Critical points of the integral map of the charged 3-body problem

TL;DR

This paper investigates the topology of integral manifolds in the charged three-body problem, focusing on ordinary critical points and their relation to central configurations, laying groundwork for understanding the system's global structure.

Contribution

It provides a detailed analysis of ordinary critical points and their connection to central configurations in the charged three-body problem, advancing the understanding of the system's integral manifolds.

Findings

Identification of ordinary critical points and their relation to central configurations

Analysis of how critical points affect the topology of integral manifolds

Foundation for future study of critical points at infinity and Hill regions

Abstract

This is the first in a series of three papers where we study the integral manifolds of the charged three-body problem. The integral manifolds are the fibers of the map of integrals. Their topological type may change at critical values of the map of integrals. Due to the non-compactness of the integral manifolds one has to take into account besides `ordinary' critical points also critical points at infinity. In the present paper we concentrate on `ordinary' critical points and in particular elucidate their connection to central configurations. In a second paper we will study critical points at infinity. The implications for the Hill regions, i.e. the projections of the integral manifolds to configuration space, are the subject of a third paper.