A Kaluza-Klein Reduction of Super-integrable Systems

TL;DR

This paper demonstrates a reverse Kaluza-Klein reduction technique to derive lower-dimensional super-integrable systems from higher-dimensional ones, revealing how isometries transform under this process.

Contribution

It introduces a novel method to reduce super-integrable systems via Kaluza-Klein construction, highlighting the role of isometries and quadratic momentum expressions in the reduction process.

Findings

Successfully reduced 3D super-integrable systems to 2D cases.

Showed isometries of reduced systems derive from higher dimensions.

Illustrated the method with two explicit examples.

Abstract

Given a super-integrable system in $n$ degrees of freedom, possessing an integral which is linear in momenta, we use the "Kaluza-Klein construction" in reverse to reduce to a lower dimensional super-integrable system. We give two examples of a reduction from 3 to 2 dimensions. The constant curvature metric (associated with the kinetic energy) is the same in both cases, but with two different super-integrable extensions. For these, we use different elements of the algebra of isometries of the kinetic energy to reduce to $2-$dimensions. Remarkably, the isometries of the reduced space can be derived from those of the $3-$dimensional space, even though it requires the use of {\em quadratic} expressions in momenta.