Dynamics of the the dihedral four-body problem

Consider four point particles with equal masses in the euclidean space, subject to the following symmetry constraint: at each instant they are symmetric with respect to the dihedral group D2, that is the group generated by two rotations of angle π around two orthogonal axes. Under a homogeneous potential of degree −α for 0<2, this is a subproblem of the four-body problem, in which all orbits have zero angular momentum and the configuration space is three-dimensional. In this paper we study the flow in McGehee coordinates on the collision manifold, and discuss the qualitative behavior of orbits which reach or come close to a total collision