Saari's conjecture is true for generic vector fields

Tanya Schmah*, Cristina Stoica

*Corresponding author for this work

    Research output: Contribution to journalArticle

    11 Citations (Scopus)

    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 languageEnglish
    Pages (from-to)4429-4448
    Number of pages20
    JournalTransactions of the American Mathematical Society
    Volume359
    Issue number9
    DOIs
    Publication statusPublished - 2007

    Keywords

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

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