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

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 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, thereby providing a complete classification of this 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 across the entire planar 4-body configuration space.

21 pages, 1 table, 8 figures

References in corpus (8)

• Symmetry of Planar Four-Body Convex Central Configurations
• Convex Four Body Central Configurations with Some Equal Masses
• Central Configurations of the Five-Body Problem with Equal Masses
• Central configurations in planar $n$-body problem with equal masses for $n=5,6,7$
• On the uniqueness of trapezoidal four-body central configurations
• Bifurcations of Central Configurations in the Four-Body Problem with some equal masses
• Classifying Four-Body Convex Central Configurations
• On the Uniqueness of Co-circular Four Body Central Configurations