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

Saari’s homographic conjecture of the three-body problem
HTML articles powered by AMS MathViewer

by Florin Diacu, Toshiaki Fujiwara, Ernesto Pérez-Chavela and Manuele Santoprete PDF

Trans. Amer. Math. Soc. 360 (2008), 6447-6473 Request permission

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

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, DOI 10.1007/s002220050200
A. Albouy and F. Yanning, Euler configurations and quasi-polynomial systems, preprint (2006) arXiv:math-ph/0603075
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
A. Chenciner, Introduction to the $N$-body problem, preprint (1997), http://www.bdl.fr/ Equipes/ASD/preprints/prep.1997/Ravello.1997.pdf
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, DOI 10.2307/2661357
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, DOI 10.1090/S0002-9947-04-03606-2
Toshiaki Fujiwara, Hiroshi Fukuda, and Hiroshi Ozaki, Choreographic three bodies on the lemniscate, J. Phys. A 36 (2003), no. 11, 2791–2800. MR 1965292, DOI 10.1088/0305-4470/36/11/310
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, DOI 10.1088/0305-4470/36/42/009
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, DOI 10.1088/0305-4470/37/44/008
Marshall Hampton and Richard Moeckel, Finiteness of relative equilibria of the four-body problem, Invent. Math. 163 (2006), no. 2, 289–312. MR 2207019, DOI 10.1007/s00222-005-0461-0
J. Llibre and E. Piña, Saari’s conjecture holds for the planar 3-body problem, preprint (2002).
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, DOI 10.1023/B:CELE.0000034500.64355.f9
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, DOI 10.1090/conm/081/986254
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. MR 2135737, DOI 10.1090/S0002-9947-04-03527-5
R. Moeckel, A proof of Saari’s conjecture for the three-body problem in $R^d$, preprint (2005).
Richard Montgomery, The $N$-body problem, the braid group, and action-minimizing periodic solutions, Nonlinearity 11 (1998), no. 2, 363–376. MR 1610784, DOI 10.1088/0951-7715/11/2/011
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, DOI 10.2307/2007159
Julian I. Palmore, Relative equilibria and the virial theorem, Celestial Mech. 19 (1979), no. 2, 167–171. MR 529313, DOI 10.1007/BF01796089
Julian I. Palmore, Saari’s conjecture revisited, Celestial Mech. 25 (1981), no. 1, 79–80. MR 648650, DOI 10.1007/BF01301808
Gareth E. Roberts, Some counterexamples to a generalized Saari’s conjecture, Trans. Amer. Math. Soc. 358 (2006), no. 1, 251–265. MR 2171232, DOI 10.1090/S0002-9947-05-03697-4
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).
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, DOI 10.1007/BF01241046
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; by the American Mathematical Society, Providence, RI, 2005. MR 2139425, DOI 10.1090/cbms/104
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, DOI 10.1023/B:CELE.0000043575.00445.21
Tanya Schmah and Cristina Stoica, Saari’s conjecture is true for generic vector fields, Trans. Amer. Math. Soc. 359 (2007), no. 9, 4429–4448. MR 2309192, DOI 10.1090/S0002-9947-07-04330-9
Aurel Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Mathematical Series, vol. 5, Princeton University Press, Princeton, N. J., 1941. MR 0005824