Saari's conjecture in celestial mechanics

Abstract

In 1969, D. Saari conjectured that the only solutions of the Newtonian n-body problem that have constant moment of inertia are relative equilibria. For n = 3, there is a computer assisted proof of this conjecture given by R. Moeckel in 2005, [10]. The collinear case was solved the same year by F. Diacu, E. Pérez-Chavela, and M. Santoprete, [4]. All the other cases are open. Denoting by U the potential energy, Saari's homographic conjecture states that if along an orbit of the n-body problem IU2 is constant, then the orbit is a homographic solution, i.e. a solution whose initial configuration remains similar to itself. In this paper, we discuss both conjectures and survey the proof of the latter for a large set of initial data. This survey follows our previous paper on this subject, [5]. © 2008 American Institute of Physics.