##
**Equilibrium, regular polygons, and Coulomb-type dynamics in different dimensions.**
*(English)*
Zbl 1478.78020

Summary: The equation of motion in \(\mathbb{R}^d\) of \(n\) generalized point charges interacting via the \(s\)-dimensional Coulomb potential, which contains for \(d=2\) a constant magnetic field, is considered. Planar exact solutions of the equation are found if either negative \(n-1>2\) charges and their masses are equal or \(n=3\) and the charges are different. They describe a motion of negative charges along identical orbits around the positive immobile charge at the origin in such a way that their coordinates coincide with vertices of regular polygons centered at the origin. Bounded solutions converging to an equilibrium in the infinite time for the considered equation without a magnetic field are also obtained. A condition permitting the existence of such solutions is proposed.

### MSC:

78A35 | Motion of charged particles |

35A01 | Existence problems for PDEs: global existence, local existence, non-existence |

35Q60 | PDEs in connection with optics and electromagnetic theory |

PDF
BibTeX
XML
Cite

\textit{W. I. Skrypnik}, Adv. Math. Phys. 2021, Article ID 6639294, 11 p. (2021; Zbl 1478.78020)

Full Text:
DOI

### References:

[1] | Vladimirov, V., Equations of Mathematical Physics (1967), Moscow: Series in Pure and applied mathematics, Moscow |

[2] | Fleischer, S.; Knauf, A., Improbability of collisions in n-body systems (2018), https://arxiv.org/abs/1802,08564v1 |

[3] | Skrypnik, W., On exact solutions of Coulomb equation of motion of planar charges, Journal of Geometry and Physics, 98, 285 (2015) · Zbl 1334.78018 |

[4] | Skrypnik, W., On regular polygon solutions of Coulomb equation of motion of n+2 charges n of which are planar, Journal of Mathematical Physics, 57, 4, article 042904 (2016) · Zbl 1342.78020 |

[5] | Skrypnik, W., Quasi-exact Coulomb dynamics of n + 1 charges n — 1 of which are equal, Reports on Mathematical Physics, 80, 73-86 (2017) · Zbl 1384.78005 |

[6] | Skrypnik, W., Mechanical systems with singular equilibria and Coulomb dynamics of three charges, Ukrainian Mathematical Journal, 70, 519 (2018) · Zbl 1426.78012 |

[7] | Siegel, C.; Moser, J., Lectures on Celestial Mechanics (1971), Berlin, Heidelberg, New-York: Springer-Verlag, Berlin, Heidelberg, New-York · Zbl 0312.70017 |

[8] | Bilogliadov, M., Equilibria of the field generated by point charges (2014), https://arxiv.org/abs/1404.7198v1 |

[9] | Gabrielov, A.; Novikov, D.; Shapiro, B., Mystery of point charges, Proceedings of the London Mathematical Society, 95, 443 (2007) · Zbl 1128.31001 |

[10] | Hartman, P., Ordinary Differential Equations (1964), New York, London, Sydney: Wiley and Sons, New York, London, Sydney · Zbl 0125.32102 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.