Action-angle variables for dihedral systems on the circle

A nonrelativistic particle on a circle and subject to a cos−2(kφ) potential is related to the two-dimensional (dihedral) Coxeter system I₂(k), for k∈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 A₂, BC₂ and G₂ three-particle rational Calogero models on R, which we also analyze.