No infinite spin for total collisions in the spatial N-body problem

Abstract

In the $n$-body problem, when bodies tend to a total collision, then its normalized shape curve converges to the set of normalized central configurations, which has $SO(3)$ symmetry in the planar case. This leaves a possibility that the normalized shape curve tends to the set obtained by rotations of some central configuration instead of a particular point on it. This is the \emph{infinite spin problem} which concerns the rotational behavior of total collision orbits in the $n$-body problem. We show that the infinite spin is not possible if the limiting shape is isolated from other connected components of the set of normalized central configurations. Our approach extends the method from recent work for total collision for the planar case by Moeckel and Montgomery. The main tool is a full reduction $\rm SO(3)$--symmetry in a context of vanishing angular momentum.