Superintegrable Smorodinsky-Winternitz potential on the N-dimensional sphere and hyperbolic space

Francisco Jose Herranz (fjherranz@ubu.es)
Universidad de Burgos
Departamento de Fisica Escuela Politecnica Superior
Avda. Cantabria s.n. 09006 Burgos SPAIN

Abstract

The well known Smorodinsky-Winternitz system on the N-dimensional Euclidean space is formed by a harmonic oscillator potential ($\sum_i x_i^2$) together with centrifugal terms ($\sum_i \alpha_i/x_i^2$). We present a construction for its non-zero curvature versions on the sphere and hyperbolic spaces for an arbitrary number $N$ of degrees of freedom. This approach makes use of the $(N+1)$-dimensional vector model in terms of Weierstrass (ambient) coordinates in $R^{N+1}$ together with a direct relationship with the Lie algebras $so(N+1)$ and $so(N,1)$; this provides simple and close general expressions depending on the curvature. Next we write such expressions in terms of two sets of coordinate systems on the proper $N$-dimensional spaces: either geodesic parallel (Cartesian) or geodesic polar coordinates. Their corresponding integrals of motion are also obtained in a very explicit way which, in turn, show that superintegrability is preserved when curvature arises. All these results generalize those already obtained for $N=2,3$ degrees of freedom and when the curvature vanishes the `flat' Smorodinsky-Winternitz system is recovered.