Gelfand, I. M., Fomin, S. V. Calculus of Variations (Variatsionnoe ischislenie), 1961. In Russian.

Гельфанд И. М., Фомин С. В. Вариационное исчисление.
Москва : Физматгиз, 1961.
228 с. : ил. Твёрдый издательский переплёт, обычный формат.
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Gelfand, I. M., Fomin, S. V. Calculus of Variations (Variatsionnoe ischislenie).
Moscow : Fizmatgiz, 1961.
228 pp.: ill. Publisher’s hardcover, ordinary format.

This 1961 first edition, published by Fizmatgiz, is a legendary cornerstone of the Soviet mathematical school. Written by the titan of 20th-century mathematics Israel Gelfand and his colleague Sergei Fomin, this volume is celebrated globally for its extraordinary clarity, elegance, and economy of prose. It remains one of the most effective gateways into a field that is vital for both pure mathematics and theoretical physics.
The book meticulously covers the fundamental methods used to find extrema of functionals. Starting with the classic problems that birthed the field—such as the brachistochrone and the isoperimetric problem—the authors lead the reader through the derivation of the Euler-Lagrange equations. The text is structured to provide a rigorous logical progression, moving from the first variation to the more complex analysis of the second variation and Jacobi's necessary conditions for an extremum.
A key strength of Gelfand and Fomin’s approach is the inclusion of direct methods in the calculus of variations and an introduction to field theory (in the context of the Weierstrass function). The authors also address problems with movable boundaries and the transition to Hamiltonian mechanics, bridging the gap between classical analysis and modern theoretical physics. The diagrams and geometric interpretations provided are specifically designed to build physical intuition alongside mathematical rigor.
Part of the famous series of Gelfand’s textbooks, this 1961 edition is a masterpiece of pedagogical design. Its influence spans decades, having been translated into numerous languages and remaining a staple on the syllabi of leading universities worldwide. For the collector of mathematical classics or the student seeking the most "Gelfandian" (concise and profound) path to mastery, this original Soviet printing is an indispensable acquisition.