Seminar

Of central importance in the N-body problem is the fact that isolated binary collisions can be regularized :  that a singular change of space and time variables (first written down by Levi-Civita) allows trajectories to pass analytically through binary collisions unscathed. The resulting flow is smooth with respect to initial conditions. Curiously, we are not so lucky with simultaneous binary collisions. In a landmark paper, Martinez and Simo gave strong evidence that the best one can hope is 8/3 differentiability of the flow in a neighborhood of simultaneous binary collisions in the 4 body problem. In this talk we follow Easton by linking regularizability to the behaviour of the flow in Conley isolating blocks around the collisions. We show the 8/3 is produced from the first resonant monomial with nonzero coefficient near a degenerate singularity formed when the two binaries are separately Levi-Civita regularized. To show this, we blow-up the singularity and study the flow near the resulting two, 3:1 resonant, normally hyperbolic manifolds connected by heteroclinics. A lengthy normal form computation confirms the conjecture.