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

DOI:
https://doi.org/10.1090/S0002-9947-07-04330-9

Published electronically:
April 6, 2007

MathSciNet review:
2309192

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The simplest non-collision 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 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.

**[A63]**R. Abraham.

Transversality in manifolds of mappings.*Bull. Amer. Math. Soc.*, 69:470-474, 1963. MR**0149495 (26:6982)****[AM78]**R. Abraham and J.E. Marsden.*Foundations of Mechanics*.

Addison-Wesley, second edition, 1978. MR**0515141 (81e:58025)****[AR67]**R. Abraham and J. Robbin.*Transversal Mappings and Flows*.

W. A. Benjamin, 1967. MR**0240836 (39:2181)****[Arn78]**V. I. Arnold.*Mathematical Methods of Classical Mechanics*.

Number 60 in Graduate Texts in Mathematics,

Springer-Verlag, 1978. MR**0690288 (57:14033b)****[Ch03]**K.-C. Chen, Open Problems. http://www.aimath.org/WWN/varcelest/articles/html/ 35a/**[CM00]**A. Chenciner and R. Montgomery [2000], A remarkable periodic solution of the three body problem in the case of equal masses,*Ann. of Math*,**152**, 881-901. MR**1815704 (2001k:70010)****[DPS05]**F. Diacu, E. Pérez-Chavela and M. Santoprete [2005], Saari's Conjecture for the collinear -body problem,*Trans. Amer. Math. Soc.***357**, 4215-4223 MR**2159707****[F77]**M.J. Field [1977], Transversality in -manifolds.*Trans. Amer. Math. Soc.*,**231**No. 2, 429-450 MR**0451276 (56:9563)****[GG73]**M. Golubitsky and V. Guillemin [1973]*Stable Mappings and Their Singularities*, Springer-Verlag MR**0341518 (49:6269)****[HM06]**M. Hampton and R. Moeckel [2006] Finiteness of relative equilibria of the four-body problem.*Invent. Math*.**163**, no. 2, 289-312. MR**2207019****[HLM05]**A. Hernández-Garduño, A., J. K. Lawson, and J. E. Marsden [2005], Relative equilibria for the generalized rigid body,*J. Geom. Phys.***53**, Issue 3, 259-274 MR**2108531 (2006a:37049)****[H76]**M. W. Hirsch.*Differentiable Topology*.

Number 33 in Graduate Texts in Mathematics. Springer-Verlag, 1976. MR**0448362 (56:6669)****[LSt07]**J. K. Lawson and C. Stoica [2007], Constant locked inertia tensor trajectories for simple mechanical systems with symmetry.*J. Geom. Phys*.**57**, 1115-1312.**[LlP02]**J. Llibre and E. Piña [2002], Saari's Conjecture holds for the planar 3-body problem, preprint**[McC04]**C. McCord [2004], Saari's conjecture for the planar three-body problem with equal masses.*Celestial Mech. Dynam. Astronom.***89**, no. 2, 99-118 MR**2086184 (2005g:70015)****[Moe05a]**R. Moeckel [2005], A computer assisted proof of Saari's Conjecture for the planar three-body problem,*Trans. Amer. Math. Soc.***357**, 3105-3117. MR**2135737 (2005m:70054)****[Moe05b]**R. Moeckel [2005], A proof of Saari's Conjecture for the Three-Body Problem in .*Preprint.***[Mey99]**K. R. Meyer [1999],*Periodic solutions of the body problem*. Springer-Verlag. MR**1736548 (2001i:70013)****[Rob06]**G. Roberts [2006], Some counterexamples to a Generalized Saari Conjecture.*Trans. Amer. Math. Soc.***358**, 251-265 MR**2171232 (2006e:70021)****[RSS05]**M. Roberts, T. Schmah and C. Stoica [2005], Relative equilibria in systems with configuration space isotropy.*J. Geom. Phys.*Available online.**[Saa70]**D. Saari [1970], On bounded solutions of the -body problem, in Giacaglia, G. (ed.),*Periodic Orbits, Stability and Resonances*, D. Riedel, Dordrecht, 76-81.**[San04]**M. Santoprete [2004], A counterexample to a Generalized Saari's Conjecture with a continuum of central configurations.*Celestial Mech. Dynam. Astronom.***89**, 357-364 MR**2104899**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
70F10,
37J05,
57N75

Retrieve articles in all journals with MSC (2000): 70F10, 37J05, 57N75

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:
https://doi.org/10.1090/S0002-9947-07-04330-9

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.