Finiteness of central configurations of five bodies in the plane

Abstract

We prove there are finitely many isometry classes of planar central configurations (also called relative equilibria) in the Newtonian 5-body problem, except perhaps if the 5-tuple of positive masses belongs to a given codimension 2 subvariety of the mass space.

Bifurcation of Relative Equilibria of the (1+3)-Body Problem

“…In a computer assisted proof Hampton and Moeckel [21] proved the finiteness of the number of central configurations for N = 4 and any choice of the masses. This result was obtained analytically by Albouy and Kaloshin [5], who also extended this result to N = 5 for almost all choice of the masses. The question about the finiteness of the number of classes of central configurations remains open for N > 5.…”

A note on the Dziobek central configurations

“…For a classical background, see the sections on central configurations in the books of Wintner [17] and Hagihara [6]. For a modern background see, for instance, the papers of Albouy and Chenciner [2], Albouy and Kaloshin [3], Hampton and Moeckel [7], Moeckel [9], Palmore [13], Saari [14], Schmidt [15], Xia [18], ... One of the reasons why central configurations are important is that they allow to obtain the unique explicit solutions in function of the time of the n-body problem known until now, the homographic solutions for which the ratios of the mutual distances between the bodies remain constant. They are also important because the total collision or the total parabolic escape at infinity in the n-body problem is asymptotic to central configurations, see for more details Dziobek [5] and [14].…”

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

“…For a classical background, see the sections on central configurations in the books of Wintner [22] and Hagihara [9]. For a modern background see, for instance, the papers of Albouy and Chenciner [2], Albouy and Kaloshin [3], Hampton and Moeckel [11], Moeckel [14], Palmore [17], Saari [18], Schmidt [19], Xia [23], ... One of the reasons why central configurations are important is that they allow to obtain the unique explicit solutions in function of the time of the n-body problem known until now, the homographic solutions for which the ratios of the mutual distances between the bodies remain constant. They are also important because the total collision or the total parabolic escape at infinity in the n-body problem is asymptotic to central configurations, see for more details Saari [18].…”

“…Recently, Hampton and Moeckel [11] proved that for any choice of four masses there exist a finite number of classes of central configurations. For five or more masses this result is unproved, but recently an important contribution to the case of five masses has been made by Albouy and Kaloshin [3].…”