Action-angle variables for dihedral systems on the circle

A nonrelativistic particle on a circle and subject to a cos−2(kφ)cos−2(kφ) potential is related to the two-dimensional (dihedral) Coxeter system I2(k)I2(k), for k∈Nk∈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 A2A2, BC2BC2 and G2G2 three-particle rational Calogero models on RR, which we also analyze.