Central Configurations in the Trapezoidal Four-Body Problems

TL;DR

This paper analyzes the central configurations of four bodies arranged in an isosceles trapezoid, deriving conditions for their existence and identifying regions in phase space where such configurations are possible.

Contribution

It provides analytical and numerical characterization of central configurations in the trapezoidal four-body problem, highlighting regions where positive masses can form such configurations.

Findings

Identified regions in phase space allowing central configurations with positive masses.

Proved the non-existence of central configurations outside these regions.

Derived analytical expressions for configurations in the trapezoidal four-body setup.

Abstract

In this paper we discuss the central configurations of the Trapezoidal four-body Problem. We consider four point masses on the vertices of an isosceles trapezoid with two equal masses $m_1=m_4$ at positions $(\mp 0.5, r_B)$ and $m_2=m_3$ at positions $(\mp \alpha/2, r_A)$. We derive, both analytically and numerically, regions of central configurations in the phase space where it is possible to choose positive masses. It is also shown that in the compliment of these regions no central configurations are possible.