Four-body central configurations with one pair of opposite sides parallel

TL;DR

This paper investigates four-body central configurations with one pair of opposite sides parallel, deriving new constraints and properties, including symmetry and mass relations, specific to trapezoidal arrangements.

Contribution

It introduces a novel constraint for these configurations, proves mass ordering properties, and characterizes symmetric and isosceles trapezoid cases.

Findings

A partial mass ordering for trapezoidal configurations.

Symmetry and kite properties when opposite masses are equal.

Conditions under which the configuration is an isosceles trapezoid.

Abstract

We study four-body central configurations with one pair of opposite sides parallel. We use a novel constraint to write the central configuration equations in this special case, using distances as variables. We prove that, for a given ordering of the mutual distances, a trapezoidal central configuration must have a certain partial ordering of the masses. We also show that if opposite masses of a four-body trapezoidal central configuration are equal, then the configuration has a line of symmetry and it must be a kite. In contrast to the general four-body case, we show that if the two adjacent masses bounding the shortest side are equal, then the configuration must be an isosceles trapezoid, and the remaining two masses must also be equal.