Calculus of variations in the large: birth, formation, applications. (Russian. English summary) Zbl 1539.49001

Boeva, Anna Vyacheslavovna; Mukhin, Ravil’ Rafkatovich

Chebyshevskiĭ Sb. 24, No. 3(89), 263-288 (2023).

Summary: The work is devoted to the evolution of the concepts and methods of the calculus of variations in the large, a branch of mathematics that is a little over a century old. The subject of this area is the study of qualitative characteristics of variational problems. In the development of the calculus of variations in the large several periods can be distinguished with features inherent in each of them. The first period is defined from the end of the 19th century. until the end of the 1940s, when the theory was born and formed, which was formed from two main parts – the Morse theory and the theory of Lyusternik-Shnirelman categories. Here, the features of traditional mathematics are still noticeable. In the next period - from the end of the 1940s to the end of the 1970s. the calculus of variation in the large was formed as a separate area of mathematics, and it acquired its modern form, based on the concepts and methods of algebraic topology. Ample opportunities opened up for solving new problems in mathematics, and a number of impressive results were obtained. The modern period can be defined from the late 1970s. until now. Its main feature is the unprecedented convergence of mathematics and the field of its applications, especially with physics. It has not always been possible to indicate a distinguishable boundary between the two fields of science; even the term “physical mathematics” has appeared. The calculus of variations in the large is included in the qualitative theory, which represents a significant part of modern mathematics and finds wide applications. But its place is even more significant, it is one of the foundations that forms our worldview.

MSC:

49-03	History of calculus of variations and optimal control
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
01A55	History of mathematics in the 19th century
01A60	History of mathematics in the 20th century
01A61	History of mathematics in the 21st century
49K10	Optimality conditions for free problems in two or more independent variables
49J10	Existence theories for free problems in two or more independent variables

Keywords:

critical point; extremal; geodesic line; topology; category; first period: end of 19th century until end of the 1940s; second period: end of 1940s to end of 1970s; modern (third) period: late 1970s until now; calculus of variations in the large; Morse theory; Lyusternik-Shnirelman categories

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