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We give the complete solution of the extremal problem posed by N.G. Tchebotaröv in 20th of the last century, and we establish explicit parametric formulae for the extremals.

The stability of the configuration of a non-linear electromagnetic field corresponding to a charged point source is studied. The static classical solution of the field equations, describing charged particles has been modified by a small perturbation obeying linearised field equations. Assuming time dependence of the perturbation in the form exp(-iwt) it has been shown that w 2 is non-negative and the classical static configuration of the field describing charged particles is stable. Detailed analysis of the linearised field equations has been performed for the Born-Infeld model of the non-linear electromagnetic field. There is only a continuous spectrum of w 2 with certain distinct values of o corresponding to the resonant states. The problem of the zero modes connected with translational invariance is also studied. Jackiw R 1977 Rev. Mod. Phys. 49 681 Nielsen H and Olesen P 1973 Nucl. Phys. B 61 45 Plebahski J 1970 Lectures on nonlinear electrodynamics, N O R D l T A lectures, Copenhagen Schwinger J 1951 Phys. Rev. 82 664

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