Action-angle variables for dihedral systems on the circle

TL;DR

This paper constructs action-angle variables for dihedral systems on the circle, relates them to Coxeter systems, and explores their quantization and supersymmetric extensions, including connections to Calogero models.

Contribution

It introduces a method to derive action-angle variables for dihedral systems and establishes their equivalence to free particles, extending to supersymmetric and Calogero models.

Findings

Action-angle variables for dihedral systems are explicitly constructed.

Dihedral systems are shown to be locally equivalent to free particles on the circle.

Quantization and supersymmetric extensions of these systems are discussed.

Abstract

A nonrelativistic particle on a circle and subject to a cos^{-2}(k phi) potential is related to the two-dimensional (dihedral) Coxeter system I_2(k), for k in N. For such `dihedral systems' we construct the action-angle variables and establish a local equivalence with a free particle on the circle. We perform the quantization of these systems in the action-angle variables and discuss the supersymmetric extension of this procedure. By allowing radial motion one obtains related two-dimensional systems, including A_2, BC_2 and G_2 three-particle rational Calogero models on R, which we also analyze.