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Regular and Chaotic Dynamics

, Volume 22, Issue 4, pp 408–434 | Cite as

Periodic orbits in the restricted three-body problem and Arnold’s J +-invariant

  • Kai CieliebakEmail author
  • Urs Frauenfelder
  • Otto van Koert
Article

Abstract

We apply Arnold’s theory of generic smooth plane curves to Stark–Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold’s J +-invariant, we introduce invariants of periodic orbits in planar Stark–Zeeman systems and study their behavior.

Keywords

generic immersions into the plane Arnold’s plane curve invariants restricted three-body problem 

MSC2010 numbers

53A04 57R42 70F05 70F07 05C10 

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References

  1. 1.
    Abraham, R. and Marsden, J.E., Foundations of Mechanics, 2nd ed., rev. and enlarg., Reading, Mass.: Benjamin/Cummings, 1978.zbMATHGoogle Scholar
  2. 2.
    Arenstorf, R. F., Periodic Solutions of the Restricted Three-Body Problem Representing Analytic Continuations of Keplerian Elliptic Motions, Amer. J. Math., 1963, vol. 85, pp. 27–35.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arnol’d, V. I., Topological Invariants of Plane Curves and Caustics, Univ. Lecture Ser., vol. 5, Providence, R.I.: AMS, 1994.zbMATHGoogle Scholar
  4. 4.
    Barrow-Green, J., Poincaré and the Three Body Problem, Hist. Math., vol. 11, Providence, R.I.: AMS, 1997.zbMATHGoogle Scholar
  5. 5.
    Bruno, A.D., The Restricted 3-Body Problem: Plane Periodic Orbits, de Gruyter Exp. Math., vol. 17, Berlin: Walter de Gruyter & Co., 1994.CrossRefGoogle Scholar
  6. 6.
    Chmutov, S., Goryunov, V., and Murakami, H., Regular Legendrian Knots and the HOMFLY Polynomial of Immersed Plane Curves, Math. Ann., 2000, vol. 317, no. 3, pp. 389–413.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Darwin, G.H., Periodic Orbits, Acta Math., 1897, vol. 21, no. 1, pp. 99–242.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Darwin, G.H., On Certain Families of Periodic Orbits, Mon. Not. R. Astron. Soc., 1909, vol. 70, pp. 108–143.CrossRefzbMATHGoogle Scholar
  9. 9.
    Friedrich, H. and Wintgen, D., The Hydrogen Atom in a Uniform Magnetic Field — an Example of Chaos, Phys. Rep., 1989, vol. 183, no. 2, pp. 37–79.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hénon, M., Numerical Exploration of the Restricted Problem: 5. Hill’s Case: Periodic Orbits and Their Stability, Astron. Astrophys., 1969, vol. 1, pp. 223–238.zbMATHGoogle Scholar
  11. 11.
    Hénon, M., Generating Families in the Restricted Three-Body Problem, Lect. Notes Phys. Monogr., vol. 52, Berlin: Springer, 1997.Google Scholar
  12. 12.
    Hietarinta, J., Direct Methods for the Search of the Second Invariant, Phys. Rep., 1987, vol. 147, no. 2, pp. 87–154.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hill, G. W., Researches in the Lunar Theory, Amer. J. Math., 1878, vol. 1, no. 1, pp. 5–26MathSciNetCrossRefzbMATHGoogle Scholar
  14. 13a.
    Hill, G. W., Researches in the Lunar Theory, Amer. J. Math., 1878, vol. 1, no. 2, pp. 129–147MathSciNetCrossRefzbMATHGoogle Scholar
  15. 13b.
    Hill, G. W., Researches in the Lunar Theory, Amer. J. Math., 1878, vol. 1, no. 3, pp. 245–260.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 14.
    Landau, L. D. and Lifshitz, E. M., Course of Theoretical Physics: Vol. 1. Mechanics, 3rd ed., Oxford: Pergamon, 1976.Google Scholar
  17. 15.
    Lyapunov, A. M., Collected Works: Vol. 1, Moscow: Akad. Nauk SSSR, 1954 (Russian).Google Scholar
  18. 16.
    Matukuma, T., On the Periodic Orbits in Hill’s Case, Proc. Imp. Acad. Japan, 1930, vol. 6, pp. 6–8MathSciNetCrossRefzbMATHGoogle Scholar
  19. 16a.
    Matukuma, T., On the Periodic Orbits in Hill’s Case: 2. Retrograde Variational Orbits, Proc. Imp. Acad. Japan, 1932, vol. 8, pp. 147–150zbMATHGoogle Scholar
  20. 16b.
    Matukuma, T., On the Periodic Orbits in Hill’s Case: 3. Periodic Ejectional Orbits, Proc. Imp. Acad. Japan, 1933, vol. 9, pp. 364–366.zbMATHGoogle Scholar
  21. 17.
    Moulton, F. R., Periodic Orbits, Washington, D.C.: Carnegie Inst. of Washington, 1920.zbMATHGoogle Scholar
  22. 18.
    Ng, L., Plane Curves and Contact Geometry, in Proc. of Gökova Geometry/Topology Conference (GGT, 2005), S. Akbulut, T. Onder, R. J. Stern (Eds.), Gokova, 2006, pp. 165–174.Google Scholar
  23. 19.
    Poincaré, H., Les méthodes nouvelles de la mécanique céleste: In 3 Vols., Paris: Gauthier-Villars, 1892, 1899.zbMATHGoogle Scholar
  24. 20.
    Ryabov, Yu.A., On a Bound for the Region of Existence of a Periodic Solution of the Hill Equation in the Problem of Lunar Motion, Bjull. Inst. Teoret. Astronom., 1962, vol. 8, no. 10, pp. 772–786 (Russian).MathSciNetGoogle Scholar
  25. 21.
    Strömgren, E., Connaissance actuelle des orbites dans le problème des trois corps, Bull. Astron., 1935, vol. 9, pp. 87–130.zbMATHGoogle Scholar
  26. 22.
    Szebehely, V., Theory of Orbits: The Restricted Problem of Three Bodies, New York: Acad. Press, 1967.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • Kai Cieliebak
    • 1
    Email author
  • Urs Frauenfelder
    • 1
  • Otto van Koert
    • 2
  1. 1.Mathematisches InstitutUniversität AugsburgAugsburgGermany
  2. 2.Department of Mathematics and Research Institute of MathematicsSeoul National UniversitySeoulSouth Korea

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