On the uniqueness of the isosceles trapezoidal central configuration in the 4-body problem for power-law potentials

Abstract

In this article we study the planar 4-body problem under homogeneous power-law potentials where the interaction between the bodies is given by r −a , a 4/3 (the Newtonian case corresponding to a = 3 and the vortex problem corresponding to a = 2). We study convex central configurations assuming two pairs of positive equal masses located at two adjacent vertices of a convex quadrilateral. Under these assumptions we prove that the isosceles trapezoid is the unique central configuration for every a 4/3. For the case 0 < a < 4/3, which is more difficult to study analytically, we present some numerical considerations.

On the Uniqueness of Co-circular Four Body Central Configurations

“…The conjecture is also known to be true if all the masses are equal [1,2], if two pairs of masses are equal [25,5,13], and for the case of three small masses [9]. Some of these results also hold for homogeneous power-law potentials [5,14].…”

On the uniqueness of trapezoidal four-body central configurations

“…The central configurations of the trapezoidal four-body problem were studied in detail in [9,28]. The uniqueness of trapezoidal central configurations was recently proved for the particular case of two pairs of equal masses in the case of power-law potentials [14]. The main goal of this paper is to prove the following theorem: Theorem 1.…”