On the C 8/3-regularisation of simultaneous binary collisions in the planar four-body problem

Abstract

The dynamics of the four-body problem allows for two binary collisions to occur simultaneously. It is known that in the collinear four-body problem this simultaneous binary collision (SBC) can be block-regularised, but that the resulting block map is only C 8/3 differentiable. In this paper, it is proved that the C 8/3 differentiability persists for the SBC in the planar four-body problem. The proof uses several geometric tools, namely, blow-up, normal forms, dynamics near normally hyperbolic manifolds of equilibrium points, and Dulac maps.