On a Class of Central Configurations in the Planar $${\varvec{3n}}$$ 3 n -Body Problem

Abstract

The existence of a central configuration of 2n bodies located on two concentric regular n-gons with the polygons which are homotetic or similar with an angle equal to π n and the masses on the same polygon, are equal, has proved by Elmabsout (C R Acad Sci 312(5): [467][468][469][470][471][472] 1991). Moreover, the existence of a planar central configuration which consists of 3n bodies, also situated on two regular polygons, the interior n-gon with equal masses and the exterior 2n-gon with masses on the 2n-gon alternating, has shown by author. Following Smale (Invent Math 11:45-64, 1970), we reduce this problem to one, concerning the critical points of some effective-type potential. Using computer assisted methods of proof we show the existence of ten classes of such critical points which corresponds to ten classes of central configurations in the planar six-body problem.

On central configurations of the κn-body problem

“…For example, Llibre and Mello [5] show the existence of some (3, 3)-crown and also some (4, 2)-crown. Siluszyk [10] shows numerically the existence of some (3, 2)-crowns where a 2 = a 3 . In Figure 1 we show two examples of crowns of three rings (with a numerical accuracy up to 10 −10 ).…”