Saari’s homographic conjecture of the three-body problem

Diacu, Florin; Fujiwara, Toshiaki; Pérez-Chavela, Ernesto; Santoprete, Manuele

doi:10.1090/S0002-9947-08-04517-0

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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
Posted: May 29, 2008
MathSciNet review: 2434294
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Abstract: Saari's homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian -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 $n\ge 3$ .

1. Alain Albouy and Alain Chenciner, Le problème des 𝑛 corps et les distances mutuelles, Invent. Math. 131 (1998), no. 1, 151–184 (French). MR 1489897 (98m:70017), http://dx.doi.org/10.1007/s002220050200
2. A. Albouy and F. Yanning, Euler configurations and quasi-polynomial systems, preprint (2006) arXiv:math-ph/0603075
3. Jean Chazy, Sur l’allure du mouvement dans le problème des trois corps quand le temps croît indéfiniment, Ann. Sci. École Norm. Sup. (3) 39 (1922), 29–130 (French). MR 1509241
4. A. Chenciner, Introduction to the -body problem, preprint (1997), http://www.bdl.fr/ Equipes/ASD/preprints/prep.1997/Ravello.1997.pdf
5. Alain Chenciner and Richard Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math. (2) 152 (2000), no. 3, 881–901. MR 1815704 (2001k:70010), http://dx.doi.org/10.2307/2661357
6. Florin Diacu, Ernesto Pérez-Chavela, and Manuele Santoprete, Saari’s conjecture for the collinear 𝑛-body problem, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4215–4223. MR 2159707 (2007k:70014), http://dx.doi.org/10.1090/S0002-9947-04-03606-2
7. Toshiaki Fujiwara, Hiroshi Fukuda, and Hiroshi Ozaki, Choreographic three bodies on the lemniscate, J. Phys. A 36 (2003), no. 11, 2791–2800. MR 1965292 (2004b:70020), http://dx.doi.org/10.1088/0305-4470/36/11/310
8. Toshiaki Fujiwara, Hiroshi Fukuda, and Hiroshi Ozaki, Evolution of the moment of inertia of three-body figure-eight choreography, J. Phys. A 36 (2003), no. 42, 10537–10549. MR 2024911 (2005b:70008), http://dx.doi.org/10.1088/0305-4470/36/42/009
9. Toshiaki Fujiwara, Hiroshi Fukuda, Atsushi Kameyama, Hiroshi Ozaki, and Michio Yamada, Synchronized similar triangles for three-body orbits with zero angular momentum, J. Phys. A 37 (2004), no. 44, 10571–10584. MR 2098052 (2005g:70014), http://dx.doi.org/10.1088/0305-4470/37/44/008
10. Marshall Hampton and Richard Moeckel, Finiteness of relative equilibria of the four-body problem, Invent. Math. 163 (2006), no. 2, 289–312. MR 2207019 (2008c:70019), http://dx.doi.org/10.1007/s00222-005-0461-0
11. J. Llibre and E. Piña, Saari's conjecture holds for the planar 3-body problem, preprint (2002).
12. Christopher McCord, Saari’s conjecture for the planar three-body problem with equal masses, Celestial Mech. Dynam. Astronom. 89 (2004), no. 2, 99–118. MR 2086184 (2005g:70015), http://dx.doi.org/10.1023/B:CELE.0000034500.64355.f9
13. Richard Moeckel, Some qualitative features of the three-body problem, Hamiltonian dynamical systems (Boulder, CO, 1987) Contemp. Math., vol. 81, Amer. Math. Soc., Providence, RI, 1988, pp. 1–22. MR 986254 (90e:58054), http://dx.doi.org/10.1090/conm/081/986254
14. Richard Moeckel, A computer-assisted proof of Saari’s conjecture for the planar three-body problem, Trans. Amer. Math. Soc. 357 (2005), no. 8, 3105–3117 (electronic). MR 2135737 (2005m:70054), http://dx.doi.org/10.1090/S0002-9947-04-03527-5
15. R. Moeckel, A proof of Saari's conjecture for the three-body problem in , preprint (2005).
16. Richard Montgomery, The 𝑁-body problem, the braid group, and action-minimizing periodic solutions, Nonlinearity 11 (1998), no. 2, 363–376. MR 1610784 (99a:70019), http://dx.doi.org/10.1088/0951-7715/11/2/011
17. F. R. Moulton, The straight line solutions of the problem of 𝑛 bodies, Ann. of Math. (2) 12 (1910), no. 1, 1–17. MR 1503509, http://dx.doi.org/10.2307/2007159
18. Julian I. Palmore, Relative equilibria and the virial theorem, Celestial Mech. 19 (1979), no. 2, 167–171. MR 529313 (80i:70014), http://dx.doi.org/10.1007/BF01796089
19. Julian I. Palmore, Saari’s conjecture revisited, Celestial Mech. 25 (1981), no. 1, 79–80. MR 648650 (83c:70011), http://dx.doi.org/10.1007/BF01301808
20. Gareth E. Roberts, Some counterexamples to a generalized Saari’s conjecture, Trans. Amer. Math. Soc. 358 (2006), no. 1, 251–265 (electronic). MR 2171232 (2006e:70021), http://dx.doi.org/10.1090/S0002-9947-05-03697-4
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. Donald G. Saari, From rotations and inclinations to zero configurational velocity surfaces. I. A natural rotating coordinate system, Celestial Mech. 33 (1984), no. 4, 299–318. MR 777381 (86g:70003), http://dx.doi.org/10.1007/BF01241046
23. Donald G. Saari, Collisions, rings, and other Newtonian 𝑁-body problems, CBMS Regional Conference Series in Mathematics, vol. 104, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2005. MR 2139425 (2006d:70030)
24. Manuele Santoprete, A counterexample to a generalized Saari’s conjecture with a continuum of central configurations, Celestial Mech. Dynam. Astronom. 89 (2004), no. 4, 357–364. MR 2104899 (2007b:70025), http://dx.doi.org/10.1023/B:CELE.0000043575.00445.21
25. Tanya Schmah and Cristina Stoica, Saari’s conjecture is true for generic vector fields, Trans. Amer. Math. Soc. 359 (2007), no. 9, 4429–4448 (electronic). MR 2309192 (2009h:70020), http://dx.doi.org/10.1090/S0002-9947-07-04330-9
26. Aurel Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Mathematical Series, v. 5, Princeton University Press, Princeton, N. J., 1941. MR 0005824 (3,215b)

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