Structure results for higher order symmetry algebras of 2D classical superintegrable systems

Abstract

Recently the authors and J.M. Kress presented a special function recurrence relation method to prove quantum superintegrability of an integrable 2D system that included explicit constructions of higher order symmetries and the structure relations for the closed algebra generated by these symmetries. We applied the method to 5 families of systems, each depending on a rational parameter k, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system. Here we work out the analogs of these constructions for all of the associated classical Hamiltonian systems, as well as for a family including the generic potential on the 2-sphere. We do not have a proof in every case that the generating symmetries are of lowest possible order, but we believe this to be so via an extension of our method.

23 pages

References in corpus (9)

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Cited by in corpus (9)

• Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials
• New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials
• Combined state-adding and state-deleting approaches to type III multi-step rationally-extended potentials: applications to ladder operators and superintegrability
• General $N^{th}$-order superintegrable systems separating in polar coordinates
• Recurrence approach and higher rank cubic algebras for the $N$-dimensional superintegrable systems
• Two-dimensional superintegrable systems from operator algebras in one dimension
• New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates
• New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies
• Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras