Convex Four Body Central Configurations with Some Equal Masses

TL;DR

This paper proves the uniqueness and symmetry properties of certain convex four-body central configurations in the planar Newtonian problem, focusing on configurations with specific mass arrangements.

Contribution

It establishes the uniqueness and geometric symmetry of convex non-collinear central configurations with some equal masses in the four-body problem.

Findings

Unique convex non-collinear configuration with two opposite equal masses

Configuration with at most one larger mass among the remaining

Existence of a unique convex configuration when opposite masses are equal

Abstract

We prove that there is a unique convex non-collinear central configuration of the planar Newtonian four-body problem when two equal masses are located at opposite vertices of a quadrilateral and, at most, only one of the remaining masses is larger than the equal masses. Such central configuration posses a symmetry line and it is a kite shaped quadrilateral. We also show that there is exactly one convex non-collinear central configuration when the opposite masses are equal. Such central configuration also posses a symmetry line and it is a rhombus.