Superintegrability and geometry: a review of the extended Hamiltonian approach

TL;DR

This paper reviews the extended Hamiltonian approach to constructing superintegrable systems, emphasizing geometric structures like warped manifolds and covering spaces, with examples including classical and quantum systems such as oscillators and Coulomb potentials.

Contribution

It introduces a geometric framework for generating superintegrable systems with high-degree polynomial integrals, expanding the understanding of their structure and symmetries.

Findings

Construction of superintegrable systems via Hamiltonian extension.

Identification of geometric structures like warped manifolds involved.

Examples include anisotropic oscillators and Coulomb systems.

Abstract

We review the results of several of our papers about the procedure of extension of Hamiltonians, allowing the construction of families of superintegrable systems with non-trivial polynomial first integrals (or symmetry operators) of arbitrarily high degree. In particular, we focus on the geometric structures involved by the procedure: warped manifolds and Riemannian coverings. Examples of superintegrable systems, classical and quantum, with the structure of extended Hamiltonians are anisotropic harmonic oscillators, Kepler-Coulomb, Tremblay-Turbiner-Winternitz and Post-Winternitz systems.