Degeneracy of Planar 4-body Kite Central Configurations: Restricted and Full Space Perspective

Abstract

Based on our previously proposed direct approach to central configurations in the full configuration space, we study the degeneracy of planar four-body kite central configurations with arbitrary positive masses both in kite symmetry restricted subspace and in full configuration space. We derive a factorization of the determinant of the Jacobian that substantially simplifies the subsequent analysis. The methods used to study the singularity of the Jacobian include rigorous interval arithmetic, the Krawczyk operator, a quantitative implicit function theorem, and computer-assisted proofs. Both convex and concave kite central configurations are considered. In the convex case, we prove the nondegeneracy in both the restricted subspace and the full configuration space. In the concave case, there is a unique smooth degeneracy curve in the restricted subspace; an additional degeneracy curve emerges in the full space setting, which is hidden in the subspace of kite configurations and makes it unavoidable to study the degeneracy in the full configuration space. We also establish several properties of these degeneracy curves. As an illustration to this complete characterization, we exam the well-known central configuration of equilateral triangle with equal masses at the vertices and a fourth mass at the center, which reveals additional features of degeneracy captured by our full-space framework.