Blow up of Total Collision in the Tetrahedral Non-Rotating Four Body Problem

Abstract

In this paper we introduce the tetrahedral four body problem where the particles form a tetrahedral configuration of variable size without rotation. In order for this configuration to be feasible, masses are equal pairwise with a parameter μ, measuring their ratio. We first perform the blow up of total collision leading to an invariant four-dimensional invariant manifold as boundary of the fixed energy levels. The flow is described at the linear level; We find exactly three central configurations, two symmetric ones in the form of a regular tetrahedron for any value of the ratio of masses and a planar rhomboidal configuration whose shape depends ond the masses. We then compute the dimensions of the invariant manifolds. Singularities are of the type of single binary collisions, simultaneous binary collisions and total collision. While single binary collisions are regularized explicitly, simultaneous binary collisions represent the main obstruction to complete the global flow on the total collision manifold.