On the singularities of the curved 𝑛-body problem

Diacu, Florin

doi:10.1090/S0002-9947-2010-05251-1

On the singularities of the curved $n$-body problem
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by Florin Diacu PDF

Trans. Amer. Math. Soc. 363 (2011), 2249-2264

Abstract:

We study singularities of the $n$-body problem in spaces of constant curvature and generalize certain results due to Painlevé, Weierstrass, and Sundman. For positive curvature, some of our proofs use the correspondence between total collision solutions of the original system and their orthogonal projection—a property that offers a new method of approaching the problem in this particular case.

References

Florin Diacu, Singularities of the $N$-body problem, Classical and celestial mechanics (Recife, 1993/1999) Princeton Univ. Press, Princeton, NJ, 2002, pp. 35–62. MR 1974779
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
Florin Diacu, Toshiaki Fujiwara, Ernesto Pérez-Chavela, and Manuele Santoprete, Saari’s homographic conjecture of the three-body problem, Trans. Amer. Math. Soc. 360 (2008), no. 12, 6447–6473. MR 2434294, DOI 10.1090/S0002-9947-08-04517-0
F. Diacu, E. Pérez-Chavela, and M. Santoprete, The $n$-body problem in spaces of constant curvature, arXiv:0807.1747 (2008), 54 pp.
Joseph L. Gerver, The existence of pseudocollisions in the plane, J. Differential Equations 89 (1991), no. 1, 1–68. MR 1088334, DOI 10.1016/0022-0396(91)90110-U
W. Killing, Die Mechanik in den nichteuklidischen Raumformen, J. Reine Angew. Math. 98 (1885), 1-48.
H. Liebmann, Die Kegelschnitte und die Planetenbewegung im nichteuklidischen Raum, Berichte Königl. Sächsischen Gesell. Wiss., Math. Phys. Klasse 54 (1902), 393-423.
H. Liebmann, Über die Zentralbewegung in der nichteuklidische Geometrie, Berichte Königl. Sächsischen Gesell. Wiss., Math. Phys. Klasse 55 (1903), 146-153.
R. Lipschitz, Extension of the planet-problem to a space of $n$ dimensions and constant integral curvature, Quart. J. Pure Appl. Math. 12 (1873), 349-370.
J. N. Mather and R. McGehee, Solutions of the collinear four body problem which become unbounded in finite time, Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974) Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975, pp. 573–597. MR 0495348
G. Mittag-Leffler, Zur Biographie von Weierstrass, Acta Math. 35 (1912), no. 1, 29–65 (German). MR 1555072, DOI 10.1007/BF02418812
P. Painlevé, Leçons sur la théorie analytique des équations différentielles, Hermann, Paris, 1897.
E. Schering, Die Schwerkraft im Gaussischen Raume, Nachr. Königl. Gesell. Wiss. Göttingen 15 (1870), 311-321.
C. L. Siegel and J. K. Moser, Lectures on celestial mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Translated from the German by C. I. Kalme; Reprint of the 1971 translation. MR 1345153
K. F. Sundman, Recherches sur le problème des trois corps, Acta. Soc. Sci. Fennicae 34, 6 (1907), 1-43.
K. F. Sundman, Nouvelles recherches sur le problème des trois corps, Acta. Soc. Sci. Fennicae 35, 9 (1909), 3-27.
Karl F. Sundman, Mémoire sur le problème des trois corps, Acta Math. 36 (1913), no. 1, 105–179 (French). MR 1555085, DOI 10.1007/BF02422379
Aurel Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Mathematical Series, vol. 5, Princeton University Press, Princeton, N. J., 1941. MR 0005824
Zhihong Xia, The existence of noncollision singularities in Newtonian systems, Ann. of Math. (2) 135 (1992), no. 3, 411–468. MR 1166640, DOI 10.2307/2946572