Advertisement

Regular and Chaotic Dynamics

, Volume 24, Issue 2, pp 202–211 | Cite as

Variational Construction of Orbits Realizing Symbolic Sequences in the Planar Sitnikov Problem

  • Mitsuru ShibayamaEmail author
Article
  • 21 Downloads

Abstract

Using the variational method, Chenciner and Montgomery (2000 Ann. Math. 152 881–901) proved the existence of an eight-shaped orbit of the planar three-body problem with equal masses. Since then a number of solutions to the N-body problem have been discovered. On the other hand, symbolic dynamics is one of the most useful methods for understanding chaotic dynamics. The Sitnikov problem is a special case of the three-body problem. The system is known to be chaotic and was studied by using symbolic dynamics (J. Moser, Stable and random motions in dynamical systems, Princeton University Press, 1973).

In this paper, we study the limiting case of the Sitnikov problem. By using the variational method, we show the existence of various kinds of solutions in the planar Sitnikov problem. For a given symbolic sequence, we show the existence of orbits realizing it. We also prove the existence of periodic orbits.

Keywords

variational methods symbolic dynamics periodic solutions 

MSC2010 numbers

70F07 37K05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chenciner, A. and Montgomery, R., A Remarkable Periodic Solution of the Three-Body Problem in the Case of Equal Masses, Ann. of Math. (2), 2000, vol. 152, no. 3, pp. 881–901.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Devaney, R. L., Triple Collision in the Planar Isosceles Three-Body Problem, Invent. Math., 1980, vol. 60, no. 3, 249–267.Google Scholar
  3. 3.
    McGehee, R., Triple Collision in the Collinear Three-Body Problem, Invent. Math., 1974, vol. 27, pp. 191–227.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Meyer, K. and Wang, Q.D., Global Phase Structure of the Restricted Isosceles Three-Body Problem with Positive Energy, Trans. Amer. Math. Soc., 1993, vol. 338, no. 1, pp. 311–336.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Sitnikov, K., The Existence of Oscillatory Motions in the Three-Body Problems, Sov. Phys. Dokl., 1960, vol. 5, pp. 647–650; see also: Dokl. Akad. Nauk SSSR, 1960, vol. 133, no. 2, pp. 303–306.MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityYoshida-Honmachi, Sakyo-ku KyotoJapan

Personalised recommendations