Calculus of Variations with Applications

370 Pages·2016·1.41 MB·English

by  George M. Ewing

CALCULUS OF
VARIATIONS
WITH
APPLICATIONS
George M. Ewing
George L. Cross Research Professor
of Mathematics Emeritus,
University of Oklahoma, Norman
DOVER PUBLICATIONS, INC.
NEW YORK
Copyright © 1969, 1985 by George M. Ewing.
All rights reserved under Pan American and International Copy­
right Conventions.
Published in Canada by General Publishing Company, Ltd., 30
Lesmill Road, Don Mills, Toronto, Ontario.
Published in the United Kingdom by Constable and Company,
Ltd., 10 Orange Street, London WC2H 7EG.
This Dover edition, first published in 1985, is an unabridged,
corrected and slightly enlarged republication of the work first pub­
lished by W. W. Norton 8c Co., New York, 1969. The author has
supplied a new Preface and a Supplementary Bibliography for this
edition.
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
Library of Congress Cataloging in Publication Data
Ewing, George M. (George McNaught), 1907-
Calculus of variations with applications.
Reprint. Originally published: New York : W. W. Norton, cl969.
With new pref. and supplemental bibliography.
Bibliography: p.
1. Calculus of variations. I. Title.
QA315.E9 1985 515'.64 84-18870
ISBN 0-486-64856-7
Contents
PREFACE TO THE DOVER EDITION vii
PREFACE TO THE ORIGINAL EDITION vii
Chapter 1, Introduction and Orientation
1.1 Prerequisites 1
1.2 Functions 1
1.3 The Extended Real Numbers 3
1.4 Bounds, Maxima, and Minima 4
1.5 Limits 5
1.6 Continuity and Semi-Continuity 8
1.7 Derivatives 9
1.8 Piecewise Continuous Functions 11
1.9 Continuous Piecewise Smooth Functions 12
1.10 Metric Spaces 13
1.11 Functions Defined Implicitly 15
1.12 Ordinary Differential Equations 16
1.13 The Riemann Integral 18
1.14 What Is the Calculus of Variations? 21
Chapter 2, Necessary Conditions for an Extremum
2.1 Introduction 26
2.2 The Fixed-Endpoint Problem in the Plane 27
2.3 Minima of Ordinary Point-Functions 28
2.4 Different Kinds of Minima ofJ (y) 29
2.5 The Lemma of du Bois Reymond 31
2.6 The Euler Necessary Condition 32
2.7 Examples 36
2.8 The Weierstrass Necessary Condition 39
2.9 The Erdmann Corner Conditions 42
2.10 The Figurative 44
2.11 The Legendre Necessary Condition 45
2.12 The Jacobi Necessary Condition 46
2.13 Other Forms of the Jacobi Condition 48
2.14 Concluding Remarks 51
Chapter 3, Sufficient Conditions for an Extremum
3.1 Introduction 53
3.2 Fields 55
3.3 The Hilbert Integral 57
iii
3.4 The Fundamental Sufficiency Theorem 58
3.5 Examples 59
3.6 Sufficient Combinations of Conditions 62
3.7 Problems for Which Condition III' Fails 64
3.8 Sufficient Conditions When There Is a Corner 69
3.9 Extensions, Other Methods 71
3.10 Convex Sets and Convex Point-Functions 72
3.11 Convexity of Integrals and Global Minima 77
3.12 A Naive Expansion Method 80
Chapter 4, Variations and Hamilton's Principle
4.1 Introduction 85
4.2 The Operator 8 86
4.3 Formal Derivation of the Euler Equation 88
4.4 The Second Variation 90
4.5 Concluding Remarks on the 8-Calculus 92
4.6 Introduction to Hamilton’s Principle 93
4.7 Examples 95
4.8 Side-Conditions and New Coordinates 97
4.9 The Generalized Hamilton Principle 101
4.10 Applications to Electric Networks 103
4.11 Concluding Remarks 104
Chapter 5, The Nonparametric Problem of Bolza
5.1 Introduction 106
5.2 Examples 107
5.3 Formulation of the Problem of Bolza 108
5.4 Alternative Forms of a Problem 110
5.5 Constrained Extrema of Point-Functions 111
5.6 Different Kinds of Extrema 113
5.7 The Multiplier Rule 115
5.8 Normality 120
5.9 Application of the Multiplier Rule to Examples 121
