New stacked central configurations for the planar 5-body problem

Abstract

Abstract. A stacked central configuration in the n-body problem is one that has a proper subset of the n-bodies forming a central configuration. In this paper we study the case where three bodies with masses m1, m2, m3 (bodies 1, 2, 3) form an equilateral central configuration, and the other two with masses m4, m5 are symmetric with respect to the mediatrix of the segment joining 1 and 2, and they are above the triangle generated by {1, 2, 3}. We show the existence and non-existence of this kind of stacked central configurations for the planar 5-body problem.

Central configurations of the five-body problem with equal masses

“…More recently, in 2005, Hampton [15] found new examples of (symmetric) central configurations in the planar five-body problem. Llibre and Mello [28] and Llibre, Mello, and Perez-Chavela [29] found some further central configurations using the same techniques employed by Hampton. Not much more is known in the planar five-body problem It has been known for a long time that, in the equal mass case, the regular pentagon and the square with one mass at each of its vertices and with the fifth mass at its center are central configurations [49].…”

A Special Family of Stacked Central Configurations: Lagrange Plus Euler in One

“…This concept was first introduced by Hampton in a seminal paper [9] by providing a family of central configurations in the planar five-body problem where if two masses are removed, the remaining three are at the vertices of an equilateral triangle. After that, several papers have shown the existence of other stacked central configurations in the planar five-body problem, see [3,5,7,11,12]. Besides planar configurations, stacked central configuration have also been found in the spatial case, see [10,14,15,19] or in the general n-body problem, see [6,20,21].…”

Stacked Central Configurations for the Spatial Nine‐Body Problem

“…The authors [7] find new classes of central configurations of the 5-body problem which are the ones studied by Hampton [6] having three bodies in the vertices of an equilateral triangle, but the other two, instead of being located symmetrically with respect to a perpendicular bisector, are on the perpendicular bisector. The stacked central configurations studied by Hampton [6] were completed by Llibre et al [8] (see also [9]).…”