The simplest non-collision solutions of the N-body problem are the “relative equilibria”, in which each body follows a circular orbit around the centre of mass and the shape formed by the N bodies is constant. It is easy to see that the moment of inertia of such a solution is constant. In 1970, D. Saari conjectured that the converse is also true for the planar Newtonian N-body problem: relative equilibria are the only constant-inertia solutions. A computer-assisted proof for the 3-body case was recently given by R. Moeckel, Trans. Amer. Math. Soc. (2005). We present a different kind of answer: proofs that several generalisations of Saari’s conjecture are generically true. Our main tool is jet transversality, including a new version suitable for the study of generic potential functions.
Schmah, Tanya and Stoica, Cristina, "Saari’s Conjecture is True for Generic Vector Fields" (2007). Mathematics Faculty Publications. 13.
This article was originally published in Transactions of the American Mathematical Society, 359(9): 4429-4448. (c) 2007 American Mathematical Society