Superintegrable oscillator and kepler systems on spaces of nonconstant curvature via the stackel transform

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Abstract

The Stackel transform is applied to the geodesic motion on Euclidean space, through the harmonic oscillator and Kepler-Coloumb potentials, in order to obtain maximally superintegrable classical systems on N-dimensional 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 Kepler-Coloumb system on a hyperbolic curved space and as the so-called Darboux III oscillator. On the other, the Kepler-Coloumb potential gives rise to an oscillator system on a spherical curved space as well as to the Taub-NUT oscillator. Their integrals of motion are explicitly given. The role of the (flat/curved) Fradkin tensor and Laplace-Runge-Lenz N-vector for all of these Hamiltonians is highlighted throughout the paper. The corresponding quantum maximally superintegrable systems are also presented. © 2010 Mathematics Subject Classification.

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Ballesteros, A., Enciso, A., Herranz, F. J., Ragnisco, O., & Riglioni, D. (2011). Superintegrable oscillator and kepler systems on spaces of nonconstant curvature via the stackel transform. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 7. https://doi.org/10.3842/SIGMA.2011.048

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