Genuine Equilibria of Three Body Coulomb Configurations

The problem analyzed is the classical non-regularized Hamiltonian formulation of a restricted three body problem under the influence of Coulomb-interactions. Completing the large literature on Helium-like systems we will consider the motions around a fixed positive point charge of (i) one negative and one positive point charge as well as the “classical” issue of (ii) two negative point charges. Thereby, all our considerations deal with arbitrary positive and negative charge values of the mass particles. Here, we give necessary and sufficient conditions for the existence of genuine equilibria in such systems—recall that there are none in the usual classical Helium-atom—and analyze their linearized stability. The thus obtained insights allow us to study the dynamics near such genuine equilibria.