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

@article{Ballesteros2009MaximalSO,
  title={Maximal superintegrability of the generalized Kepler–Coulomb system on N-dimensional curved spaces},
  author={Angel Ballesteros and Francisco J. Herranz},
  journal={Journal of Physics A},
  year={2009},
  volume={42},
  pages={245203}
}
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. 49 022902) by finding an additional (hidden) integral of motion which is quartic in the momenta. In this paper, we present the generalization of this result to the N-dimensional spherical, hyperbolic and Euclidean spaces by making use of a unified symmetry approach that makes use of the curvature… 
View PDF on arXiv
47 Citations
Highly Influential Citations
2
Background Citations
12
Methods Citations
5

47 Citations

Citation Type

Citation Type

Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the St
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…
A maximally superintegrable deformation of the N-dimensional quantum Kepler-Coulomb system
The N-dimensional quantum Hamiltonian is shown to be exactly solvable for any real positive value of the parameter η. Algebraically, this Hamiltonian system can be regarded as a new maximally…
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
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,…
Variables separation and superintegrability of the nine-dimensional MICZ-Kepler problem
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…
A new integrable anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane
A new integrable generalization to the two-dimensional (2D) sphere, , and to the hyperbolic space, , of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms is…
Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
The classical Kepler{Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under Poisson brackets. This polynomial closure is…
The anisotropic oscillator on the 2D sphere and the hyperbolic plane
An integrable generalization on the 2D sphere S2 and the hyperbolic plane H2 of the Euclidean anisotropic oscillator Hamiltonian with 'centrifugal' terms given by is presented. The resulting…
Classical and quantum superintegrability with applications
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…
Classical and quantum higher order superintegrable systems from coalgebra symmetry
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…
New superintegrable models with position-dependent mass from Bertrand's Theorem on curved spaces
A generalized version of Bertrand's theorem on spherically symmetric curved spaces is presented. This result is based on the classification of (3+1)-dimensional (Lorentzian) Bertrand spacetimes, that…
...
...

References

SHOWING 1-10 OF 68 REFERENCES
Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles
Superintegrability on N-dimensional spaces of constant curvature from so(N + 1) and its contractions
The Lie—Poisson algebra so(N + 1) and some of its contractions are used to construct a family of superintegrable Hamiltonians on the N-dimensional spherical, Euclidean, hyperbolic, Minkowskian, and…
Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature
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…
Maximal superintegrability on N-dimensional curved spaces
A unified algebraic construction of the classical Smorodinsky–Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie groups SO(N + 1), ISO(N) and SO(N, 1) is presented.…
Superintegrable systems on the two-dimensional sphere S2 and the hyperbolic plane H2
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…
Dynamical symmetries in a spherical geometry. I
The two potentials for which a particle moving non-relativistically in a spherical space under the action of conservative central force executes closed orbits are found. When the curvature is zero…
Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S2 and the hyperbolic plane H2
The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane. First, the theory of central potentials on spaces of…
Completeness of multiseparable superintegrability on the complex 2-sphere
The possibility that Schrodinger's equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability. Here we demonstrate,…
A new superintegrable Hamiltonian
We identify a new superintegrable Hamiltonian in three degrees of freedom, obtained as a reduction of pure Keplerian motion in six dimensions. The new Hamiltonian is a generalization of the Keplerian…
On superintegrable symmetry-breaking potentials in N-dimensional Euclidean space
We give a graphical prescription for obtaining and characterizing all separable coordinates for which the Schr?dinger equation admits separable solutions for one of the superintegrable potentials…
...
...