Simultaneous Binary Collisions in the Collinear N-Body Problem

Abstract

1. We show that in the collinear N-body problem any simultaneous binary collision singularity (SBC) of L pairs of particles, with no other collisions, is C 1 block-regularizable, that is, we show that near such an orbit there is a C 1 diffeomorphism from collision and near collision orbits to ejection and near ejection orbits. 2. We also show that collision and ejection orbits form a real analytic submanifold in certain spatial variables that behaves like t 1 3 . 3. Finally we introduce a new method for blowing up the SBC singularity and replace it by a collision manifold in such a way that different SBC orbits correspond to different rest points on the collision manifold. Moreover, each collision restpoint is connected to the corresponding ejection restpoint by the flow on the collision manifold.