Central configurations for the planar Newtonian Four-Body problem

TL;DR

This paper analyzes and computes central configurations in the planar Newtonian four-body problem, introducing new numerical methods and exploring connections with three-body configurations and special cases like coorbital problems.

Contribution

It develops a novel numerical algorithm for four-body central configurations and clarifies the connection between three- and four-body configurations as one mass approaches zero.

Findings

New properties of symmetric and non-symmetric configurations identified

Explicit connection between three- and four-body configurations established

Numerical algorithm successfully constructs general four-body configurations

Abstract

The plane case of central configurations with four different masses is analyzed theoretically and is computed numerically. We follow Dziobek's approach to four body central configurations with a direct implicit method of our own in which the fundamental quantities are the quotient of the directed area divided by the corresponding mass and a new simple numerical algorithm is developed to construct general four body central configurations. This tool is applied to obtain new properties of the symmetric and non-symmetric central configurations. The explicit continuous connection between three body and four body central configurations where one of the four masses approaches zero is clarified. Some cases of coorbital 1+3 problems are also considered.