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Shooting for the Eight

A Topological Existence Proof for a Figure-Eight Orbit of the Three-Body Problem
  • Richard MoeckelEmail author
Chapter
Part of the Abel Symposia book series (ABEL, volume 5)

Abstract

A topological existence proof is given for a figure-eight periodic solution of the equal mass three-body problem. The proof is based on the construction of a Wazewski set W in the phase space. The figure-eight solution is then found by a kind of shooting argument in which symmetrical initial conditions entering W are followed under the flow until they exit W. A linking argument shows that the image of the symmetrical entrance states under this flow map must intersect an appropriate set of symmetrical exit states.

Keywords

Celestial mechanics three-body problem symmetrical periodic solutions topological methods 

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References

  1. 1.
    R. Cabanela, The retrograde solutions of the planar three-body problem in the neighborhood of the restricted problem via a submanifold convex to the flow, Thesis, University of Minnesota (1995).Google Scholar
  2. 2.
    C.C. Conley, The retrograde circular solutions of the restricted three-body problem via a submanifold convex to the flow, SIAM J. Appl. Math., 16, (1968) 620–625.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    C.C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series, 38, American Mathematical Society (1978).Google Scholar
  4. 4.
    C.C. Conley and R.W. Easton, Isolated invariant sets and isolating blocks, Trans. AMS, 158, 1 (1971) 35–60.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Annals of Math., 152 (2000) 881–901.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    R. W. Easton, Existence of invariant sets inside a submanifold convex to the flow, JDE, 7 (1970) 54–68.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    T. Kapela and P. Zgliczynński, The existence of simple choreographies for N-body problem - a computer assisted proof, Nonlinearity 16 (2003) 1899–1918.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    R. McGehee, Triple collision in the collinear three-body problem, Inv. Math, 27 (1974) 191–227.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    R. Montgomery, Infinitely many syzygies, Arch. Rat. Mech., 164, 4 (2002) 311–340.zbMATHCrossRefGoogle Scholar
  10. 10.
    C.L. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer, New York (1971).zbMATHGoogle Scholar
  11. 11.
    T. Wazewski, Sur un principe topologique de l'examen de l'allure asymptotiques des intègrales des équations différentielles ordinaires, Ann. Soc. Pol. Math. 20 (1947) 279–313.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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