Advanced engineering analysis : the calculus of variations and functional analysis with applications in mechanics

500 Pages·2012·2.26 MB·English

by  Michael J

AdvAnced
engineering
AnAlysis
The Calculus of Variations
and Functional Analysis with
Applications in Mechanics
This page intentionally left blank
AdvAnced
engineering
AnAlysis
The Calculus of Variations
and Functional Analysis with
Applications in Mechanics
Leonid P. Lebedev
Department of Mathematics,
National University of Colombia, Colombia
Michael J. Cloud
Department of Electrical and Computer Engineering,
Lawrence Technological University, USA
Victor A. Eremeyev
Institute of Mechanics, Otto von Guericke University Magdeburg, Germany
South Scientifc Center of RASci
and South Federal University, Rostov on Don, Russia
World Scientifc
NEW J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
ADVANCED ENGINEERING ANALYSIS
The Calculus of Variations and Functional Analysis with
Applications in Mechanics
Copyright © 2012 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
Desk Editor: Tjan Kwang Wei
ISBN-13 978-981-4390-47-7
ISBN-10 981-4390-47-X
Printed in Singapore.
KwangWei - Advanced Engineering Analysis.pm1d 2/14/2012, 11:29 AM
September 30, 2011 8:42 World Scientific Book - 9in x 6in aea
Preface
A little over half a century ago, it was said that even an ingenious per-
son could not be an engineer unless he had nearly perfect skills with the
logarithmic slide rule. The advent of the computer changed this situa-
tion crucially; at present, many young engineers have never heard of the
slide rule. The computer has profoundly changed the mathematical side
of the engineering profession. Symbolic manipulation programs can cal-
culate integrals and solve ordinary differential equations better and faster
than professional mathematicians can. Computers also provide solutions
to differential equations in numerical form. The easy availability of mod-
ern graphics packages means that many engineers prefer such approximate
solutions even when exact analytical solutions are available.
Because engineering courses must provide an understanding of the fun-
damentals, they continue to focus on simple equations and formulas that
are easy to explain and understand. Moreover, it is still true that stu-
dents must develop some analytical abilities. But the practicing engineer,
armed with a powerful computer and sophisticated canned programs, em-
ploys models of processes and objects that are mathematically well beyond
the traditional engineering background. The mathematical methods used
by engineers have become quite sophisticated. With insufficient base knowl-
edge to understand these methods, engineers may come to believe that the
computer is capable of solving any problem. Worse yet, they may decide
to accept nearly any formal result provided by a computer as long as it was
generated by a program of a known trademark.
But mathematical methods are restricted. Certain problems may ap-
pear to fall within the nominal solution capabilities of a computer program
and yet lie well beyond those capabilities. Nowadays, the properties of so-
phisticated models and numerical methods are explained using terminology
v
September 30, 2011 8:42 World Scientific Book - 9in x 6in aea
vi Advanced Engineering Analysis
from functional analysis and the modern theory of differential equations.
Without understanding terms such as “weak solution” and “Sobolev space”,
one cannot grasp a modern convergence proof or follow a rigorous discus-
sion of the restrictions placed on a mathematical model. Unfortunately, the
mathematical portion of the engineering curriculum remains preoccupied
with 19th century topics, even omitting the calculus of variations and other
classical subjects. It is, nevertheless, increasingly more important for the
engineer to understand the theoretical underpinning of his instrumentation
than to have an ability to calculate integrals or generate series solutions of
differential equations.
The present text offers rigorous insight and will enable an engineer to
communicate effectively with the mathematicians who develop models and
methods for machine computation. It should prove useful to those who
wish to employ modern mathematical methods with some depth of under-
standing.
The book constitutes a substantial revision and extension of the earlier
book The Calculus of Variations and Functional Analysis, written by the
first two authors. A new chapter (Chapter 2) provides applications of the
calculus of variations to nonstandard problems in mechanics. Numerous
exercises (most with extensive hints) have been added throughout.
The numbering system is as follows. All definitions, theorems, corol-
laries, lemmas, remarks, conventions, and examples are numbered consecu-
tively by chapter (thus Definition 1.7 is followed by Lemma 1.8). Equations
