Symmetric planar central configurations of five bodies: Euler plus two

Abstract

Abstract. We study planar central configurations of the five-body problem where three of the bodies are collinear, forming an Euler central configuration of the three-body problem, and the two other bodies together with the collinear configuration are in the same plane. The problem considered here assumes certain symmetries. From the three bodies in the collinear configuration, the two bodies at the extremities have equal masses and the third one is at the middle point between the two. The fourth and fifth bodies are placed in a symmetric way: either with respect to the line containing the three bodies, or with respect to the middle body in the collinear configuration, or with respect to the perpendicular bisector of the segment containing the three bodies. The possible stacked five-body central configurations satisfying these types of symmetries are: a rhombus with four masses at the vertices and a fifth mass in the center, and a trapezoid with four masses at the vertices and a fifth mass at the midpoint of one of the parallel sides.

New stacked central configurations for the planar 5-body problem

“…Also considering the particles endowed with masses and charges, Alfaro and Perez-Chavela (2002) proved the existence of a continuum of central configurations in a particular 4-body problem. Other recent papers on central configurations are due to Corbera, Cors and Llibre (2010), , Gidea and Llibre (2010), Piña and Lonngi (2010), ... Recently the first and second author of this paper gave new examples of stacked central configurations of the 5-bodies which, as the ones studied by Hampton (2005), have three bodies in the vertices of an equilateral triangle, but the other two are on the perpendicular bisector (Llibre and Mello 2008).…”

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].…”

Generic Finiteness for a Class of Symmetric Planar Central Configurations of the Six-Body Problem and the Six-Vortex Problem

“…Albouy, Fu, and Su provided the necessary and sufficient condition for a planar convex four-body central configuration be symmetric with respect to one of its diagonals [2]. Problems involving existence or enumeration of symmetric central configurations satisfying some geometrical constraints were considered for many researchers (See for instance [4], [6], [10], [16], [22], and [29]). Montaldi proved that there is a central configuration for every choice of a symmetry type and symmetric choice of mass [20].…”