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

Abstract

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).

arXiv comment14 pages. Based on the contribution presented at the "XII International Conference on Symmetry Methods in Physics", Yerevan (Armenia), July 2006. To appear in Physics of Atomic Nuclei

References in corpus (15)

• Completeness of superintegrability in two-dimensional constant curvature spaces
• Path Integral Discussion for Smorodinsky-Winternitz Potentials: I.\ Two- and Three Dimensional Euclidean Space
• Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$
• Relation of the oscillator and Coulomb systems on spheres and pseudospheres
• Quantum superintegrability and exact solvability in N dimensions
• Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature
• Maximal superintegrability on N-dimensional curved spaces
• Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry
• A maximally superintegrable system on an n-dimensional space of nonconstant curvature
• Conformal compactification and cycle-preserving symmetries of spacetimes
• Coulomb-oscillator duality in spaces of constant curvature
• Casimir invariants for the complete family of quasi-simple orthogonal algebras
• The Coulomb-Oscillator Relation on n-Dimensional Spheres and Hyperboloids
• Integrable potentials on spaces with curvature from quantum groups
• Maximally superintegrable Smorodinsky-Winternitz systems on the N-dimensional sphere and hyperbolic spaces

Cited by in corpus (4)

• Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles
• Maximal superintegrability of the generalized Kepler--Coulomb system on N-dimensional curved spaces
• N-dimensional integrability from two-photon coalgebra symmetry
• Superintegrable potentials on 3D Riemannian and Lorentzian spaces with non-constant curvature