The Non-collision Sigularities of the 5 body Problem

  • Zhihong Xia
Conference paper
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 22)

Abstract

This paper is a survey on the author’s work on the non-collision singularities in the Newtonian n-body problem. The non-collision singularity in the n-body system corresponds to the solution which blow up to infinity in finite time. The question whether there exists such solution was first raised by Painleve in the last century and since then, it has been open. Here, we show that such solutions do exist in a 5-body problem. The method we use is based on careful analysis of near collisions orbits and McGehee’s technique of blowing up collision singularities.

Keywords

Manifold 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Devaney, Triple collision in the planar isosceles three-body problem, Inv. Math. 60 (1980), 249–267.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    R. Devaney, Singularities in classical mechanical systems, Ergodic Theory and Dynamical Systems (1981), Birkhauser, Boston.Google Scholar
  3. [3]
    J. Mather and R. McGehee, Solutions of the collinear four body problem which become unbounded in finite time, Lecture Notes in Physics 38 (J. Moser, ed.) (1975), 573–597, Springer-Verlag, Berlin Heidelberg New York.Google Scholar
  4. [4]
    R. McGehee, Triple collision in the collinear three-body problem, Inventiones Math. 27 (1974), 191–227.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    R. McGehee, Von Zeipe Vs theorem on singularities in celestial mechanics, Expo. Math. 4 (1986), 335–345.MathSciNetMATHGoogle Scholar
  6. [6]
    R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem, SI AM J. Math. Analysis 15 (1984), 857–876.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    P. Painlevé, “Leçons sur la théorie analytique des équations différentielles,” A. He- mann, Paris, 1897.Google Scholar
  8. [8]
    H. Pollard and D. Saari, Singularities of the n-body problem, I, Arch. Rational Mech. and Math. 30 (1968), 263–269.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    H. Pollard and D. Saari, Singularities of the n-body problem, II, Inequalities-II (1970), 255–259, Academic Press.Google Scholar
  10. [10]
    D. Saari, Singularities and collisions of Newtonian gravitational systems, Archive for Rational Mechanics and Analysis 49, no. 4 (1973), 311–320.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    D. Saari, Singularities of Newtonian gravitational systems, Proceedings of Symposium on Global Analysis, Dynamical Systems and Celestial Mechanics (1971), Brazil, August.Google Scholar
  12. [12]
    Z. Xia, The Existence of Non-collision Singularities In Newtonian System, Thesis (1988), Northwestern University.Google Scholar
  13. [13]
    H. von Zeipel, Sur les singularités du problème des n corps, Arkiv for Matematik, Astronomi och Fysik 4, 32 (1908), 1–4.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1991

Authors and Affiliations

  • Zhihong Xia
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

Personalised recommendations