Superintegrable Systems on 3 Dimensional Conformally Flat Spaces

TL;DR

This paper constructs and analyzes a broad class of superintegrable Hamiltonian systems on 3D conformally flat spaces, using conformal algebra to find integrals and reducing systems to 2D for detailed study.

Contribution

It introduces a method to generate superintegrable systems on 3D conformally flat spaces using conformal algebra and provides universal reductions to 2D systems.

Findings

Constructed superintegrable systems with quadratic integrals.

Derived the complete Poisson algebra of first integrals.

Reduced 3D systems to 2D Darboux-Koenigs type systems.

Abstract

We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional {\em quadratic} first integrals, thus constructing a large class of superintegrable systems and the complete Poisson algebra of first integrals. We then use the isometries to reduce our systems to 2 degrees of freedom. For each isometry algebra we give a {\em universal} reduction of the corresponding general Hamiltonian. The superintegrable specialisations reduce, in this way, to systems of Darboux-Koenigs type, whose integrals are reductions of those of the 3 dimensional system.