Classical and Quantum Super-integrability: From Lissajous Figures to Exact Solvability

The first part of this paper explains what super-integrability is and how it differs in the classical and quantum cases. This is illustrated with an elementary example of the resonant harmonic oscillator. For Hamiltonians in "natural form", the kinetic energy has geometric origins and, in the flat and constant curvature cases, the large isometry group plays a vital role. We explain how to use the corresponding first integrals to build separable and super-integrable systems. We also show how to use the automorphisms of the symmetry algebra to help build the Poisson relations of the corresponding non-Abelian Poisson algebra. Finally, we take both the classical and quantum Zernike system, recently discussed by Pogosyan, et al, and show how the algebraic structure of its super-integrability can be understood in this framework.