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Coulomb Dynamics Near Equilibrium of Two Equal Negative Charges in the Field of Two Fixed Identical Positive Charges

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Ukrainian Mathematical Journal Aims and scope

Periodic and quasiperiodic solutions of the Coulomb equation of motion of two equal negative charges in the field of two fixed identical positive charges are found with the help of the Lyapunov center theorem.

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References

  1. W. Skrypnik, “Periodic and bounded solutions of the Coulomb equation of motion of two and three point charges with equilibrium on the line,” Ukr. Math. J., 66, No. 5, 668–682 (2014).

    Article  MathSciNet  Google Scholar 

  2. C. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer-Verlag, Berlin etc. (1971).

  3. A. Weinstein, “Normal modes for nonlinear Hamiltonian systems,” Invent. Math., 98, 47–57 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Moser, “Periodic orbits near an equilibrium and a theorem by A. Weinstein,” Comm. Pure Appl. Math., 29, 727–747 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Moser, “On the generalization of a theorem of A. Liapunoff,” Comm. Pure Appl. Math., 11, 257–271 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Lyapunov, General Problem of Stability of Motion, Moscow (1950); English translation: Int. J. Contr., 55, No. 3, 521–790 (1992).

  7. M. S. Berger, Nonlinearity and Functional Analysis. Lectures on Nonlinear Problems in Mathematical Analysis, Academic Press, New York, etc. (1977).

  8. J. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer, New York (1976).

    Book  MATH  Google Scholar 

  9. V. Nemytsky and V. Stepanov, Qualitative Theory of Differential Equations, Moscow; Leningrad (1947).

  10. W. Skrypnik, “On exact solutions of the Coulomb equation of motion of planar charges,” J. Geom. Phys., 98, 285–291 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  11. W. Skrypnik, “On regular polygon solutions of the Coulomb equation of motion of n + 2 charges n of which are planar,” J. Math. Phys., 57, No. 10 / doi: 10.1063/1,4947421 (2004).

  12. V. Arnold, V. Kozlov, and A. Neishtadt, Mathematical Aspects of the Classical and Celestial Mechanics, Moscow (2002).

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 9, pp. 1273–1285, September, 2016.

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Skrypnik, W.I. Coulomb Dynamics Near Equilibrium of Two Equal Negative Charges in the Field of Two Fixed Identical Positive Charges. Ukr Math J 68, 1466–1480 (2017). https://doi.org/10.1007/s11253-017-1307-9

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  • DOI: https://doi.org/10.1007/s11253-017-1307-9

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