Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 048, 15 pages      arXiv:1103.4554      http://dx.doi.org/10.3842/SIGMA.2011.048
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform

Ángel Ballesteros a, Alberto Enciso b, Francisco J. Herranz a, Orlando Ragnisco c and Danilo Riglioni c
a) Departamento de Física, Universidad de Burgos, E-09001 Burgos, Spain
b) Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UC3M), Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera 14-16, E-28049 Madrid, Spain
c) Dipartimento di Fisica, Università di Roma Tre and Istituto Nazionale di Fisica Nucleare sezione di Roma Tre, Via Vasca Navale 84, I-00146 Roma, Italy

Received March 18, 2011, in final form May 12, 2011; Published online May 14, 2011

Abstract
The Stäckel 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.

Key words: coupling constant metamorphosis; integrable systems; curvature; harmonic oscillator; Kepler-Coulomb; Fradkin tensor; Laplace-Runge-Lenz vector; Taub-NUT; Darboux surfaces.

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