On exact solutions of Coulomb equation of motion of planar charges

For n >= 3 point planar charges e(j) < 0, j = 1,..., n - 1, e(n) > 0 exact Lagrange-type solutions of their Coulomb equation of motion are found. For n > 3 all the negative charges are identical and their masses are equal. These solutions describe a motion of the negative charges along keplerian orbits around the immobile positive charge in such a way that their coordinates coincide with vertices of a regular polygon. It is established that there exist equilibrium configurations such that the equal negative charges are located at the vertices of regular polygons centered at the positive charge. It is shown that there are no Lagrange-type triangular solutions for Coulomb equation of motion of three charges. The rectilinear Lagrange-type solution is shown to exist for it. (C) 2015 Elsevier B.V. All rights reserved.