The Co-circular Central Configurations of the $$5$$ 5 -Body Problem

Abstract

Abstract. Chenciner in 2001 asked: Is the regular n-gon with equal masses the unique central configuration such that all the bodies lie on a circle, and the center of mass coincides with the center of the circle? This question has a positive answer for n = 3. Hampton in 2003 proved that also this question has a positive answer for n = 4. Here we provide a positive answer for n = 5.

On the equilateral pentagonal central configurations

“…Other studies have focused on restricting the problem to a particular shape, in [15] it was proved that the unique co-circular central configuration in the planar 5-body problem is the regular 5-gon with equal masses, while in [14] was proved the existence of three families of planar central configurations where three bodies are at the vertices of an equilateral triangle and the other two bodies are on a perpendicular bisector. Later on in [21] was studied the central configuration in a symmetric 5-body problem with three masses on an axis of symmetry and two other masses outside this axis, placed in symmetric positions.…”

On Centered Co-circular Central Configurations of the n-Body Problem

“…Hampton in [15] proved that the question has positive answer for n = 4. For n = 5 the problem has been studied in [21]. Many other authors have been interested in proving the positive answer of the question for n > 5.…”

Uniqueness results for co-circular central configurations for power-law potentials

“…, θ n + β) (taken mod 2π) for any β. This is due to the SO(2) symmetry of equations (1) and (10). We can specify a unique member of this one-parameter family of critical points by requiring θ 1 = 0.…”

“…The first to provide an answer to Chenciner's question was Hampton, who proved that the only four-body co-circular central configuration with center of mass coinciding with the center of the circle is the square with equal masses [8]. Llibre and Valls have announced that the regular pentagon (again with equal masses) is the only co-circular central configuration with this special property for n = 5 [10]. This question is listed as Problem 12 in a collection of important open problems in celestial mechanics compiled by Albouy, Cabral and Santos [1].…”