Giorgi Khimshiashvili
Maxwell's conjecture for three aligned point charges
We will present several results concerned with the J.C. Maxwell's conjecture on the number of equilibrium points of a system of point charges. Specifically, we will show that this conjecture holds true for three aligned point charges of arbitrary signs and magnitudes. To this end, we use an analytic approach developed in our two joint papers with G. Giorgadze published in J. Math. Phys. (vol. 62, no. 5, 2021) and in Georgian Math. J. (vol. 29, no. 4, 2022). More concretely, for a triple of non-aligned points, we compute the so-called canonical stationary charges and describe their behavior under the deformations of the reference triangle, which is the crucial point in the proof. We also establish functorial properties of the nonlinear system of coordinates in the ambient plane given by the canonical charges and present some geometric applications of such coordinates.
