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Geometric characterizations for variational minimizing solutions of charged 3-body problems

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Abstract

We study the charged 3-body problem with the potential function being (−α)-homogeneous on the mutual distances of any two particles via the variational method and try to find the geometric characterizations of the minimizers. We prove that if the charged 3-body problem admits a triangular central configuration, then the variational minimizing solutions of the problem in the \(\tfrac{\tau } {2}\)-antiperiodic function space are exactly defined by the circular motions of this triangular central configuration.

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References

  1. Arnold V I. Mathematical Methods of Classical Mechanics. Berlin: Springer, 1978

    Book  MATH  Google Scholar 

  2. Coti Zelati V. Periodic solutions for N-body type problems. Ann IHP Analyse non linéaire, 1990, 7(5): 477–492

    MathSciNet  MATH  Google Scholar 

  3. Gordon W. A minimizing property of Keplerian orbits. Amer J Math, 1977, 99(5): 961–971

    Article  MathSciNet  MATH  Google Scholar 

  4. Hardy G, Littlewood J, Pólya G. Inequalities. 2nd ed. Cambridge: Cambridge Univ Press, 1952

    MATH  Google Scholar 

  5. Long Y. Lectures on Celestial Mechanics and Variational Methods. Preprint. 2012

    Google Scholar 

  6. Long Y, Zhang S. Geometric characterizations for variational minimization solutions of the 3-body problem. Acta Math Sin (Engl Ser), 2000, 16: 579–592

    Article  MathSciNet  MATH  Google Scholar 

  7. Meyer K, Hall G. Introduction to Hamiltonian Systems and the N-body Problems. Berlin: Springer, 1992

    Book  MATH  Google Scholar 

  8. Moeckel R. On central configurations. Math Z, 1990, 205: 499–517

    Article  MathSciNet  MATH  Google Scholar 

  9. Perez-Chavela E, Saari D G, Susin A, Yan Z. Central configurations in the charged three body problem. Contemp Math, 1996, 198: 137–154

    Article  MathSciNet  MATH  Google Scholar 

  10. Rabinowitz P, Periodic solutions of Hamiltonian systems. Comm Pure Appl Math, 1978, 31: 157–184

    Article  MathSciNet  Google Scholar 

  11. Zhou Q, Long Y. Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems. J Differential Equations, 2015, 258: 3851–3879

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Chern Institute of Mathematics, Nankai University, Tianjin, 300071, China

    Wentian Kuang & Yiming Long

  2. Key Laboratory of Pure Mathematics and Combinatorics of the Ministry of Education, Nankai University, Tianjin, 300071, China

    Yiming Long

Authors
  1. Wentian Kuang
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  2. Yiming Long
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Correspondence to Yiming Long.

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Kuang, W., Long, Y. Geometric characterizations for variational minimizing solutions of charged 3-body problems. Front. Math. China 11, 309–321 (2016). https://doi.org/10.1007/s11464-016-0514-2

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  • DOI: https://doi.org/10.1007/s11464-016-0514-2

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