Simultaneous binary collisions in the planar four body problem

Regina Mart\'{\i}nez, Departament de Matem\`atiques Universitat Aut\`onoma
   de Barcelona, Bellaterra, Barcelona, Spain

Carles Sim\'o,  Dept. de Matem\`atica Aplicada i An\`alisi, Univ. de
   Barcelona, Gran Via 585, 08007 Barcelona,Spain

Abstract

We consider simultaneous binary collisions in the general planar four
body problem. We prove they are regularizable in the sense of
continuity with respect initial conditions using a blow up of the
singularity. Furthermore, numerical evidence is given that the
differentiability of the regularization should be, in general,
less than $ C^{8/3} $. A simple example, the double isosceles four body 
problem, displays that kind of behaviour.
