Periodic Coulomb dynamics of two equal negative charges in the field of six equal positive fixed charges
Abstract
UDC 517.9
Periodic solutions of the Coulomb $d$-dimensional $(d=1,2,3)$ equation of motion for two equal negative point charges in the field of six equal positive point charges fixed at vertices of a convex symmetric hexagon and octahedron are found.
These systems possess an equilibrium configuration.
Their periodic solutions are obtained with the help of the Lyapunov central limit theorem.
References
W. Skrypnik, Periodic and bounded solutions of the Coulomb equation of motion of two and three point charges with equilibrium on line, Ukr. Math. J., 66, № 5, 668 – 682 (2014), https://doi.org/10.1007/s11253-014-0970-3 DOI: https://doi.org/10.1007/s11253-014-0970-3
W. Skrypnik, Kulonivs`ka dinamika dvokh ta tr`okh rivnikh vid'yemnikh zaryadiv na ploshhini u poli fiksovanikh dvokh rivnikh dodatnikh zaryadiv, Ukr. mat. zhurn., 68, № 11, 1528 – 1539 (2016).
W. Skrypnik, Coulomb dynamics near equilibrium of two equal negative charges in the field of fixed two equal positive charges, Ukr. Math. J., 68, № 9, 1273 – 1285 (2016), https://doi.org/10.1007/s11253-017-1307-9 DOI: https://doi.org/10.1007/s11253-017-1307-9
W. Skrypnik, Coulomb dynamics of three equal negative charges in field of fixed two equal positive charges, J. Geom. and Phys., 127, 101 – 111 (2018), https://doi.org/10.1016/j.geomphys.2018.02.006 DOI: https://doi.org/10.1016/j.geomphys.2018.02.006
A. Lyapunov, General problem of stability of motion, Moscow, (1950); English translation: Internat. J. Control, 55, № 3, 521 – 790 (1992), https://doi.org/10.1080/00207179208934253
M. S. Berger, Nonlinearity and functional analysis. Lectures on Nonlinear Problems in Math. Analysis, Acad. Press, New York ect. (1977).
J. Marsden, M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag, New York (1976). DOI: https://doi.org/10.1007/978-1-4612-6374-6
C. Siegel, J. Moser, Lectures on celestial mechanics, Springer-Verlag, Berlin ect. (1971). DOI: https://doi.org/10.1007/978-3-642-87284-6
V. Nemy`czkij, V. Stepanov, Kachestvennaya teoriya differenczial`ny`kh uravnenij, Moskva, Leningrad (1947).
W. Skrypnik, Coulomb planar periodic motion of n equal charges n the field of n equal positive charges fixed at a line and constant magnetic field, Adv. Math. phys., 2018, Article ID 2548074, (2018), 10 p., https://doi.org/10.1155/2018/2548074 DOI: https://doi.org/10.1155/2018/2548074
C. Siegel, Über eine periodische Lösung im ebenen Dreikörperproblem. (German), Math. Nachr., 4, 28 – 35 (1950-1951), DOI: https://doi.org/10.1002/mana.19500040104
Copyright (c) 2020 В. І. Скрипник
This work is licensed under a Creative Commons Attribution 4.0 International License.
