#### Document Type

Article

#### Publication Date

9-2007

#### Department

Mathematics

#### Abstract

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.

#### Recommended Citation

Schmah, Tanya and Stoica, Cristina, "Saari’s Conjecture is True for Generic Vector Fields" (2007). *Mathematics Faculty Publications*. 13.

https://scholars.wlu.ca/math_faculty/13

## Comments

This article was originally published in

Transactions of the American Mathematical Society, 359(9): 4429-4448. (c) 2007 American Mathematical Society