The Poincaré compactification of the MIC-Kepler problem with positive energies

Abstract

The Poincar? compactification and the symplectic reduction methods are first reviewed and then used to study the behaviour at infinity of the MIC (McIntosh-Cisneros)-Kepler problem at positive energies. The hyperbolic orbits leave the unstable equilibrium point set at infinity and tend eventually to the stable equilibrium point set at infinity. Both of these equilibrium point sets are diffeomorphic with S2, the unit sphere in R3. The hyperbolic orbits determine a map of the unstable equilibrium point set to the stable equilibrium point set in such a manner that the initial point (or the limit point as t?-∞) of an orbit is mapped to its final point (or the limit point as t?∞). This map is found explicitly as a rotation matrix which depends on the energy and the angular momentum of the orbits.