No infinite spin for partial collisions converging to isolated central configurations on the plane

TL;DR

This paper proves that in the planar n-body problem, partial collision orbits cannot exhibit infinite rotational spin if their limiting central configuration is isolated, extending previous total collision results.

Contribution

It establishes the non-existence of infinite spin for partial collisions with isolated limiting configurations, using an extension of the center manifold and shadowing methods.

Findings

Infinite spin does not occur for isolated limiting central configurations.

The approach combines center manifold theory, Łojasiewicz inequality, and shadowing near hyperbolic manifolds.

Results extend previous total collision analysis to partial collisions.

Abstract

In the $n$-body problem, when a~cluster of bodies tends to a collision, then its normalized shape curve converges to the set of normalized central configurations, which has $SO(2)$ symmetry in the planar case. This leaves a possibility that the normalized shape curve tends to the circle obtained by rotation 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. The question also makes sense for partial collision. We show that the infinite spin is not possible if the limiting circle is isolated from other connected components of the set of normalized central configurations. Our approach extends the method from recent work for total collision by Moeckel and Montgomery, which was based on a combination of the center manifold theorem with {\L}ojasiewicz inequality. To that we add a shadowing result for pseudo-orbits near normally hyperbolic manifold and careful estimates on the influence of other bodies on the cluster of colliding bodies.