Ж.-Л. Лагранж Как Один Из Основоположников Теории Экстремумов Функций Многих Переменных

Abstract

In the article we carried out a detailed analysis of the results obtained by J.-L. Lagrange in his first. The theory of extrema of functions of many variables, as part of mathematical analysis, refers to the mathematical foundations of the study of operations. In turn, many optimization problems are actually problems on the conditional extremum of the function of many variables. The relevance of this topic is determined by the fact that the methods for solving problems on the extremum of the function of many variables obtained in the mid 18th - early 20th centuries are used in solving modern problems. A special place here is occupied by L. Euler and J.-L. Lagrange. The aim of the article is to study the conditions for the maximum and minimum functions of many variables obtained by J.-L. Lagrange, and a comparison of its results with the presentation of this topic in modern textbooks on higher mathematics and mathematical analysis. It was established that in his first printed work he first formulated and proved sufficient conditions for the existence of an extremum of the function of many variables by actually establishing a criterion for the positive (negative) definiteness of quadratic forms, long before it appeared in J. Sylvester in the mid-19th century. A comparative analysis of the results of L. Euler and J.-L. Lagrange. It was found that sufficient conditions for the existence of an extremum of functions of many variables obtained in the first printed work of the young Lagrange are included in all modern textbooks in virtually the same form. The examples shown illustrate his theory. These are tasks of geometric and physical content. Special cases are considered in detail: functions of two and three variables. It is noted that this article became programmatic for the young Lagrange, although it remained unnoticed by his contemporaries. Subsequently, based on the method he obtained, he created the variational calculus, using the principle of least action and the theory of extrema, derived the basic laws of mechanics, the rule of factors for finding the conditional extremum of functions of many variables, which is named after him.