spaces of constant curvature from so(N+1) and its contractions Superintegrability on N-dimensional spaces of constant curvature from so(N+1) and its contractions

2007-07-25

arxiv.org/abs/0707.3772v1

Phys.Atom.Nucl.71:905-913,2008

Physics

Mathematical Physics

14 pages. Based on the contribution presented at the "XII International Conference on Symmetry Methods in Physics", Yerevan (A

Scientific paper

10.1134/S1063778808050207

The Lie-Poisson algebra so(N+1) and some of its contractions are used to construct a family of superintegrable Hamiltonians on the ND spherical, Euclidean, hyperbolic, Minkowskian and (anti-)de Sitter spaces. We firstly present a Hamiltonian which is a superposition of an arbitrary central potential with N arbitrary centrifugal terms. Such a system is quasi-maximally superintegrable since this is endowed with 2N-3 functionally independent constants of the motion (plus the Hamiltonian). Secondly, we identify two maximally superintegrable Hamiltonians by choosing a specific central potential and finding at the same time the remaining integral. The former is the generalization of the Smorodinsky-Winternitz system to the above six spaces, while the latter is a generalization of the Kepler-Coulomb potential, for which the Laplace-Runge-Lenz N-vector is also given. All the systems and constants of the motion are explicitly expressed in a unified form in terms of ambient and polar coordinates as they are parametrized by two contraction parameters (curvature and signature of the metric).

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