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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2022
• Main Authors: Yurduşen, İsmet, Escobar-Ruiz, Adrián Mauricio, Palma y Meza Montoya, Irlanda
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2022
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211629
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Doubly Exotic th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates. İsmet Yurduşen, Adrián Mauricio Escobar-Ruiz and Irlanda Palma y Meza Montoya. SIGMA 18 (2022), 039, 20 pages

Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space ₂ are explored. The study is restricted to Hamiltonians allowing separation of variables (, ) = ₁() + ₂() in Cartesian coordinates. In particular, the Hamiltonian ℋ admits a polynomial integral of order > 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 = 5, where doubly exotic confining potentials appear for the first time, is completely solved to illustrate the present approach. The general case > 2 and a formulation of the inverse problem in superintegrability are briefly discussed as well.