5.10 Further Necessary Conditions, Sufficient
Conditions for Local Extrema 129
5.11 Sufficient Conditions for Global Extrema 129
5.12 Analysis of a Problem from Rocket Propulsion 134
5.13 Concluding Remarks 138
Chapter 6, Parametric Problems
6.1 Introduction 140
6.2 What Is a Curve? 141
6.3 Frechet Distance between Mappings 142
6.4 Frechet Distance between Curves 146
6.5 Piecewise Smooth Curves 147
6.6 Parametric Integrals and Problems 148
6.7 Homogeneity of Parametric Integrands 150
6.8 Consequences of the Homogeneity of F 152
6.9 The Classical Fixed-Endpoint Parametric Problem 155
6.10 The Classical Parametric Problem of Bolza 156
6.11 The Euler Necessary Condition 156
6.12 Necessary Conditions of Weierstrass and Legendre 159
6.13 Related Parametric and Nonparametric Problems
of Like Dimensionality 163
6.14 An Addendum to the Euler Condition for a
Nonparametric Integral 165
6.15 Related Parametric and Nonparametric Problems
of Different Dimensionality 166
6.16 Concluding Remarks 167
Chapter 7, Direct Methods
7.1 Introduction 168
7.2 Global Extrema of Real-Valued Functions 169
7.3 Length of a Mapping 170
7.4 Lower Semi-Continuity of Length 171
7.5 Length of a Curve 173
7.6 The Representation in Terms of Length 173
7.7 The Hilbert Compactness Theorem 177
7.8 The Ascoli-Arzela Theorem 181
7.9 The Helly Compactness Theorem 183
7.10 The Weierstrass Integral 187
7.11 Existence Theorems for Parametric Problems 192
7.12 Nonparametric Weierstrass Integrals 200
Chapter 8, Measure, Integrals, and Derivatives
8.1 Introduction 202
8.2 Linear Lebesgue Outer Measure 203
8.3 Lebesgue Measurability and Measure 204
8.4 Measurable Functions 210
8.5 The Lebesgue Integral 214
8.6 Convergence Theorems 220
8.7 Other Properties of Integrals 224
8.8 Functions of Bounded Variation 227
8.9 The Vitali Covering Theorem 231
8.10 Derivatives of Functions of Bounded Variation 234
8.11 Indefinite Integrals 237
Chapter 9, Variational Theory in Terms of
Lebesgue Integrals
9.1 Introduction 245
9.2 Variational Integrals of the Lebesgue Type 246
9.3 The Lebesgue Length-Integral 251
9.4 Convergence in the Mean and in Length 255
9.5 Integrability of Parametric and Nonparametric
Integrands; Weierstrass Integrals 256
9.6 Normed Linear Spaces 259
9.7 The ¿¿-Spaces 261
9.8 Separability of the Space Lp([a,b]) 265
9.9 Linear Functionals and Weak Convergence 270
9.10 The Weak Compactness Theorem 274
9.11 Applications 280
9.12 Concluding Remarks 285
Chapter 10, A Miscellany of Nonclassical Problems
10.1 Introduction 286
10.2 Problems Motivated by Rocket Propulsion 287
10.3 A Least-Squares Estimation 294
10.4 Design of a Solenoid 296
10.5 Conflict Analysis, Games 301
10.6 Problems with Stochastic Ingredients 303
10.7 Problems with Lags 306
10.8 Concluding Remarks 310
Chapter 11, Hamilton—Jacobi Theory
11.1 Introduction 311
11.2 The Canonical Form of the Euler Condition 312
11.3 Transversals to a Field 314
11.4 The Formalism of Dynamic Programming 316
11.5 Examples 318
11.6 The Pontryagin Maximum Principle 321
11.7 Concluding Remarks 323
Chapter 12, Conclusion and Envoy
12.1 Comments and Suggestions 324
12.2 Generalized Curves 325
12.3 The Calculus of Variations in the Large 326
12.4 The Theory of Area 327
12.5 Multiple Integral Problems 328
12.6 Trends 329
BIBLIOGRAPHY 331
SUPPLEMENTARY BIBLIOGRAPHY 340
INDEX 341
Preface to the Dover Edition
This is a minor revision of the original 1969 publication. Some false
statements have been changed. Clarifying remarks have been added in
certain places and a short supplementary bibliography has been added.
Misspellings, imperfect mathematical symbols, and numerous other
errors have been corrected.
