Quasi-homogeneous two-body problem

Abstract

The quasi-homogeneous two-body problem aims at studying the interaction between two point particles under a prescribed potential in the form of [Formula: see text], where A, B > 0 are constants and r is the mutual distance between two particles. Important examples include the Manev potential ( a = 1, b = 2) and the Schwarzschild potential ( a = 1, b = 3). It is well known that power two serves as a threshold value for the homogeneous potential: One is able to observe significant differences regarding the solution dynamics as the power of the homogeneous potential exceeds two from below. This phenomenon remains observable for quasi-homogeneous potentials. In this paper, we shall provide a complete characterization of the whole phase space of the quasi-homogeneous two-body problem in terms of global existence and singularity for all the possible b > a > 0. In particular, one is able to generalize the result of the Manev and Schwarzschild two-body problem to all the quasi-homogeneous potentials. Two techniques are presented in this paper: One is the variational method based on the energy, and the other is a direct computation of collision time based on the integrability of two-body systems.