Collision]Ejection Manifold and Collective Analytic Continuation of Simultaneous Binary Collisions in the Planar N-Body Problem*

We study simultaneous binary collision SBC.singularities of L binaries in the planar N-body problem, 2LFN. 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. Let Rc Re. be the collection of SBC SBE. orbits in phase space, i.e,. in the original variables. Then, both Rcand Reare 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. Let C;Rc E;Re. be a submanifold of initial conditions that end start. in a simultaneous binary collision ejection.singularity. We show that C E.can be chosen to be a real analytic submanifold of codimension 1 in Rc Re., and that the correspondence in item 3. above defines a real analytic section mapping from C to E. 5. Collision and ejection orbits can be collectively analytically continued, i.e., each collision]ejection orbit can be written as a convergent power series in t1r3, with coefficients that depend real analytically on the initial conditions in C. 6. SBC orbits of J-L binaries do not accumulate on any SBC orbit of L binaries