Saari's conjecture is true for generic vector fields
Authors:
Tanya Schmah and Cristina Stoica
Journal:
Trans. Amer. Math. Soc. 359 (2007), 44294448
MSC (2000):
Primary 70F10; Secondary 37J05, 57N75
Published electronically:
April 6, 2007
MathSciNet review:
2309192
Fulltext PDF Free Access
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Abstract: The simplest noncollision solutions of the 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 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 body problem: relative equilibria are the only constantinertia solutions. A computerassisted proof for the 3body 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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 M. Hampton and R. Moeckel [2006] Finiteness of relative equilibria of the fourbody problem. Invent. Math. 163, no. 2, 289312. MR 2207019
 [HLM05]
 A. HernándezGarduño, A., J. K. Lawson, and J. E. Marsden [2005], Relative equilibria for the generalized rigid body, J. Geom. Phys. 53, Issue 3, 259274 MR 2108531 (2006a:37049)
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 J. K. Lawson and C. Stoica [2007], Constant locked inertia tensor trajectories for simple mechanical systems with symmetry. J. Geom. Phys. 57, 11151312.
 [LlP02]
 J. Llibre and E. Piña [2002], Saari's Conjecture holds for the planar 3body problem, preprint
 [McC04]
 C. McCord [2004], Saari's conjecture for the planar threebody problem with equal masses. Celestial Mech. Dynam. Astronom. 89, no. 2, 99118 MR 2086184 (2005g:70015)
 [Moe05a]
 R. Moeckel [2005], A computer assisted proof of Saari's Conjecture for the planar threebody problem, Trans. Amer. Math. Soc. 357, 31053117. MR 2135737 (2005m:70054)
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 R. Moeckel [2005], A proof of Saari's Conjecture for the ThreeBody Problem in . Preprint.
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 K. R. Meyer [1999], Periodic solutions of the body problem. SpringerVerlag. MR 1736548 (2001i:70013)
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 G. Roberts [2006], Some counterexamples to a Generalized Saari Conjecture. Trans. Amer. Math. Soc. 358, 251265 MR 2171232 (2006e:70021)
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 M. Roberts, T. Schmah and C. Stoica [2005], Relative equilibria in systems with configuration space isotropy. J. Geom. Phys. Available online.
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 M. Santoprete [2004], A counterexample to a Generalized Saari's Conjecture with a continuum of central configurations. Celestial Mech. Dynam. Astronom. 89, 357364 MR 2104899
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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/S0002994707043309
PII:
S 00029947(07)043309
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 (HPRNCT20000013) as a postdoctoral 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.
