byĬhoosing the function z (a, y) that is involved in the equation of this surface z = z(a, y). It is fully determined by choosing a definite surface, i.e. Likewise, the area 8 of a surface is a functional. As soon as the equation of the curve 1J = y(a ) is given, the value Z can be calculated, namely, '!Jįor fustance, the length Zof a curve joining two given points on the plane is a functional, because this length is folly determined by choosing a definite function 'JI = y (a ), the graph of which passes through these points (Fig. Functionals a.re variable values which depend on a variable running through a set of functions, or on a finite number of such variables, and which are completely determined by a definite choice of these variable function~. INTRODUCTION Along with the problems of discovering the maximum or minimum values of a given function z = f (a ), in engineering practice we often have to find the maxima or the minima of the values of mathematical entities called f ul¥UJtionals. large number of problems some of which a.re taken from existing textbooks. The aim of this book is to provide engineers and students of colleges of technology with the opportunity of becoming familiar with the basic notions and standard methods of the calculus of variations including the direct methods of solution of the variational problems, which are important from the practical point of view. THE PREFACE TO THE FIRST RUSSIAN EDITION There ha.s been in recent yea.rs a wide variety of applications of variational methods to various fields of mechanics and technology a.nd this is why engineers, of many kinds, are faced with the necessity of learning the fundamentals of the calculus of variations. DIRECT METHODS OF SOLVING VARIATIONAi.A PROBLEl\iIS 1. Constraints of the form 'P (z, yi, y2 ,ģ. VAltf.A.TIONAL PROBLEMS OF CONSTRAINED EXTREMA 1. SUFFICIENCY CONDITIONS FOR AN EXTREMUM 1. Problems with movable boundaries for functionals of the ZlĤ. Problems with movable boundaries for fnnetionals of
VARIATIONAL PROBLEMS WITH MOVABLE BOUNDARIES AND SOME OTHER PROBLEl\IS 1. Parametric representation of variational problems 7. Functionals depending on funetions of several independent variables. Functional.P., involving derivatives of higher order. International Standard Book Number: 0-48fr45 799-0 Manufactured in the United States of America Dover Publications, Inc., 31East2nd Street, Mineola, N.Y 11501ĬONTENTS From the preface to the first Russian edition. In this edition, two blank pages at the beginning of the original edition have been omitted, so that the present book begins on page 3.
Mineola, New Yorkīibliographical Note This Dover edition, first published in 2007, is an unabridged republication of the work published by Pergamon Press, NewYork, 1961.
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