Non-integrability of charged three-body problem

Abstract: We consider the problem of $n$ points with positive masses interacting pairwise with forces inversely proportional to the distance between them. In particular, it is the classical gravitational, Coulomb or photo-gravitational $n$-body problem. Under this general form of interaction, we investigate the integrability problem of three bodies. We show that the system is not integrable except in one case when two among three interaction constants vanish. In our investigation, we used the Morales-Ramis theorem concerning the integrability of a natural Hamiltonian system with a homogeneous potential and its generalization.