The Anisotropic Schwarzschild-Type Problem, Main Features

Abstract

The two-body problem associated to an anisotropic Schwarzschild-type field is being tackled. Both the motion equations and the energy integral are regularized via McGehee-type transformations. The regular vector field exhibits nice symmetries that form a commutative group endowed with an idempotent structure. The physically fictitious flows on the collision and infinity manifolds, as well as the local flows in the neighbourhood of these manifolds, are fully described. Homothetic, spiral, and oscillatory orbits are pointed out. Some features of the global flow are depicted for all possible levels of energy. For the negative-energy case, few things have been done. The positive-energy global flow does not have zero-velocity curves; every orbit is of the type ejection – escape or capture – collision. In the zero-energy case, the collision and infinity manifolds have a very similar structure. The existence of eight trajectories that connect the equilibria on these manifolds is proved. The projectability of the zero-energy global flow completes the full understanding of the problem in this case.