Collective branch regularization of simultaneous binary collisions in the 3D N-body problem

Abstract

In this work we study simultaneous binary collision (SBC) singularities of M⩽N∕2 binaries in the three dimensional classical gravitational N-body problem. We show the following: (1) In the generalized Kustaanheimo–Stiefel variables, the totality of SBC orbits, the totality of simultaneous binary collisions (SBE) orbits, and the collision singularity itself together form a real analytic submanifold which we call the collision-ejection manifold. (2) We use the collision-ejection manifold to show geometrically, without writing down any power series, that SBC solutions can be collectively analytically continued. That is, all SBC orbits, not just a single orbit, can be written as a convergent power series in s=t1∕3 with coefficients that depend real analytically on initial conditions that lie in a real analytic submanifold. (3) There are two important ingredients in our work. (i) We use the intrinsic energies and properly rescaled intrinsic angular momenta of the binaries as variables in order to reduce the order of the singularity and to parametrize (distinguish between different) collision orbits that constitute the stable manifolds of the rest points that appear on the collision manifold in the McGehee coordinates. (ii) We use what we call the Kustaanheimo–Stiefel-projective transformation near a SBC singularity to resolve the singularity and isolate collision and ejection orbits from nearby near-collision and near-ejection orbits. We will see that quaternionic multiplication and quaternionic projective spaces are not suitable.