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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2020
• ISSN: 1815-0659
• Main Authors: Chanu, Claudia Maria, Rastelli, Giovanni
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2020
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210698
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces. Claudia Maria Chanu and Giovanni Rastelli. SIGMA 16 (2020), 052, 16 pages

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 also show 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.