The problem of infinite Spin for parabolic and collision solutions in the planar $n$-body problem

TL;DR

This paper investigates the infinite spin problem in the planar n-body problem, extending previous results to parabolic and collision solutions, and demonstrating that infinite spin cannot occur under certain conditions.

Contribution

It generalizes Moeckel and Montgomery's results to include parabolic and partial collision solutions, providing new insights into the infinite spin problem.

Findings

Infinite spin does not occur for complete and partially parabolic solutions under certain conditions.

The results extend to partial collision solutions, broadening the understanding of the problem.

The approach builds on and generalizes previous work by Moeckel and Montgomery.

Abstract

In the planar $n$-body problem, the problem of infinite spin occurs for both parabolic and collision solutions. Recently Moeckel and Montgomery \cite{MM25} showed that there is no infinite spin for total collision solutions, when the reduced and normalized configuration converges to an isolated central configuration. Following their approach, we show it can not happen for both complete and partially parabolic solutions, under similar conditions. Our approach also allows us to generalize Moeckel and Montgomery's result to partial collision solutions under similar conditions.