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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 173639, 8 pages
http://dx.doi.org/10.1155/2013/173639
Research Article

Periodic Solutions for Circular Restricted -Body Problems

1College of Mathematics, Sichuan University, Chengdu 610064, China
2School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China

Received 18 February 2013; Accepted 3 April 2013

Academic Editor: Baodong Zheng

Copyright © 2013 Xiaoxiao Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Chenciner and R. Montgomery, “A remarkable periodic solution of the three-body problem in the case of equal masses,” Annals of Mathematics, vol. 152, no. 3, pp. 881–901, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. A. Chenciner, “Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry,” in Proceedings of the International Mathematical Union (ICM '02), pp. 255–264, Beijing, China, 2002.
  3. A. Venturelli, “Une caractérisation variationnelle des solutions de Lagrange du problème plan des trois corps,” Comptes Rendus de l'Académie des Sciences, vol. 332, no. 7, pp. 641–644, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. C. Marchal, “How the method of minimization of action avoids singularities,” Celestial Mechanics & Dynamical Astronomy, vol. 83, no. 1–4, pp. 325–353, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. C. Moore, “Braids in classical dynamics,” Physical Review Letters, vol. 70, no. 24, pp. 3675–3679, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. C. Simó, “New families of solutions in n-body problems,” Progress in Mathematics, vol. 201, pp. 101–115, 2001. View at MathSciNet
  7. D. L. Ferrario and S. Terracini, “On the existence of collisionless equivariant minimizers for the classical N-body problem,” Inventiones Mathematicae, vol. 155, no. 2, pp. 305–362, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  8. F. Y. Li, S. Q. Zhang, and X. X. Zhao, “Periodic solutions for spatial restricted N+1-body problems,” Preprint.
  9. S. Zhang and Q. Zhou, “Symmetric periodic noncollision solutions for n-body-type problems,” Acta Mathematica Sinica, vol. 11, no. 1, pp. 37–43, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  10. S. Q. Zhang and Q. Zhou, “A minimizing property of Lagrangian solutions,” Acta Mathematica Sinica, vol. 17, no. 3, pp. 497–500, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S. Zhang and Q. Zhou, “Variational methods for the choreography solution to the three-body problem,” Science in China A, vol. 45, no. 5, pp. 594–597, 2002. View at Zentralblatt MATH · View at MathSciNet
  12. S. Zhang and Q. Zhou, “Nonplanar and noncollision periodic solutions for n-body problems,” Discrete and Continuous Dynamical Systema A, vol. 10, no. 3, pp. 679–685, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  13. S. Zhang, Q. Zhou, and Y. Liu, “New periodic solutions for 3-body problems,” Celestial Mechanics & Dynamical Astronomy, vol. 88, no. 4, pp. 365–378, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. U. Bessi and V. Coti Zelati, “Symmetries and noncollision closed orbits for planar n-body-type problems,” Nonlinear Analysis, vol. 16, no. 6, pp. 587–598, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  15. X. X. Zhao and S. Q. Zhang, “Periodic solutions for circular restricted 4-body problems with Newtonian potentials,” Preprint.
  16. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74, Springer, New York, NY, USA, 1989. View at MathSciNet
  17. M. Struwe, Variational Methods, Springer, Berlin, Germany, 3rd edition, 1990. View at MathSciNet
  18. R. S. Palais, “The principle of symmetric criticality,” Communications in Mathematical Physics, vol. 69, no. 1, pp. 19–30, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. W. B. Gordon, “A minimizing property of Keplerian orbits,” The American Journal of Mathematics, vol. 99, no. 5, pp. 961–971, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Y. Long and S. Zhang, “Geometric characterizations for variational minimization solutions of the 3-body problem,” Acta Mathematica Sinica, vol. 16, no. 4, pp. 579–592, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet