Symmetry of Planar Four-Body Convex Central Configurations

TL;DR

This paper explores the symmetry properties of convex four-body central configurations in the plane, establishing a link between mass equality and geometric symmetry, and extends findings to five-body configurations.

Contribution

It proves a necessary and sufficient condition for symmetry in convex four-body configurations and extends the analysis to non-planar five-body configurations.

Findings

Convex four-body configurations are symmetric with respect to a diagonal if and only if the masses on the opposite diagonal are equal.

When the masses are unequal, the less massive particle is closer to the diagonal.

Results are extended to non-planar five-body central configurations.

Abstract

We study the relationship between the masses and the geometric properties of central configurations. We prove that in the planar four-body problem, a convex central configuration is symmetric with respect to one diagonal if and only if the masses of the two particles on the other diagonal are equal. If these two masses are unequal, then the less massive one is closer to the former diagonal. Finally, we extend these results to the case of non-planar central configurations of five particles.