Superintegrable systems on the two-dimensional sphere S2 and the hyperbolic plane H2

Abstract

The existence of superintegrable systems with n=2 degrees of freedom possessing three independent globally defined constants of motion which are quadratic in the velocities is studied on the two-dimensional sphere S2 and on the hyperbolic plane H2. The approach used is based on enforcing the conditions for the existence of two independent integrals (further than the energy). This is done in a way which allows us to discuss at once the cases of the sphere S2 and the hyperbolical plane H2, by considering the curvature κ as a parameter. Different superintegrable potentials are obtained as the solutions of certain systems of two κ-dependent second order partial differential equations. The Euclidean results are directly recovered for κ=0, and the superintegrable potentials on either the standard unit sphere (radius R=1) or the unit Lobachewski plane (“radius” R=1) appear as the particular values of the κ-dependent superintegrable potentials for the values κ=1 and κ=−1. Some new superintegrable potentials are found, both on S2 and H2. The correspondence between superintegrable systems in spaces of zero and nonzero curvature is discussed.