On the singularities of the curved n-body problem

TL;DR

This paper investigates the nature of singularities in the n-body problem within curved spaces, extending classical results and introducing new methods for analyzing collision solutions in positively curved spaces.

Contribution

It generalizes classical singularity results to curved spaces and introduces a novel approach using orthogonal projections for positive curvature cases.

Findings

Extended singularity analysis to spaces of constant curvature

Established correspondence between collision solutions and projections in positive curvature

Provided new methods for studying the n-body problem in curved geometries

Abstract

We study singularities of the n-body problem in spaces of constant curvature and generalize certain results due to Painleve, Weierstrass, and Sundman. For positive curvature, some of our proofs use the correspondence between total collision solutions of the original system and their orthogonal projection--a property that offers a new method of approaching the problem in this particular case.