# Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the St

@inproceedings{Ballesteros2011SuperintegrableOA, title={Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the St}, author={{\'A}ngel Ballesteros and Alberto Enciso and Francisco J. Herranz and Orlando Ragnisco and Danilo Riglioni}, year={2011} }

The Stackel transform is applied to the geodesic motion on Euclidean space, through the harmonic oscillator and Kepler{Coloumb potentials, in order to obtain ma- ximally superintegrable classical systems on N-dimensional Riemannian spaces of noncon- stant 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…

## 18 Citations

Superintegrability on the three-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the sphere S 3 and on the hyperbolic space H 3

- Mathematics, PhysicsJournal of Physics A: Mathematical and Theoretical
- 2021

The superintegrability of several Hamiltonian systems defined on three-dimensional configuration spaces of constant curvature is studied. We first analyze the properties of the Killing vector fields,…

Classical and quantum superintegrability with applications

- Mathematics
- 2013

A superintegrable system is, roughly speaking, a system that allows more integrals of motion than degrees of freedom. This review is devoted to finite dimensional classical and quantum…

Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability

- Mathematics
- 2017

The Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian H = T + V into a geodesic Hamiltonian T with one additional degree of freedom, is applied to the four families of…

Superintegrability of three-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems

- Mathematics, Physics
- 2021

We study four particular three-dimensional natural Hamiltonian systems defined in conformally Euclidean spaces. We prove their superintegrability and we obtain, in the four cases, the maximal number…

Superintegrable systems in non-Euclidean plane: Hidden symmetries leading to linearity

- MathematicsJournal of Mathematical Physics
- 2021

Nineteen classical superintegrable systems in two-dimensional non-Euclidean spaces are shown to possess hidden symmetries leading to their linearization. They are the two Perlick systems [A.…

Variables separation and superintegrability of the nine-dimensional MICZ-Kepler problem

- Mathematics, Physics
- 2018

The nine-dimensional MICZ-Kepler problem is of recent interest. This is a system describing a charged particle moving in the Coulomb field plus the field of a SO(8) monopole in a nine-dimensional…

Classical and quantum higher order superintegrable systems from coalgebra symmetry

- Mathematics, Physics
- 2013

The N-dimensional generalization of Bertrand spaces as families of maximally superintegrable (M.S.) systems on spaces with a nonconstant curvature is analyzed. Considering the classification of…

Examples of complete solvability of 2D classical superintegrable systems

- Mathematics
- 2015

Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very…

Curvature as an Integrable Deformation

- MathematicsIntegrability, Supersymmetry and Coherent States
- 2019

The generalization of (super)integrable Euclidean classical Hamiltonian systems to the two-dimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed.…

## References

SHOWING 1-10 OF 55 REFERENCES

Maximal superintegrability of the generalized Kepler–Coulomb system on N-dimensional curved spaces

- Physics, Mathematics
- 2009

The superposition of the Kepler–Coulomb potential on the 3D Euclidean space with three centrifugal terms has recently been shown to be maximally superintegrable (Verrier and Evans 2008 J. Math. Phys.…

Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability

- Physics
- 2011

Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

- Mathematics
- 2007

An infinite family of quasi-maximally superintegrable Hamiltonians with a com- mon set of (2N 3) integrals of the motion is introduced. The integrability properties of all these Hamiltonians are…

Path integral approach for superintegrable potentials on spaces of nonconstant curvature: I. Darboux spaces D I and D II

- Mathematics
- 2007

In this paper, the Feynman path integral technique is applied for superintegrable potentials on two-dimensional spaces of nonconstant curvature: these spaces are Darboux spaces DI and DII. On DI,…

Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature

- Mathematics
- 2007

An infinite family of classical superintegrable Hamiltonians defined on the N-dimensional spherical, Euclidean and hyperbolic spaces are shown to have a common set of (2N − 3) functionally…

Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles

- Physics, Mathematics
- 2008

Superintegrable systems on the two-dimensional sphere S2 and the hyperbolic plane H2

- Mathematics
- 1999

The existence of superintegrable systems with n=2 degrees of freedom possessing three independent globally defined constants of motion which are quadratic in the velocities is studied on the…

Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds

- Mathematics
- 2008

Bertrand’s theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is…

Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: II. Darboux Spaces DIII and DIV

- Mathematics
- 2006

In this paper the Feynman path integral technique is applied for superintegrable potentials on two-dimensional spaces of non-constant curvature: these spaces are Darboux spaces D_I and D_II,…

Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform

- Mathematics
- 2005

This paper is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems, both classical and quantum, in conformally flat spaces. Here…