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Symmetric Periodic Orbits and Schubart Orbits in The Charged Collinear Three-Body Problem

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Abstract

Using analytic continuation of periodic orbits and symmetries we construct symmetric periodic solutions in the charged collinear three-body problem for the case where the outer particles are sufficiently small and have small charge. The results obtained allows to construct Schubart-like periodic solutions for different values of masses and charges in this setting and for the case of equal masses and charges of the outer bodies we have explicit Schubart periodic solutions.

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References

  1. Alfaro, F., Perez-Chavela, E.: Linear stability of relative equilibria in the charged three-body problem. J. Differ. Equ. 245, 1923–1944 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atela, P., McLachlan, R.: Global behavior of the charged isosceles three-body problem. Int. J. Bifurc. Chaos 4, 865–884 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Casasayas, J., Nunes, J.: A restricted charged four-body problem. Celest. Mech. Dyn. Astron. 47, 245–266 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Mansilla, J.E., Vidal, C.: Geometric interpretation for the spectral stability in the charged three-body problem. Celest. Mech. Dyn. Astron. 113, 205–213 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Antonacopoulus, G., Dionysiou, D.: Relativistic dynamics for three charged particles. Celest. Mech. Dyn. Astron. 23, 109–117 (1981)

    Article  Google Scholar 

  6. Castro Ortega, A., Lacomba, E.A.: Non-hyperbolic equilibria in the charged collinear three-body problem. J. Dyn. Differ. Equ. 24, 85–100 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. McGehee, R.: Triple collision in the collinear three body problem. Invent. Math. 27, 191–227 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  8. Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste, Dover 1892–1899 (1957)

  9. Broucke, R., Walker, D.E.: Numerical explorations of the rectilinear problem of three bodies. Celest. Mech. Dyn. Astron. 21, 73–81 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kammeyer, P.C.: Symmetric rectilinear periodic orbits of three bodies. Celest. Mech. Dyn. Astron. 30, 329–342 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Corbera, M., Llibre, J.: Symmetric periodic orbits for the collinear three-body problem via the continuation method. Hydrodyn. Dyn. Syst. C 113–137 (2003)

  12. Schubart, J.: Numerische Aufsuchung periodischer Lsungen im Dreikrperproblem. Astron. Nachr. 283, 17–22 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  13. Moeckel, R.: A topological existence proof for the Schubart orbits in the collinear three-body problem. Discr. Contin. Dyn. Syst. Ser. B 10, 609–620 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Venturelli, A.: A variational proof for the existence of Von Schubart’s orbit. Discr. Contin. Dyn. Syst. Ser. B 10(2–3), 699–717 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Levi-Civita, T.: Sur la regularisation du probleme des trois corps. Acta Math. 42, 99–144 (1920)

    Article  MathSciNet  Google Scholar 

  16. Szebehely, V.: Theory of Orbits. Academic Press, New York (1967)

    Google Scholar 

Download references

Acknowledgments

The first author is pleased to acknowledge the financial support from DGAPA which allows him a postdoctoral stay in the Department of Mathematics of the Faculty of Sciences, UNAM. The second author acknowledge the partial support from PAPIIT IN111410.

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Authors and Affiliations

  1. Depto. de Matemáticas, Fac. de Ciencias, UNAM, C. Universitaria, México, D.F., 04510, Mexico

    Alberto Castro Ortega & Manuel Falconi

  2. Depto. de Matemáticas, UAM-Iztapalapa, México, D.F., 09340, Mexico

    Ernesto A. Lacomba

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  1. Alberto Castro Ortega
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  2. Manuel Falconi
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  3. Ernesto A. Lacomba
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Correspondence to Alberto Castro Ortega.

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Castro Ortega, A., Falconi, M. & Lacomba, E.A. Symmetric Periodic Orbits and Schubart Orbits in The Charged Collinear Three-Body Problem. Qual. Theory Dyn. Syst. 13, 181–196 (2014). https://doi.org/10.1007/s12346-014-0112-1

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  • DOI: https://doi.org/10.1007/s12346-014-0112-1

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