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 (17)

• Superintegrability with third order invariants in quantum and classical mechanics
• Superintegrable Systems in Darboux spaces
• Superintegrability in a two-dimensional space of non-constant curvature
• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Quantum superintegrability and exact solvability in N dimensions
• 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
• On maximally superintegrable systems
• On superintegrable systems separable in Cartesian coordinates
• General $N^{th}$-order superintegrable systems separating in polar coordinates
• Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials
• The Tremblay-Turbiner-Winternitz system as extended Hamiltonian
• Fourth-order superintegrable systems separating in Polar Coordinates. II. Standard Potentials
• New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates
• Second Order Superintegrable Systems in Three Dimensions
• Higher order symmetries for linear and nonlinear Schroedinger equations

Cited by in corpus (2)

• Superintegrable quantum mechanical systems with position dependent masses invariant with respect to two parametric Lie groups
• Polynomially Superintegrable Hamiltonians Separating in Cartesian Coordinates