## 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.

Original language | English |
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Pages (from-to) | 4429-4448 |

Number of pages | 20 |

Journal | Transactions of the American Mathematical Society |

Volume | 359 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2007 |

## Keywords

- Jet transversality
- N-body problem
- Saari's conjecture