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Jump to Page. Search inside document. Englewood Cliffs, N. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. However, the book goes considerably beyond the material actually presented in the lectures.

Our aim is to give a treatment of the ele- ments of the calculus of variations in a form which is both easily understandable and sufficiently modern. Considerable attention is devoted to physical applica- tions of variational methods, e. The reader who merely wishes to become familiar with the most basic concepts and methods of the calculus of variations need only study the first chapter.

The first three chapters, taken together, form a more compre- hensive course on the elements of the calculus of varia- tions,-but one which is still quite elementary involving only necessary conditions for extrema. The first six chapters contain, more or less, the material given in the usual university course in the calculus of variations with applications to the mechanics of systems with a finite number of degrees of freedom , including the theory of fields presented in a somewhat novel way and sufficient conditions for weak and strong extrema.

Chapter 7 is devoted to the application of variational methods to the study of systems with infinitely many degrees of freedom. Chapter 8 contains a brief treat- ment of direct methods in the calculus of variations.

The authors are grateful to M. Yevgrafov and A. Kostyuchenko, who read the book in manuscript and made many useful comments. The present edition is rather different from the Russian original. In so doing, I have had two special assets: 1 A substantial list of revisions and corrections from Professor S.

Fomin himself, and 2 A variety of helpful suggestions from Professor J. Schwartz of New York University, who read the entire translation in typescript. The problems appearing at the end of each of the eight chapters and two appendices were made specifically for the English edition, and many of them comment further on the corresponding parts of the text.

A variety of Russian sources have played an important role in the synthesis of this material. In particular, I have consulted the textbooks on the calculus of variations by N. Akhiezer, by L. Elsgolts, and by M. Lavrentev and L. Lyusternik, as well as Volume 2 of the well- known problem collection by N. Gyunter and R. Kuzmin, and Chapter 3 of G.

This list is not intended as an exhaustive cata- log of the literature, and is in fact confined to books available in English.

Some Simple Variational Problems, 1. A Necessary Condition for an Extremum, 8. Problems, Variational Problems in Parametric Form, The Second Variation of a Func- tional, More on Conjugate Points, General Definition of a Field, Sufficient Conditions for a Strong Extremum, Some Simple Variational Problems Variable quantities called functionals play an important role in many problems arising in analysis, mechanics, geometry, etc.

By a functional, we mean a correspondence which assigns a definite real number to each function or curve belonging to some class. Thus, one might say that a functional is a kind of function, where the independent variable is itself a function or curve. The following are examples of functionals: 1. Consider the set of all rectifiable plane curves. Thus, the length of a curve is a functional defined on the set of rectifiable curves. Suppose that each rectifiable plane curve is regarded as being made out of some homogeneous material.

Then if we associate with each such curve the ordinate of its center of mass, we again obtain a functional. Consider all possible paths joining two given points A and B in the plane. Suppose that a particle can move along any of these paths, and let the particle have a definite velocity v x, y at the point x, y.

Then we obtain a functional by associating with each path the time the particle takes to traverse the path. If this limit exists and is finite, the curve is said to be rectifiable. As a more general example, let F x, y, z be a continuous function of three variables. By choosing different functions F x, y,z , we obtain different functionals. In what follows, we shall be concerned mainly with functionals of the form 1.

Particular instances of problems involving the concept of a functional were considered more than three hundred years ago, and in fact, the first important results in this area are due to Euler We now indicate some typical examples of variational problems, by which we mean problems involving the determination of maxima and minima of functionals.

The curve in question turns out to be the straight line segment joining A and B. Thus, as an approxi- mation, we can regard the variational problem as the problem of finding the extrema of the function J ;, In solving variational problems, Euler made extensive use of this method of finite differences. By replacing smooth curves by polygonal lines, he reduced the problem of finding extrema of a functional to the problem of finding extrema of a function of n variables, and then he obtained exact solutions by passing to the limit as n— oo.

Function Spaces In the study of functions of n variables, it is convenient to use geometric language, by regarding a set of n numbers y;, In just the same way, geometric language is useful when studying functionals.

Thus, we shall regard each function x belonging to some class as a point in some space, and spaces whose elements are functions will be called function spaces. In the study of functions of a finite number n of independent variables, it is sufficient to consider a single space, i. In fact, the nature of the problem under consideration determines the choice of the function space. For example, if we are dealing with a functional of the form [ Fes yy ax, it is natural to regard the functional as defined on the set of all functions with a continuous first derivative, while in the case of a functional of the form Pf Fes xy.

Therefore, in studying functionals of various types, it is reasonable to use various function spaces. The concept of continuity plays an important role for functionals, just as it does for the ordinary functions considered in classical analysis. This is most conveniently done by introducing the concept of the norm of a function, analogous to the concept of the distance between a point in Euclidean space and the origin of coordinates.

Although in what follows we shall always be concerned with function spaces, it will be most convenient to introduce the concept of a norm in a more general and abstract form, by introducing the concept of a normed linear space. Silverman, Prentice-Hall, Inc. The elements of a normed linear space can be objects of any kind, e. The following normed linear spaces are important for our subsequent purposes: 1.

The space , or more precisely a, b , consisting of all continuous functions y x defined on a closed interval [a,b]. By addition of elements of and multiplication of elements of by numbers, we mean ordinary addition of functions and multiplication of functions by numbers, while the norm is defined as the maximum of the absolute value, i.

This contradiction proves the lemma. If a x is continuous in [a, b , and if. The next lemma will be needed in Chapter 8: Lemma 3. Lemma 4. We emphasize that the differentiability of the function B x was not assumed in advance. We now introduce the concept of the variation or differential of a functional.

If y is fixed, AJ[h] is a functional of A, in general a nonlinear functional.

After that, going from two to three was just more algebra and more complicated pictures. The first part starts with the statement of the fundamental variational problem and its solution via the Euler-Lagrange equation. An excellent introduction to the calculus of variations with application to various problems of physics. The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Exercises throughout. Unfortunately, a few books that are available are written at a level which is not easily comprehensible for postgraduate students. This book, written by a highly respected academic, presents the materials.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. It should be fairly complete; build up gradually from the very basics; offer detailed explanations; have some emphasis on applications of variational methods. This book may be worth checking out. I began learning from it and found it pretty thorough.

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CALCULUS. OF VARIATIONS. I. M. GELFAND. S. V. FOMIN. Moscow State University. Revised English Edition. Translated and Edited by. Richard A. Silverman.

This concise text offers both professionals and students an introduction to the fundamentals and standard methods of the calculus of variations. Logan - Applied Mathematics, Second Edition -John Wiley At this introductory course we will focus on the origins of calculus of variations: the study of the extrema1 of functionals de ned on in nite dimensional function vector spaces with real Calculus of Variations and Integral Equations. The main aim of the paper is to start the higher-order calculus of variations and the theory of optimal control for GSF. Total no.

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The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals , to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functions that maximize or minimize functionals may be found using the Euler—Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics.

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*Беккер посмотрел вниз, на свои ноги.*