Skip to main content
SpringerLink
Log in

Non-Hyperbolic Equilibria in the Charged Collinear Three-Body Problem

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

Consider three charged masses moving along the line. For this model we study the solutions near total collision using blow up techniques obtaining that for given masses and charges the vector field on the collision manifold has a non-hyperbolic equilibrium point. To study this situation the vector field is written in normal form and the center manifold theory is used obtaining that all nonzero solutions near the origin escape to infinity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alfaro F., Pérez-Chavela E.: The rhomboidal charged four body problem. Hamiltonian systems and celestial mechanics (HAMSYS 98). World Sci. Monogr. Series Math. 6, 1–19 (2000)

    Article  Google Scholar 

  2. Arnold V.I.: Dynamical systems I. Springer-Verlag, New York (1981)

    Google Scholar 

  3. Ashkenazi, M., Chow, S.: Normal forms near critical points for differential equations and maps. IEEE Transactions on circuits and systems, 35 (1998)

  4. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Carr J.: Applications of center manifold theory. Springer-Verlag, New York (1981)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen G., Dora, J.Della: Normal forms for differentiable maps near a fixed point. Numer. Algoritm. 22, 213–230 (1999)

  9. Devaney, R.: Singularities in Classical and celestial mechanics, In: Ergodic theory and dynamical systems, I. Progress in mathematics, vol. 10, pp. 211–233. Birkhauser, Boston (1981)

  10. Lacomba E.A.: Infinity manifolds for positive energy in celestial mechanics. Contemp. Math. 58, 193–201 (1987)

    MathSciNet  Google Scholar 

  11. McGehee R.: A stable manifold theory in celestial mechanics. J. Differ. Equ. 14, 70–88 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  12. McGehee R.: Triple collision in the collinear three-body problem. Invent. Math. 27, 192–226 (1974)

    Article  MathSciNet  Google Scholar 

  13. Sano M.: The classical Couloumb three-body problem in the collinear eZe configuration. J. Phys. A 37, 803–822 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pérez-Chavela E., Saari D., Susin A., Yan Z.: Central configurations in the charged three-body problem. Contemp. Math. 198, 137–155 (1996)

    Article  Google Scholar 

  15. Wiggins S.: Introduction to applied nonlinear dynamical systems and chaos. Springer-Verlag, New York (1990)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, México, 09340, DF, México

    Alberto Castro Ortega & Ernesto A. Lacomba

Authors
  1. Alberto Castro Ortega
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ernesto A. Lacomba
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Castro Ortega.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Castro Ortega, A., Lacomba, E.A. Non-Hyperbolic Equilibria in the Charged Collinear Three-Body Problem. J Dyn Diff Equat 24, 85–100 (2012). https://doi.org/10.1007/s10884-011-9237-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-011-9237-y

Keywords

Search

Navigation