Skip to main content
Log in

Painlevé’s conjecture

  • Article
  • Published:
The Mathematical Intelligencer Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. V. I. Arnold,Dynamical Systems III, New York: Springer-Verlag, 1988.

    Book  Google Scholar 

  2. H. Bruns, Über die Integrale des Vielkörper-Problems,Acta Math. 11 (1887), 25–96.

    Article  MathSciNet  Google Scholar 

  3. J. Chazy, Sur les singularités impossibles du problème desn corps,C. R. Hebdomadaires Séances Acad. Sci. Paris 170 (1920), 575–577.

    MATH  Google Scholar 

  4. F. N. Diacu, Regularization of partial collisions in the N-body problem,Diff. Integral Eq. 5 (1992), 103–136.

    MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. L. Gerver, The existence of pseudocollisions in the plane,J. Diff. Eq. 89 (1991), 1–68.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Chapter  Google Scholar 

  8. R. McGehee, Triple collision in the collinear three-body problem,Invent. Math. 27 (1974), 191–227.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. McGehee, Triple collision in Newtonian gravitational systems,Dynamical Systems Theory and Applications (J. Moser, ed.), Berlin: Springer-Verlag, 1975, 550–572.

    Chapter  Google Scholar 

  10. R. McGehee, Von Zeipel’s theorem on singularities in celestial mechanics,Expo. Math. 4 (1986), 335–345.

    MATH  MathSciNet  Google Scholar 

  11. P. Painlevé,Leçons sur la théorie analytique des équations différentielles, Paris: Hermann, 1897.

    Google Scholar 

  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.

    MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  Google Scholar 

  16. D. G. Saari, Singularities and collisions in Newtonian gravitational systems,Arch. Rational Mech. Anal. 49 (1973), 311–320.

    Article  MATH  MathSciNet  Google Scholar 

  17. D. G. Saari, Collisions are of first category,Proc. Amer. Math. Soc. 47 (1975), 442–445.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  19. C. L. Siegel and J. K. Moser,Lectures on Celestial Mechanics, Berlin: Springer-Verlag, 1971.

    Book  MATH  Google Scholar 

  20. H. J. Sperling, On the real singularities of the N-body problem,J. Reine Angew. Math. 245 (1970), 15–40.

    MATH  MathSciNet  Google Scholar 

  21. V. Szebehely, Burrau’s problem of the three bodies,Proc. Nat. Acad. Sci. USA 58 (1967), 60–65.

    Article  MATH  Google Scholar 

  22. J. Waldvogel, The close triple approach,Celestial Mech. 11 (1975), 429–432.

    Article  MATH  MathSciNet  Google Scholar 

  23. J. Waldvogel, The three-body problem near triple collision,Celestial Mech. 14 (1976), 287–300.

    Article  MATH  MathSciNet  Google Scholar 

  24. A. Wintner,The Analytical Foundations of Celestial Mechanics, Princeton, NJ: Princeton University Press, 1941.

    Google Scholar 

  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.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Diacu, F.N. Painlevé’s conjecture. The Mathematical Intelligencer 15, 6–12 (1993). https://doi.org/10.1007/BF03024186

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03024186

Keywords

Navigation