Geodesic
On three point charges on a flexible arc
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 177 (2020), pp. 63-68
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We consider equilibrium configurations of three mutually repelling point charges with the Coulomb interaction confined to a simple arc of a constant length with fixed positions of ends. For given values of the charges, the length of the arc, and the distance between its ends, we calculate all possible equilibrium configurations. We also study the behavior of equilibrium configurations for variable values of charges and show that a unique possible bifurcation is the pitchfork bifurcation. Similar results are presented for elastic loop obeying Hooke's law and for charges interacting via a Riesz potential.
Mots-clés : equilibrium configuration, bifurcation
Keywords: Riesz potential, point charge. 
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     title = {On three point charges on a flexible arc},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
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     year = {2020},
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}
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G. Giorgadze; G. N. Khimshiashvili. On three point charges on a flexible arc. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 177 (2020), pp. 63-68. http://geodesic.mathdoc.fr/item/INTO_2020_177_a5/

[1] Exner P., “An isoperimetric problem for point interactions”, J. Phys. A: Math. Gen., A38 (2005), 4795–4802 | DOI | MR | Zbl

[2] Giorgadze G., Khimshiashvili G., “Concyclic and aligned equilibrium configurations of point charges”, Proc. I. Vekua Inst. Appl. Math., 67 (2017), 20–33 | MR | Zbl

[3] Khimshiashvili G., “Equilibria of constrained point charges”, Bull. Georgian Natl. Acad. Sci., 7:2 (2013), 15–20 | MR | Zbl

[4] Khimshiashvili G., “Equilibria of point charges on elastic contour”, Bull. Georgian Natl. Acad. Sci., 11:4 (2017), 7–12 | MR | Zbl

[5] Khimshiashvili G., Panina G., and Siersma D., “Equilibria of three constrained point charges”, J. Geom. Phys., 98:2 (2015), 110–117 | DOI | MR | Zbl

[6] Stieltjes T. J., “Sur les racines de l'equation $X_n=0$”, Acta Math., 9 (1886), 385–400 | DOI | MR