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References
- 1.
V. I. Arnold,Dynamical Systems III, New York: Springer-Verlag, 1988.
- 2.
H. Bruns, Über die Integrale des Vielkörper-Problems,Acta Math. 11 (1887), 25–96.
- 3.
J. Chazy, Sur les singularités impossibles du problème desn corps,C. R. Hebdomadaires Séances Acad. Sci. Paris 170 (1920), 575–577.
- 4.
F. N. Diacu, Regularization of partial collisions in the N-body problem,Diff. Integral Eq. 5 (1992), 103–136.
- 5.
J. L. Gerver, A possible model for a singularity without j collisions in the five-body problem,J. Diff. Eq. 52 (1984), 76–90.
- 6.
J. L. Gerver, The existence of pseudocollisions in the plane,J. Diff. Eq. 89 (1991), 1–68.
- 7.
J. Mather and R. McGehee, Solutions of the collinear ! four-body problem which become unbounded in finite time,Dynamical Systems Theory and Applications (J. Moser, ed.), Berlin: Springer-Verlag, 1975, 573–589.
- 8.
R. McGehee, Triple collision in the collinear three-body problem,Invent. Math. 27 (1974), 191–227.
- 9.
R. McGehee, Triple collision in Newtonian gravitational systems,Dynamical Systems Theory and Applications (J. Moser, ed.), Berlin: Springer-Verlag, 1975, 550–572.
- 10.
R. McGehee, Von Zeipel’s theorem on singularities in celestial mechanics,Expo. Math. 4 (1986), 335–345.
- 11.
P. Painlevé,Leçons sur la théorie analytique des équations différentielles, Paris: Hermann, 1897.
- 12.
Oeuvres de Paul Painlevé, Tome I, Paris Ed. Centr. Nat. Rech. Sci., 1972.
- 13.
H. Poincaré, Sur le problème des trois corps et les équations de la dynamique,Acta Math. 13 (1890), 1–271.
- 14.
H. Poincaré,Les nouvelles méthodes de la mécanique céleste, Paris: Gauthier-Villar et Fils, vol. I (1892), vol. II (1893), vol. III (1899).
- 15.
D. G. Saari, Improbability of collisions in Newtonian gravitational systems,Trans. Amer. Math. Soc. 162 (1971), 267–271; 168 (1972), 521; 181 (1973), 351-368.
- 16.
D. G. Saari, Singularities and collisions in Newtonian gravitational systems,Arch. Rational Mech. Anal. 49 (1973), 311–320.
- 17.
D. G. Saari, Collisions are of first category,Proc. Amer. Math. Soc. 47 (1975), 442–445.
- 18.
D. G. Saari, The manifold structure for collisions and for hyperbolic parabolic orbits in the n-body problem,J. Diff. Eq. 41 (1984), 27–43.
- 19.
C. L. Siegel and J. K. Moser,Lectures on Celestial Mechanics, Berlin: Springer-Verlag, 1971.
- 20.
H. J. Sperling, On the real singularities of the N-body problem,J. Reine Angew. Math. 245 (1970), 15–40.
- 21.
V. Szebehely, Burrau’s problem of the three bodies,Proc. Nat. Acad. Sci. USA 58 (1967), 60–65.
- 22.
J. Waldvogel, The close triple approach,Celestial Mech. 11 (1975), 429–432.
- 23.
J. Waldvogel, The three-body problem near triple collision,Celestial Mech. 14 (1976), 287–300.
- 24.
A. Wintner,The Analytical Foundations of Celestial Mechanics, Princeton, NJ: Princeton University Press, 1941.
- 25.
Z. Xia, The existence of noncollision singularities in the N-body problem.Ann. Math, (in press).
- 26.
H. von Zeipel, Sur les singularités du problème des corps,Arkiv för Mat. Astron. Fys. 4, (1908), 1–4.
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Correspondence to Florin N. Diacu.
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Diacu, F.N. Painlevé’s conjecture. The Mathematical Intelligencer 15, 6–12 (1993). https://doi.org/10.1007/BF03024186
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Keywords
- Celestial Mechanic
- Dynamical System Theory
- Binary Collision
- Triple Collision
- Collision Singularity