Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Saari's homographic conjecture of the three-body problem


Authors: Florin Diacu, Toshiaki Fujiwara, Ernesto Pérez-Chavela and Manuele Santoprete
Journal: Trans. Amer. Math. Soc. 360 (2008), 6447-6473
MSC (2000): Primary 70F10, 70H05
DOI: https://doi.org/10.1090/S0002-9947-08-04517-0
Published electronically: May 29, 2008
MathSciNet review: 2434294
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Saari's homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian $ n$-body problem with constant configurational measure are homographic. In other words, if the mutual distances satisfy a certain relationship, the configuration of the particle system may change size and position but not shape. We prove this conjecture for large sets of initial conditions in three-body problems given by homogeneous potentials, including the Newtonian one. Some of our results are true for $ n\ge 3$.


References [Enhancements On Off] (What's this?)

  • 1. A. Albouy and A. Chenciner, Le probléme des $ n$ corps et les distances mutuelles, Inventiones Mathematicae 131, 151-184 (1998). MR 1489897 (98m:70017)
  • 2. A. Albouy and F. Yanning, Euler configurations and quasi-polynomial systems, preprint (2006) arXiv:math-ph/0603075
  • 3. J. Chazy, Sur l'allure du mouvement dans le probléme des trois corps quand le temps croît indéfiniment, Annales Scientifiques de l'École Normale Supérieure Sér. 3, 39, 29-130 (1922). MR 1509241
  • 4. A. Chenciner, Introduction to the $ N$-body problem, preprint (1997), http://www.bdl.fr/ Equipes/ASD/preprints/prep.1997/Ravello.1997.pdf
  • 5. A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Mathematics 152, 881-901 (2000). MR 1815704 (2001k:70010)
  • 6. F. Diacu, E. Pérez-Chavela and M. Santoprete, Saari's conjecture of the $ N$-body problem in the collinear case, Trans. Amer. Math. Soc. 357, 4215-4223 (2005). MR 2159707 (2007k:70014)
  • 7. T. Fujiwara, H. Fukuda and H. Ozaki, Choreographic three bodies on the lemniscate, J. Phys. A: Math. Gen. 36, 2791-2800 (2003). MR 1965292 (2004b:70020)
  • 8. T. Fujiwara, H. Fukuda and H. Ozaki, Evolution of the moment of inertia of three-body figure-eight choreography, J. Phys. A: Math. Gen. 36, 10537-10549 (2003). MR 2024911 (2005b:70008)
  • 9. T. Fujiwara, H. Fukuda, A. Kameyama, H. Ozaki and M. Yamada, Synchronized similar triangles for three-body orbits with zero angular momentum, J. Phys. A: Math. Gen. 37, 10571-10584 (2004). MR 2098052 (2005g:70014)
  • 10. M. Hampton and R. Moeckel, Finiteness of Relative Equilibria of the Four-Body Problem, preprint (2004). MR 2207019
  • 11. J. Llibre and E. Piña, Saari's conjecture holds for the planar 3-body problem, preprint (2002).
  • 12. C. McCord, Saari's conjecture for the planar three-body problem with equal masses, Celestial Mechanics 89, 2, 99-118 (2004). MR 2086184 (2005g:70015)
  • 13. R. Moeckel, Some qualitative properties of the three body problem, Contemp. Math. 81, 1-21 (1988). MR 986254 (90e:58054)
  • 14. R. Moeckel, A computer-assisted proof of Saari's conjecture for the planar three-body problem, Trans. Amer. Math. Soc. 357, 3105-3117 (2005). MR 2135737 (2005m:70054)
  • 15. R. Moeckel, A proof of Saari's conjecture for the three-body problem in $ R^d$, preprint (2005).
  • 16. R. Montgomery, The $ N-$body problem, the braid group, and action minimizing periodic solutions, Nonlinearity 11, 363-376 (1998). MR 1610784 (99a:70019)
  • 17. F. Moulton, The straight line solutions of the problem on $ N$ bodies, Ann. of Math. 2-12, 1-17 (1910). MR 1503509
  • 18. J. I. Palmore, Relative equilibria and the virial theorem, Celestial Mechanics 19, 167-171 (1979). MR 529313 (80i:70014)
  • 19. J. I. Palmore, Saari's conjecture revisited, Celestial Mechanics 25, 79-80 (1981). MR 648650 (83c:70011)
  • 20. G. Roberts, Some counterexamples to a generalized Saari's conjecture, Trans. Amer. Math. Soc. 358, 251-265 (2006). MR 2171232 (2006e:70021)
  • 21. D. Saari, On bounded solutions of the n-body problem, Periodic Orbits, Stability and Resonances, G.E.O., Giacaglia (Ed.), D. Riedel, Dordrecht, 76-81 (1970).
  • 22. D. Saari, From rotations and inclinations to zero configurational velocity surfaces I, a natural rotating coordinate system, Celestial Mechanics 33, 105-119 (1985). MR 777381 (86g:70003)
  • 23. D. Saari, Collisions, Rings, and Other Newtonian $ N$-Body Problems, American Mathematical Society, Regional Conference Series in Mathematics, No. 104, Providence, RI, 2005. MR 2139425 (2006d:70030)
  • 24. M. Santoprete, A counterexample to a generalized Saari's conjecture with a continuum of central configurations, Celestial Mechanics 89, 4, 357-364 (2004). MR 2104899 (2007b:70025)
  • 25. T. Schmah and C. Stoica, Saari's conjecture is true for generic vector fields, Trans. Amer. Math. Soc. 359, no. 9, 4429-4448 (2007). MR 2309192
  • 26. A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Univ. Press, Princeton, NJ, 1941. MR 0005824 (3:215b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 70F10, 70H05

Retrieve articles in all journals with MSC (2000): 70F10, 70H05


Additional Information

Florin Diacu
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada
Email: diacu@math.uvic.ca

Toshiaki Fujiwara
Affiliation: College of Liberal Arts and Sciences, Kitasato University, 1-15-1 Kitasato, Sagamihara, Kanagawa 228-8555, Japan
Email: fujiwara@clas.kitasato-u.ac.jp

Ernesto Pérez-Chavela
Affiliation: Departamento de Matemáticas, UAM–Iztapalapa, A.P. 55–534, 09340 Iztapalapa, Mexico, D.F., Mexico
Email: epc@xanum.uam.mx

Manuele Santoprete
Affiliation: Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada
Email: msantoprete@wlu.ca

DOI: https://doi.org/10.1090/S0002-9947-08-04517-0
Keywords: Three-body problem, homographic solutions, central configurations
Received by editor(s): November 27, 2006
Published electronically: May 29, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society