On the history of variational methods of non-linear equations investigations and the contribution of Soviet scientists (1920s-1950s).

Egor Mikhailovich Bogatov. "On the history of variational methods of non-linear equations investigations and the contribution of Soviet scientists (1920s-1950s).." Antiquitates Mathematicae (2020): null. <http://eudml.org/doc/295116>.

@article{EgorMikhailovichBogatov2020,
abstract = {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.},
author = {Egor Mikhailovich Bogatov},
journal = {Antiquitates Mathematicae},
keywords = {history of nonlinear functional analysis, variational methods, Dirichlet principle, Hammerstein equation, Ritz method, potential operators, Lyusternik-Shnirelman theory, Sobolev space, semi-continuous functionals, justification of linearization in the bif},
language = {eng},
pages = {null},
title = {On the history of variational methods of non-linear equations investigations and the contribution of Soviet scientists (1920s-1950s).},
url = {http://eudml.org/doc/295116},
year = {2020},
}

TY - JOUR
AU - Egor Mikhailovich Bogatov
TI - On the history of variational methods of non-linear equations investigations and the contribution of Soviet scientists (1920s-1950s).
JO - Antiquitates Mathematicae
PY - 2020
SP - null
AB - 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.
LA - eng
KW - history of nonlinear functional analysis, variational methods, Dirichlet principle, Hammerstein equation, Ritz method, potential operators, Lyusternik-Shnirelman theory, Sobolev space, semi-continuous functionals, justification of linearization in the bif
UR - http://eudml.org/doc/295116
ER -