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

@article{Cariena2017SuperintegrableSO,
  title={Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability},
  author={Jos{\'e} F. Cari{\~n}ena and Francisco J. Herranz and Manuel F Ra{\~n}ada},
  journal={Journal of Mathematical Physics},
  year={2017},
  volume={58},
  pages={022701}
}
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 quadratically superintegrable systems with multiple separabilities in the Euclidean plane. First, the separability and superintegrability of such four geodesic Hamiltonians T r (r = a, b, c, d) in a three-dimensional curved space are studied and then these four systems are modified with the addition… 

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References

SHOWING 1-10 OF 99 REFERENCES
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
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
Superintegrability in a two-dimensional space of nonconstant curvature
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
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
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.
Completeness of superintegrability in two-dimensional constant-curvature spaces
We classify the Hamiltonians H = px2 + py2 + V(x,y) of all classical superintegrable systems in two-dimensional complex Euclidean space with two additional second-order constants of the motion. We
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
...
...