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

Abstract

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.

Superintegrable and shape invariant systems with position dependent mass

“…f ′ x which is compatible with (16) and (17) in two cases: either ξ n = µ kn k (27) or the vectorξ n = ξ n − µ kn k satisfies the following condition f ′ x which is the necessary condition for coefficients of the first order integrals of motion [15]. Since such integrals of motion had been already classified in [15], we will setξ a = 0, i.e., impose the condition (27) on coefficients ξ n .…”

“…Thus the problem of classification of rotationally invariant PDM systems admitting second order integrals of motion is reduced to search for inequivalent solutions of equations (16), (17) and (18) for unknowns f ,Ṽ , ξ a and η for all versions of functions µ ab enumerated in (21)- (25). The corresponding calculations are outlined in Appendix, while the classification results are presented in the following section.…”

“…The rotationally invariant and superintegrable PDM systems were discussed in numerous interesting papers by A. Ballesteros, A. Enciso, F. J. Herranz, O. Ragnisco and D. Riglioni. Selected papers of this Spanish-Italian team are presented in the reference list, see [16]- [19]. Thanks to efforts of the mentioned authors and their collaborators, such systems became a well studied field.…”

“…rotations iff µ ab is reduced to tensor µ ab 1 given by equation (20). Substituting µ ab = µ ab 1 into equation (16) we obtain the following solution for f :…”

Are all classical superintegrable systems in two-dimensional space linearizable?

“…in two-dimensional space with non-constant curvature. Examples of such systems are the Perlick system [25], the Taub-NUT system [26], superintegrable systems for the Darboux space of Type I [11], and others [27], [28].…”