Examples of Complete Solvability of 2D Classical Superintegrable Systems

TL;DR

This paper demonstrates how 2D classical superintegrable systems can be explicitly solved algebraically and geometrically, illustrating detailed trajectory determination through examples and exploring algebraic structures and special cases.

Contribution

It provides explicit examples and methods for solving 2D superintegrable systems, including new insights into their algebraic structures and special trajectory types.

Findings

Trajectories can be determined explicitly using algebraic and geometric methods.

Symmetry algebra structures reveal detailed information about system trajectories.

Identification of unique trajectories like 'metronome orbits'.

Abstract

Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved algebraically. In this paper we show explicitly, mostly through examples of 2nd order superintegrable systems in 2 dimensions, how the trajectories can be determined in detail using rather elementary algebraic, geometric and analytic methods applied to the closed quadratic algebra of symmetries of the system. We treat a family of 2nd order degenerate systems: oscillator analogies on Darboux, nonzero constant curvature, and flat spaces, related to one another via contractions, and obeying Kepler's laws. Then we treat two 2nd order nondegenerate systems, an analogy of a caged Coulomb problem on the 2-sphere and its contraction to a Euclidean space caged Coulomb problem. In all cases the symmetry algebra structure provides detailed information about the trajectories. An interesting example is the occurrence of ''metronome orbits'', trajectories confined to an arc rather than a loop, which are indicated clearly from the structure equations but might be overlooked using more traditional methods. We also treat the Post-Winternitz system, an example of a classical 4th order superintegrable system that cannot be solved using separation of variables. Finally we treat a superintegrable system, related to the addition theorem for elliptic functions, whose constants of the motion are only rational in the momenta, a system of special interest because its constants of the motion generate a closed polynomial algebra. This paper contains many new results but we have tried to present most of the materials in a fashion that is easily accessible to nonexperts, in order to provide entr\'ee to superintegrablity theory.