Collision–Ejection Manifold and Collective Analytic Continuation of Simultaneous Binary Collisions in the PlanarN-Body Problem

Abstract

We study simultaneous binary collision (SBC) singularities ofLbinaries in the planarN-body problem, 2L≤N. We introduce the generalized Levi-Civita transformation and follow it by a new transformation which we call the projective transformation near a SBC singularity. We use this transformation to show the following near a SBC singularity:(1) In the generalized Levi-Civita variables, near a SBC singularity, the collection of collision and ejection orbits together with the singularity form a real analytic submanifold, which we call the collision–ejection (CE) manifold.(2) LetRc(Re) be the collection of SBC (SBE) orbits in phase space, i.e., in the original variables. Then, bothRcandReare real analytic.(3) Each collision orbit corresponds to a unique ejection orbit. Together, they form a real analytic orbit in the generalized Levi-Civita variables, which we call a collision–ejection orbit.(4) LetC⊂Rc(E⊂Re) be a submanifold of initial conditions that end (start) in a simultaneous binary collision (ejection) singularity. We show thatC(E) can be chosen to be a real analytic submanifold of codimension 1 inRc(Re), and that the correspondence in item (3) above defines a real analytic section mapping fromCtoE.(5) Collision and ejection orbits can be collectively analytically continued, i.e., each collision–ejection orbit can be written as a convergent power series int1/3, with coefficients that depend real analytically on the initial conditions inC.(6) SBC orbits ofJ<Lbinaries do not accumulate on any SBC orbit ofLbinaries.(7) A single binary collision singularity in theN-body problem is real analytic block-regularizable.We also give the asymptotic behaviour of collision and ejection orbits.