A family of stacked central configurations in the planar five-body problem

Abstract

The final publication is available at link.springer.com via http://dx.doi.org/10.1007/s10569-017-9782-8We study planar central configurations of the five-body problem where three bodies, (Formula presented.) and (Formula presented.), are collinear and ordered from left to right, while the other two, (Formula presented.) and (Formula presented.), are placed symmetrically with respect to the line containing the three collinear bodies. We prove that when the collinear bodies form an Euler central configuration of the three-body problem with (Formula presented.), there exists a new family, missed by Gidea and Llibre (Celest Mech Dyn Astron 106:89–107, 2010), of stacked five-body central configuration where the segments (Formula presented.) and (Formula presented.) do not intersect.Peer ReviewedPostprint (author's final draft

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].…”

On the equilateral pentagonal central configurations

“…Hence as in (5) we have that m 4 = m 3 . Therefore we do not need to consider λ = −1/(8x 3 3 ), and in what follows we consider that m 4 = m 3 , such that, e 6 − e 7 = 0.…”

“…This concept was introduced by Hampton [12], who was arguably the first to find stacked central configurations in the 5-body problem, where two bodies can be removed and the remaining three bodies are already in a central configuration. After, several papers have been published showing the existence of other stacked central configurations in the planar 5-body problem; see, among others, [5], [6,8,11,16,14].…”

“…Let us prove for instance that the value y 5 = y 5 (5) corresponding to t = t 5 ≈ 0.1871837, together with x 3 = x + 3 (5) = Ψ + (y 5 (5)) does not provide a solution of our system. From the Sturm sequence we get that 1871 10000 < t 5 < 1872 10000 .…”

Planar N-body central configurations with a homogeneous potential

“…In the Newtonian case, the earliest systematic attempt was by Williams [139], who attempted to extend the approach that MacMillan and Bartky [86] pioneered for N = 4 on convex configurations for general (not necessarily equal) masses; the work of Williams was later improved by Chen and Hsiao [29]. There are limited results on configurations with particular symmetries [56,81,57,51,83,32]. Albouy and Kaloshin proved that there are finitely many fivebody central configurations in the Newtonian case, apart from some exceptional cases determined by polynomials in the mass parameters for which the result is unknown [10].…”