ИСТИНА

The possibility of using the homotopy (deformation) method for studying the invariance of extremals for the generalized system of equations of relativistic electrodynamics is considered in this paper. This method makes it possible to investigate the stability of extremals of the action functional. It is known ([1-2]) that a generalized system of equations of relativistic electrodynamics can be obtained by solving the variational extremal problem for the sum of three action functionals, the first of which determines the freely moving charged mass of a particle in an electromagnetic field. The second functional gives the sum of the contributions determined by the four-dimensional electromagnetic Minkowski potential and the third one takes into account the more complete dynamics of the electromagnetic (or gravitational) field. The solution of this variational problem allows to obtain a generalized (Hamiltonian) system of equations of the relativistic dynamics [1]. An analysis of the Lyapunov’s stability of solutions of this system is carried out by a deformation (homotopy) method for Hamiltonian systems [3]. References 1. Eduard R. Smolyakov. Theory of the search for exact solutions of equations and the laws of motion. Moscow. Published by Russian Encyclopedia (2012). 162 P. (in Russian) 2. L.D.Landau, E.M.Lifshitz. The Classical Theory of Fields. Vol. 2 (4th ed.). 1975. Butterworth-Heinemann. 3. S.V.Emelyanov, S.K.Korovin, N.A.Bobylev and A.B.Bulatov. Homotopies of extremal problems. Moscow. Published by “Nauka” (2001). 350 P. (in Russian)