"Twist Vectors" for Central Configuration Equations

TL;DR

This paper introduces a novel coordinate system on the tangent space to simplify calculations related to central configurations and relative equilibria in the planar N-body problem, demonstrating its utility through four-body configuration problems.

Contribution

A new coordinate system on the tangent space is proposed to streamline calculations in the N-body problem with homogeneous potentials, including Newtonian gravity.

Findings

Simplified calculations for central configurations using the new coordinates.

Application of coordinates to four-body central configuration problems.

Demonstrated utility in specific N-body problem scenarios.

Abstract

A new coordinate system on the tangent space to planar configurations is introduced to simplify some calculations on central configurations and relative equilibria in the $N$-body problem with a homogeneous potential, which includes the case of Newtonian gravity. These coordinates are applied to some problems on four-body central configurations to illustrate their utility.