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.
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