Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold

TL;DR

This paper classifies all two-dimensional superintegrable systems with quadratic integrals of motion on manifolds into six fundamental classes using Poisson algebra, providing explicit formulas and unifying known systems.

Contribution

It introduces a unified classification scheme for superintegrable systems with quadratic integrals on 2D manifolds, encompassing all known cases.

Findings

Six fundamental classes of superintegrable systems identified

Explicit formulas for integrals of motion derived

All known systems are special cases of the six classes

Abstract

In this paper we prove that the two dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the Poisson algebra of the integrals of motion. There are six general fundamental classes of superintegrable systems. Analytic formulas for the involved integrals are calculated in all the cases. All the known superintegrable systems are classified as special cases of these six general classes.