No Infinite Spin for Planar Total Collision

TL;DR

This paper proves that in the planar n-body problem, total collision solutions cannot exhibit infinite spin and converge to a circle of central configurations if that circle is isolated, using advanced mathematical tools.

Contribution

It establishes a new result ruling out infinite spin convergence to isolated circles of central configurations in the planar n-body problem.

Findings

Infinite spin does not occur for total collision solutions under certain conditions.

Total collision solutions converge to a specific point on the circle of central configurations.

The proof combines the center manifold theorem with the Lojasiewicz gradient inequality.

Abstract

The infinite spin problem concerns the rotational behavior of total collision orbits in the $n$-body problem. It has long been known that when a solution tends to total collision then its normalized configuration curve must converge to the set of normalized central configurations. In the planar n-body problem every normalized configuration determines a circle of rotationally equivalent normalized configurations and, in particular, there are circles of normalized central configurations. It's conceivable that by means of an infinite spin, a total collision solution could converge to such a circle instead of to a particular point on it. Here we prove that this is not possible, at least if the limiting circle of central configurations is isolated from other circles of central configurations. (It is believed that all central configurations are isolated, but this is not known in general.) Our proof relies on combining the center manifold theorem with the Lojasiewicz gradient inequality.