George M. Ewing
Norman, Oklahoma
Preface to the Original Edition
The name calculus of variations comes from procedures of Lagrange
involving an operator 8 called a variation, but this restricted meaning has
long been outgrown. The calculus of variations broadly interpreted
includes all theory and practice concerning the existence and charac­
terization of minima, maxima, and other critical values of a real-valued
functional. To say much less would exclude works of eminent authors
whose titles indicate contributions to variational theory but whose meth­
ods include no calculus in the early sense.
This book is an introduction, not a treatise. It is motivated by potential
applications but is not a mere compendium of partially worked examples.
It selects a path through classical conditions for an extremum and mod­
ern existence theory to problems of recent origin and with novel features.
Although it begins with mild presuppositions, the intent is to expose the
reader progressively to more substantial and more recent parts of the
theory so as to bring him to a point where he can begin to understand
specialized books and research papers. This entails compromise. Less
than the traditional space is devoted to necessary conditions and suffi­
ciency for local extrema of a succession of problems to give more atten­
tion to global extrema, to so-called direct methods, and to other twen­
tieth-century topics. An introduction to Hamilton—Jacobi Theory makes
contact with the Dynamic Programming of R. Bellman and the Max­
imum Principle of L. S. Pontryagin.
Chapters 1 through 6 have been used with classes including members
with no special preparation beyond a course in advanced calculus. There
are accordingly numerous elaborative comments and warnings against
pitfalls. Certain prerequisite materials are collected in Chapter 1 for
vii
ready reference. The major objective is insight, not practice in writing
Euler equations or in other techniques; hence emphasis is on conceptual
and logical features of the subject. Nevertheless, the often formidable
gap between theory and the analysis of particular problems, is bridged by
treatment of a number of examples and many exercises for the reader.
Chapters 7 through 12 require more mathematical maturity or else
willingness to supplement the text as individual needs may require. The
exposition is, however, largely self-contained. A brief treatment of the
Lebesgue theory of integration, which is essential for important parts of
modern variational theory, is in Chapter 8 for those who need it. A
number of the cited books and some of the cited articles can be used in
direct support of material in the text, but others begin at or beyond
positions covered here and are listed in the bibliography as information
on recent trends and names associated therewith. Previous experience in
modern real analysis, theory of differential equations, functional analy­
sis, or topology will be helpful, but a reader with serious intentions who
lacks this advantage can still make effective use of much of the second
half of the book.
Variational theory has connections with such fields as mathematical
physics, differential geometry, mathematical statistics, conflict analysis,
and the whole area of optimal design and performance of dynamical
systems. These interrelations suggest the importance of the subject, why
one book cannot be comprehensive, and why this is not an easy subject
for the beginner. One never has adequate preparation for all the things
with which he may be confronted under variational theory and its ap­
plications.
The author is indebted to many sources, particularly to works of G. A.
Bliss, E. J. McShane, and L. Tonelli; to his association with W. T. Reid; to
Marston Morse, under whose encouragement he was privileged to spend
a postdoctoral year; to his teacher, W. D. A. Westfall; and to various
colleagues, friends, and students.
Thanks are extended to W. T. Reid and D. K. Hughes for identifying
flaws in parts of the manuscript, but this is not to suggest a shared
responsibility for such flaws as may remain.
This book was sponsored in its initial stage during the summer of 1964
by the Office of Scientific Research of the Air Research and Development
Command through Grant AF-AFOSR-211-63 to the University of
Oklahoma Research Institute, for which the author expresses his appre­
ciation.
George M. Ewing
Norman, Oklahoma
Chapter 1
IN T R O D U C T IO N
AND O R IE N T A T IO N
1.1 PREREQUISITES
The reader is assumed to be familiar with concepts and methods usually
covered by courses called advanced calculus or introduction to real
analysis. Among the things presupposed are elementary set theory,
real numbers, various kinds of limits and continuity, derivatives, ordinary
differential equations of the first and second order, functions defined
implicitly, and the Riemann integral.
A resume of such topics is given in this chapter for review and
reference and to introduce terminology, notations, and points of view
to be found throughout the book. It is suggested that the chapter be
read quickly for content* then returned to later for more details as needs
may arise. Development of variational theory begins with Chapter 2.
1.2 FUNCTIONS
Given two nonempty sets X and Y of any nature, a function traditionally
has been described as a correspondence under which to each x E X is
associated y E Y. This lacks the precision of a definition and is indeed
i