The Rhomboidal Four Body Problem. Global Flow on the Total Collision Manifold

Abstract

In this work, we have considered a particular case of the planar four-body problem, obtained when the masses form a rhomboidal configuration. If we take the ratio of the masses α as a parameter, this problem is a one parameter family of non-integrable Hamiltonian systems with two degrees of freedom. We use the blow up method introduced by McGhee to study total collision. This singularity is replaced by an invariant two-dimensional manifold, called the total collision manifold. Using numerical methods we prove first that there are two equilibrium points for the flow on this manifold, and second, that there are only two values of a for which there is a connection between the invariant submanifolds of the equilibrium points. For these values of a the problem is not regularizable.