The Mathematical Intelligencer

, Volume 15, Issue 2, pp 6–12 | Cite as

Painlevé’s conjecture

  • Florin N. Diacu


Celestial Mechanic Dynamical System Theory Binary Collision Triple Collision Collision Singularity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. I. Arnold,Dynamical Systems III, New York: Springer-Verlag, 1988.CrossRefGoogle Scholar
  2. 2.
    H. Bruns, Über die Integrale des Vielkörper-Problems,Acta Math. 11 (1887), 25–96.CrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Chazy, Sur les singularités impossibles du problème desn corps,C. R. Hebdomadaires Séances Acad. Sci. Paris 170 (1920), 575–577.zbMATHGoogle Scholar
  4. 4.
    F. N. Diacu, Regularization of partial collisions in the N-body problem,Diff. Integral Eq. 5 (1992), 103–136.zbMATHMathSciNetGoogle Scholar
  5. 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.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    J. L. Gerver, The existence of pseudocollisions in the plane,J. Diff. Eq. 89 (1991), 1–68.CrossRefzbMATHMathSciNetGoogle Scholar
  7. 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.CrossRefGoogle Scholar
  8. 8.
    R. McGehee, Triple collision in the collinear three-body problem,Invent. Math. 27 (1974), 191–227.CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    R. McGehee, Triple collision in Newtonian gravitational systems,Dynamical Systems Theory and Applications (J. Moser, ed.), Berlin: Springer-Verlag, 1975, 550–572.CrossRefGoogle Scholar
  10. 10.
    R. McGehee, Von Zeipel’s theorem on singularities in celestial mechanics,Expo. Math. 4 (1986), 335–345.zbMATHMathSciNetGoogle Scholar
  11. 11.
    P. Painlevé,Leçons sur la théorie analytique des équations différentielles, Paris: Hermann, 1897.Google Scholar
  12. 12.
    Oeuvres de Paul Painlevé, Tome I, Paris Ed. Centr. Nat. Rech. Sci., 1972.Google Scholar
  13. 13.
    H. Poincaré, Sur le problème des trois corps et les équations de la dynamique,Acta Math. 13 (1890), 1–271.zbMATHGoogle Scholar
  14. 14.
    H. Poincaré,Les nouvelles méthodes de la mécanique céleste, Paris: Gauthier-Villar et Fils, vol. I (1892), vol. II (1893), vol. III (1899).Google Scholar
  15. 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.CrossRefMathSciNetGoogle Scholar
  16. 16.
    D. G. Saari, Singularities and collisions in Newtonian gravitational systems,Arch. Rational Mech. Anal. 49 (1973), 311–320.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    D. G. Saari, Collisions are of first category,Proc. Amer. Math. Soc. 47 (1975), 442–445.CrossRefzbMATHMathSciNetGoogle Scholar
  18. 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.CrossRefMathSciNetGoogle Scholar
  19. 19.
    C. L. Siegel and J. K. Moser,Lectures on Celestial Mechanics, Berlin: Springer-Verlag, 1971.CrossRefzbMATHGoogle Scholar
  20. 20.
    H. J. Sperling, On the real singularities of the N-body problem,J. Reine Angew. Math. 245 (1970), 15–40.zbMATHMathSciNetGoogle Scholar
  21. 21.
    V. Szebehely, Burrau’s problem of the three bodies,Proc. Nat. Acad. Sci. USA 58 (1967), 60–65.CrossRefzbMATHGoogle Scholar
  22. 22.
    J. Waldvogel, The close triple approach,Celestial Mech. 11 (1975), 429–432.CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    J. Waldvogel, The three-body problem near triple collision,Celestial Mech. 14 (1976), 287–300.CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    A. Wintner,The Analytical Foundations of Celestial Mechanics, Princeton, NJ: Princeton University Press, 1941.Google Scholar
  25. 25.
    Z. Xia, The existence of noncollision singularities in the N-body problem.Ann. Math, (in press).Google Scholar
  26. 26.
    H. von Zeipel, Sur les singularités du problème des corps,Arkiv för Mat. Astron. Fys. 4, (1908), 1–4.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 1993

Authors and Affiliations

  • Florin N. Diacu
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada

Personalised recommendations