Variational methods for solving nonlinear equations (differential, integral, etc.) are perhaps the most common methods at the present time. However, the history of their origin and development, both in the USSR and in the whole world, has not been studied enough. The author attempts to fill this gap, limiting himself mainly to 1920's-1950's, starting with Hilbert’s works on justification of Dirichlet principle and with Poincaré’s work on the study of closed geodesics on convex surfaces. The article includes an analysis of the results of Soviet mathematicians - L.А. Lyusternik, L.G. Shnirelman, S.L. Sobolev, M.A. Krasnoselskii, M.M. Vainberg et al., obtained both in the field of solvability and direct methods, and in the field of qualitative analysis. Their achievements are gauged in the context of the development of this topic in the works of their foreign colleagues - L. Tonelli, W. Ritz, L. Lichtenstein, G.D. Birkhoff, M. Morse, A. Hammerstein, M. Golomb, etc. Along the way, the question of the origin of Sobolev spaces and the mutual influence of functional analysis and variational methods for solving operator equations is investigated. In conclusion, the author gives an example of applying the qualitative theory of variational methods to the problems of nonlinear mechanics that was realized by the Soviet scientists I.I. Vorovich.
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