Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
Abstract
The Stäckel transform is applied to the geodesic motion on Euclidean space, through the harmonic oscillator and KeplerColoumb potentials, in order to obtain maximally superintegrable classical systems on Ndimensional Riemannian spaces of nonconstant curvature. By one hand, the harmonic oscillator potential leads to two families of superintegrable systems which are interpreted as an intrinsic KeplerColoumb system on a hyperbolic curved space and as the socalled Darboux III oscillator. On the other, the KeplerColoumb potential gives rise to an oscillator system on a spherical curved space as well as to the TaubNUT oscillator. Their integrals of motion are explicitly given. The role of the (flat/curved) Fradkin tensor and LaplaceRungeLenz Nvector for all of these Hamiltonians is highlighted throughout the paper. The corresponding quantum maximally superintegrable systems are also presented.
 Publication:

SIGMA
 Pub Date:
 May 2011
 DOI:
 10.3842/SIGMA.2011.048
 arXiv:
 arXiv:1103.4554
 Bibcode:
 2011SIGMA...7..048B
 Keywords:

 coupling constant metamorphosis;
 integrable systems;
 curvature;
 harmonic oscillator;
 KeplerCoulomb;
 Fradkin tensor;
 LaplaceRungeLenz vector;
 TaubNUT;
 Darboux surfaces;
 Mathematical Physics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 37J35;
 70H06;
 81R12
 EPrint:
 SIGMA 7:048,2011