Planar Central Configurations of Four Equal Masses: Classification and Morse Indices

Abstract

We study the planar central configurations of four equal positive masses under homogeneous interactions with force exponent $A>2$. For $2<A\le 3$, we prove that there are exactly four types of configurations up to similarity and relabeling: the square, the centered equilateral triangle, a unique nontrivial concave kite, and a unique collinear configuration. Combined with known results for $A>3$, this completes the classification for all $A>2$ and gives an affirmative answer to Problem 11 of Albouy, Cabral, and Santos. We also establish the nondegeneracy modulo rotation of these configurations and determine their Morse indices, confirming Hampton&#39;s Conjecture 2. Our proofs are based on analytic arguments and exact algebraic calculations.