Near-horizon dynamics of particle in extreme Reissner-Nordstr\"om and Cl\'ement-Gal'tsov black hole backgrounds: action-angle variables

We analyze the periodic motion in the conformal mechanics describing the particles moving near the horizon of extreme Reissner-Nordstr\"om and axion-dilaton (Cl\'ement-Gal'tsov) black holes. For this purpose we extract the (two-dimensional) compact ("angular") parts of these systems and construct their action-angle variables. In the first case we get the well-known spherical Landau problem, which possesses hidden $so(3)$ symmetry, while in the latter case the system does not have hidden constant of motion. In both cases we indicate the existence of "critical points", separating the regions of periodic motions with qualitatively different properties.