# 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

@article{Cariena2021SuperintegrabilityOT, title={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}, author={Jos{\'e} F. Cari{\~n}ena and Manuel F Ra{\~n}ada and Mariano Santander}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2021}, volume={54} }

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, Noether symmetries and Noether momenta. Then we study the superintegrability of the harmonic oscillator, the Smorodinsky–Winternitz system and the harmonic oscillator with ratio of frequencies 1:1:2 and additional nonlinear terms on the three-dimensional sphere S 3 (κ > 0) and on the hyperbolic…

## One Citation

Reduction and integrability: a geometric perspective

- Mathematics
- 2022

A geometric approach to integrability and reduction of dynamical system is developed from a modern perspective. The main ingredients in such analysis are the inﬁnitesimal symmetries and the tensor…

## References

SHOWING 1-10 OF 79 REFERENCES

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…

Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S2 and the hyperbolic plane H2

- Mathematics, Physics
- 2005

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…

On harmonic oscillators on the two-dimensional sphere S2 and the hyperbolic plane H2. II.

- Mathematics, Physics
- 2002

The properties of several noncentral n=2 harmonic oscillators are examined on spaces of constant curvature. All the mathematical expressions are presented using the curvature κ as a parameter, in…

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…

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.…

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

- Physics
- 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…

Superintegrability in a two-dimensional space of nonconstant curvature

- Mathematics
- 2002

A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of…

New superintegrable models on spaces of constant curvature

- Mathematics, PhysicsAnnals of Physics
- 2020

Maximal superintegrability on N-dimensional curved spaces

- Mathematics
- 2003

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.…