Abstract
We discuss equilibrium configurations of the Coulomb potential of positive point charges with positions satisfying certain quadratic constraints in the plane and three-dimensional Euclidean space. The main attention is given to the case of three point charges satisfying a positive definite quadratic constraint in the form of equality or inequality. For a triple of points on the boundary of convex domain, we give a geometric criterion of the existence of positive point charges for which the given triple is an equilibrium configuration. Using this criterion, rather comprehensive results are obtained for three positive charges in the disc, ellipse, and three-dimensional ball. In the case of the circle, we strengthen these results by showing that any configuration consisting of an odd number of points on the circle can be realized as an equilibrium configuration of certain nonzero point charges and give a simple criterion for existence of positive charges with this property. Similar results are obtained for three point charges each of which belongs to one of the three concentric circles. Several related problems and possible generalizations are also discussed.
Similar content being viewed by others
References
J. M. Aguirregabiria, A. Hernandez, and M. Rivas, “On the equilibrium configuration of point charges placed on an ellipse,” Comput. Phys. 4, 960–963 (1990).
V. Arnol’d, A. Varchenko, and S. Gusein-Zade, Singularities of Differentiable Mappings [in Russian], Moscow (2005).
H. Aspden, “Earnshaw’s theorem,” Am. J. Phys., 55, No. 3, 199–200 (1987).
A. Berezin, “The distribution of charges in classical electrostatics,” Nature, 317, 208–210 (1985).
P. Exner, “An isoperimetric problem for point interactions,” J. Phys. A: Math. Gen. A38, 4795–4802 (2005).
A Gabrielov, D. Novikov, and B. Shapiro, “Mystery of point charges,” Proc. London Math. Soc., 95, No. 2, 443-472 (2007).
G. Giorgadze and G. Khimshiashvili, “On nondegeneracy of certain constrained extrema,” Dokl. Math., 92, No. 3, 691-694 (2015).
G. Giorgadze and G. Khimshiashvili, “Equilibria of point charges in convex domains,” Bull. Georgian Natl. Acad. Sci., 9, No. 2, 19–26 (2015).
C. Hassell and E. Rees, “The index of a constrained critical point,” Am. Math. Mon., 100, No. 8, 772–778 (1993).
G. Khimshiashvili, “Extremal problems on configuration spaces,” Proc. A. Razmadze Math. Inst., 155, 147–151 (2011).
G. Khimshiashvili, “Equilibria of constrained point charges,” Bull. Georgian Natl. Acad. Sci., 7, No. 2, 15–20 (2013).
G. Khimshiashvili, G. Panina, and D. Siersma, ‘Coulomb control of polygonal linkages,” J. Dynam. Control Syst., 14, No. 4, 491–501 (2014).
G. Khimshiashvili, G. Panina, and D. Siersma, “Equilibria of point charges on convex curves,” J. Geom. Phys., 98, No. 2, 110–117 (2015).
G. Khimshiashvili, G. Panina, and D. Siersma, “Equilibria of three constrained point charges,” J. Geom. Phys., 106, No. 1, 42–50 (2016).
H. Munera, “Properties of discrete electrostatic systems,” Nature, 320, 597–600 (1986).
K. Nurmela, “Minimum-energy point configurations on a circular disk,” J. Phys. A, 31, No. 3, 1035–1047 (1998).
J. Palmore, “Relative equilibria of vortices in two dimensions,” Proc. Natl. Acad. Sci. USA, 79, 716-718 (1982).
W. Paul, “Electromagnetic traps for charged and neutral particles,” Rev. Mod. Phys., 62, 531-540 (1990).
S. Webb, “Minimum-energy configurations for charges on the surface of a sphere,” Chem. Phys. Lett., 129, No. 3, 310–314 (1986).
S. Webb, ‘Minimum-Coulomb-energy electrostatic configurations,” Nature, 323, No. 20, 211–215 (1986).
L. Whyte, “Unique arrangements of points on a sphere,” Am. Math. Mon., 59, No. 9, 606–611 (1952).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 102, Complex Analysis, 2017.
Rights and permissions
About this article
Cite this article
Giorgadze, G., Khimshiashvili, G. Equilibria of Three Point Charges with Quadratic Constraints. J Math Sci 237, 110–125 (2019). https://doi.org/10.1007/s10958-019-4144-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-019-4144-6