On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces

Abstract

We prove the integrability and superintegrability of a family of natural Hamiltonians which includes and generalises those studied in some literature, originally defined on the 2D Minkowski space. Some of the new Hamiltonians are a perfect analogy of the well-known superintegrable system on the Euclidean plane proposed by Tremblay-Turbiner-Winternitz and they are defined on Minkowski space, as well as on all other 2D manifolds of constant curvature, Riemannian or pseudo-Riemannian. We show also how the application of the coupling-constant-metamorphosis technique allows us to obtain new superintegrable Hamiltonians from the previous ones. Moreover, for the Minkowski case, we show the quantum superintegrability of the corresponding quantum Hamiltonian operator.Our results are obtained by applying the theory of extended Hamiltonian systems, which is strictly connected with the geometry of warped manifolds.

References in corpus (5)

• Factorizations of one dimensional classical systems
• Superintegrable 3-body systems on the line
• The Tremblay-Turbiner-Winternitz system as extended Hamiltonian
• Superintegrable Extensions of Superintegrable Systems
• Extended Hamiltonians and shift, ladder functions and operators

Cited by in corpus (3)

• Superintegrable families of magnetic monopoles with non-radial potential in curved background
• Superintegrability of 3-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems
• Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$