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

Abstract

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) = 𝑉₁(𝑥) + 𝑉₂(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 the inverse problem in superintegrability are briefly discussed as well.