The Hessian of Planar Central Configurations in Pair Space: Decomposition, Morse Index and Symmetry Reduction
We give a variational derivation of the central configuration equations in pair space, where the relative position vectors between pairs of bodies serve as the primary variables. Vector Lagrange multipliers enforce the linear triangle relations required for these pair vectors to be realizable in the plane. For arbitrary $N$, we show that the constrained Hessian decomposes as $H_{\mathcal C}=L^Δ+\widetilde L$, where the gap Laplacian $L^Δ$ is positive semidefinite and the transverse Laplacian $\widetilde L$ is signed and contains all possible negative directions. For non-collinear planar four-body central configurations, the signed part has rank two. This yields a new proof of the known bound that the Morse index is at most two. We then characterize positive definiteness and degeneracy of the Hessian by the eigenvalues of an explicit $2\times2$ matrix or, equivalently, by two generalized eigenvalues of a pencil of symmetric matrices. We illustrate the general criterion using the equal-mass square central configuration. For reflection-symmetric configurations, the generalized eigenvalue problem decomposes into two independent smaller problems. This reduction applies to kite and isosceles trapezoidal configurations. For the trapezoid, however, the reflection acts non-orthogonally on the pair coordinates, giving a different, non-orthogonal reduction. Applying the reflection-symmetry reduction to the rhombus family, we prove that the Hessian is positive definite modulo the rotational zero mode; equivalently, every rhombus central configuration is nondegenerate modulo rotations and has Morse index zero.