Periodic Coulomb dynamics of three equal negative charges in the field of six equal positive charges fixed in octahedron vertices

  • W. Skrypnik Institute of Mathematics of the National Academy of sciences of Ukraine
Keywords: Coulomb interaction, fixed charges, octahedron, periodic motion

Abstract

UDC 517.9

We find periodic solutions of the Coulomb equations of motion for three equal negative point charges in the field of six equal positive point charges fixed at the  vertices of a octahedron.  The system possesses an equilibrium configuration.  The center Lyapunov  theorem is applied.

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). DOI: https://doi.org/10.1007/s11253-014-0970-3

W. Skrypnik, Coulomb planar dynamics of two and three equal negative charges in field of fixed two equal positive charges, Ukr. Math. J., 68, № 11, 1528–1539 (2016). DOI: https://doi.org/10.1007/s11253-017-1326-6

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). 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). DOI: https://doi.org/10.1016/j.geomphys.2018.02.006

В. Скрипник, Періодична кулонівська динаміка двох рівних негативних зарядів у полі фіксованих чотирьох рівних позитивних зарядів, Укр. мат. журн., 72, № 10, 1432–1442 (2020); DOI: 1037863/umzh.v72i10.742. DOI: https://doi.org/10.37863/umzh.v72i10.741

В. Скрипник, Періодична кулонівська динаміка двох рівних негативних зарядів у полі фіксованих шістьох рівних позитивних зарядів, Укр. мат. журн., 72, № 12, 1682–1696 (2020); DOI: 1037863/umzh.v72i12.917. DOI: https://doi.org/10.37863/umzh.v72i12.917

W. Skrypnik, Periodic Coulomb dynamics of three equal negative charges in the field of equal positive charges fixed in octagon vertices, Adv. Math. Phys., 2020, Article ID 35467136 (2020); https://doi.org/10.1155/2020/3547136. DOI: https://doi.org/10.1155/2020/3547136

A. Lyapunov, General problem of stability of motion, Moscow (1950); English translation: Internat. J. Control, 55, № 3, 521–790 (1992).

M. S. Berger, Nonlinearity and functional analysis, Lectures Nonlinear Problems in Mathematical Analysis, Acad. Press, New York etc. (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 etc. (1971). DOI: https://doi.org/10.1007/978-3-642-87284-6

V. Nemytskii, V. Stepanov, Qualitative theory of differential equations, Moscow, 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); https://doi.org/10.1155/2548074.

C. Siegel, Über eine periodische Lösung in ebenen Drei Körper Problem, Math. Nachr., 4, 28–35 (1950–1951). DOI: https://doi.org/10.1002/mana.19500040104

W. Skrypnik, Mechanical systems with singular equilibria and Coulomb dynamics of three charges, Ukr. Math. J., 70, 519–533 (2018). DOI: https://doi.org/10.1007/s11253-018-1519-7

В. Скрипник, Періодична кулонівська динаміка трьох рівних негативних зарядів у полі фіксованих чотирьох рівних позитивних зарядів, Укр. мат. журн., 73, № 12, 1698–1713 (2021). DOI: https://doi.org/10.37863/umzh.v73i12.6550

Published
02.03.2023
How to Cite
Skrypnik, W. “Periodic Coulomb Dynamics of Three Equal Negative Charges in the Field of Six Equal Positive Charges Fixed in Octahedron Vertices”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 2, Mar. 2023, pp. 230 -46, doi:10.37863/umzh.v75i2.7263.
Section
Research articles