Doubly Exotic Nth-Order Superintegrable Classical Systems Separating in Cartesian Coordinates

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YURDUŞEN İ., Mauricio Escobar-Ruiz A., Palma Y Meza Montoya I.

SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, vol.18, 2022 (SCI-Expanded) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 18
  • Publication Date: 2022
  • Doi Number: 10.3842/sigma.2022.039
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, Directory of Open Access Journals
  • Keywords: integrability in classical mechanics, higher-order superintegrability, separation, Key integrability of variables, exotic potentials, 3RD-ORDER INTEGRALS, QUANTUM, POTENTIALS, SYMMETRIES, PATH, HAMILTONIANS, OSCILLATOR, CONSTANTS, SEARCH, MOTION
  • Hacettepe University Affiliated: Yes


Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space E2 are explored. The study is restricted to Hamiltonians allowing separation of variables V(x, y) = V-1(x) + V-2(y) in Cartesian coordinates. In particular, the Hamiltonian H admits a polynomial integral of order N > 2. Only doubly exotic potentials are considered. These are potentials where none of their separated parts obey any linear ordinary differential equation. An improved procedure to calculate these higher-order superintegrable systems is described in detail. The two basic building blocks of the formalism are non-linear compatibility conditions and the algebra of the integrals of motion. The case N = 5, where doubly exotic confining potentials appear for the first time, is completely solved to illustrate the present approach. The general case N > 2 and a formulation of inverse problem in superintegrability are briefly discussed as well.