Symmetric Central Configurations in the Concave 4-Body Problem with Two Pairs of Equal Masses

TL;DR

This paper proves the existence and classification of symmetric concave four-body configurations with two pairs of equal masses, revealing bifurcation phenomena and providing a comprehensive bifurcation diagram for these configurations.

Contribution

It introduces a rigorous computer-assisted analytical method to classify symmetric concave 4-body configurations with two pairs of equal masses and analyzes their bifurcation structure.

Findings

Number of configurations is 0, 1, or 2 for any mass ratio.

Identifies a fold bifurcation point for a specific configuration.

Provides a complete bifurcation diagram for symmetric and asymmetric cases.

Abstract

We establish the existence of a single-parameter family of the concave kite central configurations in the 4-body problem with two pairs of equal masses. In such configurations, one pair of the masses must lie on the base of an isosceles triangle, and the other pair on its symmetric axis with one mass positioned inside the triangle formed by the other three. Using a rigorous computer-assisted analytical approach, we prove that for any non-negative mass ratio, the number of such configurations is either zero, one, or two, which can be viewed as a complete classification of this particular family. Furthermore, we show that the unique configuration corresponding to a specific mass ratio is a fold-type bifurcation point within the reduced subspace. We also give a clear and complete bifurcation picture for both symmetric and asymmetric cases of this concave type in the whole planar 4-body configuration space.