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Saari's Conjecture is True for Generic Vector Fields

Tanya Schmah·Cristina Stoica
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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. 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.

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Tanya Schmah, Cristina Stoica. 2005-10-01. Saari's Conjecture is True for Generic Vector Fields. https://arxiv.org/abs/math/0510012

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