On the uniqueness of trapezoidal four-body central configurations

Abstract

We study central configurations of the Newtonian four-body problem that form a trapezoid. Using a topological argument we prove that there is at most one trapezoidal central configuration for each cyclic ordering of the masses.

Some Polynomial Conditions of Cyclic Quadrilaterals, Tilted Kites and Other Quadrilaterals

“…The polynomials studied by Pech [9] have been useful in studying cyclic central configurations of four bodies in Celestial Mechanics [2,11]. Similar polynomials, related to another type of quadrilaterals, namely trapezoids, have also been found to be useful in studying central configurations of four bodies [10,12]. In light of this, finding additional polynomial conditions characterizing various configurations of four points is not only interesting from a geometry standpoint, but also from the perspective of potential applications to Celestial Mechanics.…”