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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Saari's conjecture is true for generic vector fields


Authors: Tanya Schmah and Cristina Stoica
Journal: Trans. Amer. Math. Soc. 359 (2007), 4429-4448
MSC (2000): Primary 70F10; Secondary 37J05, 57N75
Published electronically: April 6, 2007
MathSciNet review: 2309192
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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, 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.


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Additional Information

Tanya Schmah
Affiliation: Department of Mathematics, Division of ICS, Macquarie University, NSW 2109, Australia
Email: schmah@maths.mq.edu.au

Cristina Stoica
Affiliation: Department of Mathematics, Imperial College London, SW7 2AZ London, United Kingdom – and – Wilfrid Laurier University, 75 University Avenue W., Waterloo Ontario, Canada N2L 3C5
Email: cstoica@wlu.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04330-9
PII: S 0002-9947(07)04330-9
Keywords: Saari's conjecture, $N$-body problem, jet transversality
Received by editor(s): January 5, 2005
Received by editor(s) in revised form: September 27, 2005
Published electronically: April 6, 2007
Additional Notes: The first author thanks the University of Surrey, the Bernoulli Centre at the Ecole Polytechnique Fédérale de Lausanne, and Wilfrid Laurier University, for their hospitality.
The second author was supported by the MASIE (Mechanics and Symmetry in Europe) Research Training Network of the European Union (HPRN-CT-2000-0013) as a post-doctoral fellow at the University of Surrey, and thanks Macquarie University for their hospitality.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.