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

TL;DR

This paper investigates two-dimensional superintegrable classical systems with higher-order polynomial integrals, focusing on doubly exotic potentials in Cartesian coordinates, and provides a detailed method for analyzing such systems, including a complete solution for the case N=5.

Contribution

It introduces an improved method for analyzing higher-order superintegrable systems with doubly exotic potentials and fully solves the case N=5, expanding understanding of these complex systems.

Findings

Complete solution for N=5 superintegrable systems with doubly exotic potentials.

Development of a formalism based on non-linear compatibility conditions.

Insights into the algebraic structure of integrals of motion for these systems.

Abstract

Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space $E_2$ 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 $\mathcal H$ 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.