Le problème des n corps et les distances mutuelles

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].…”

Saari’s conjecture for the collinear $n$-body problem

“…The first part of the proof follows immediately from Theorem 1 and Proposition 2.14 in [1], which shows that if the potential is homogeneous of degree α = −2, then the homographic solutions are homographic with central configurations. The second part follows from Corollary 1.…”

“…Therefore the solutions are homographic (as proved in [1]) and since U is homogeneous of degree α = −2 ([1], Proposition 2.14), they are either solutions with central configurations or rigid motions. In the latter case the moment of inertia I is constant and we can show as in Corollary 1 that the solutions are relative equilibria.…”

Finiteness of spatial central configurations in the five-body problem

“…Recently Alain Albouy and Vadim Kaloshin have proved that the planar five-body central configurations are finite apart from some explicitly given special cases (Albouy and Kaloshin 2010).…”