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

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.

References in corpus (7)

• Superintegrable Systems in Darboux spaces
• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Integrable and superintegrable quantum systems in a magnetic field
• Integrable and superintegrable systems with spin
• Polynomial Poisson Algebras for Classical Superintegrable Systems with a Third Order Integral of Motion
• Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials
• The Tremblay-Turbiner-Winternitz system as extended Hamiltonian