are numbered independently, again by chapter.
We would like to thank ourWorld Scientific editor, Mr. Yeow-Hwa Quek.
Leonid P. Lebedev
Department of Mathematics, National University of Colombia, Colombia
Michael J. Cloud
Department of Electrical and Computer Engineering, Lawrence Technolog-
ical University, USA
Victor A. Eremeyev
Institute of Mechanics, Otto von Guericke University Magdeburg, Germany
South Scientific Center of RASci and South Federal University, Rostov on
Don, Russia
September 30, 2011 8:42 World Scientific Book - 9in x 6in aea
Contents
Preface v
1. Basic Calculus of Variations 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Euler’s Equation for the Simplest Problem . . . . . . . . . 15
1.3 Properties of Extremals of the Simplest Functional . . . . 21
1.4 Ritz’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Natural Boundary Conditions . . . . . . . . . . . . . . . . 31
1.6 Extensions to More General Functionals . . . . . . . . . . 34
1.7 Functionals Depending on Functions in Many Variables . 43
1.8 A Functional with Integrand Depending on Partial Deriva-
tives of Higher Order . . . . . . . . . . . . . . . . . . . . . 49
1.9 The First Variation . . . . . . . . . . . . . . . . . . . . . . 54
1.10 Isoperimetric Problems . . . . . . . . . . . . . . . . . . . . 65
1.11 General Form of the First Variation . . . . . . . . . . . . 72
1.12 Movable Ends of Extremals . . . . . . . . . . . . . . . . . 76
1.13 Broken Extremals: Weierstrass–Erdmann Conditions and
Related Problems . . . . . . . . . . . . . . . . . . . . . . . 80
1.14 Sufficient Conditions for Minimum . . . . . . . . . . . . . 85
1.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2. Applications of the Calculus of Variations in Mechanics 99
2.1 Elementary Problems for Elastic Structures . . . . . . . . 99
2.2 Some Extremal Principles of Mechanics . . . . . . . . . . 108
2.3 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . 127
2.4 Conservation Laws and Noether’s Theorem . . . . . . . . 131
vii
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viii Advanced Engineering Analysis
2.5 Functionals Depending on Higher Derivatives of y . . . . 139
2.6 Noether’s Theorem, General Case . . . . . . . . . . . . . . 143
2.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . 147
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3. Elements of Optimal Control Theory 159
3.1 A Variational Problem as an Optimal Control Problem . . 159
3.2 General Problem of Optimal Control . . . . . . . . . . . . 161
3.3 Simplest Problem of Optimal Control . . . . . . . . . . . 164
3.4 Fundamental Solution of a Linear Ordinary Differential
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.5 The Simplest Problem, Continued . . . . . . . . . . . . . 171
3.6 Pontryagin’s Maximum Principle for the Simplest Problem 173
3.7 Some Mathematical Preliminaries . . . . . . . . . . . . . . 177
3.8 General Terminal Control Problem . . . . . . . . . . . . . 189
3.9 Pontryagin’s Maximum Principle for the Terminal Optimal
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
3.10 Generalization of the Terminal Control Problem . . . . . 198
3.11 Small Variations of Control Function for Terminal Control
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
3.12 A Discrete Version of Small Variations of Control Function
for Generalized Terminal Control Problem . . . . . . . . . 205
3.13 Optimal Time Control Problems . . . . . . . . . . . . . . 208
3.14 Final Remarks on Control Problems . . . . . . . . . . . . 212
3.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
4. Functional Analysis 215
4.1 A Normed Space as a Metric Space . . . . . . . . . . . . . 217
4.2 Dimension of a Linear Space and Separability . . . . . . . 223
4.3 Cauchy Sequences and Banach Spaces . . . . . . . . . . . 227
4.4 The Completion Theorem . . . . . . . . . . . . . . . . . . 238
p
4.5 L Spaces and the Lebesgue Integral . . . . . . . . . . . . 242
4.6 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . 248
4.7 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . 250
4.8 Inner Product Spaces, Hilbert Spaces . . . . . . . . . . . . 260
4.9 Operators and Functionals . . . . . . . . . . . . . . . . . . 264
4.10 Contraction Mapping Principle . . . . . . . . . . . . . . . 269
4.11 Some Approximation Theory . . . . . . . . . . . . . . . . 276
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Contents ix
4.12 Orthogonal Decomposition of a Hilbert Space and the
Riesz Representation Theorem . . . . . . . . . . . . . . . 280
4.13 Basis, Gram–Schmidt Procedure, and Fourier Series in
Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . 284
4.14 Weak Convergence . . . . . . . . . . . . . . . . . . . . . . 291
4.15 Adjoint and Self-Adjoint Operators . . . . . . . . . . . . . 298
4.16 Compact Operators . . . . . . . . . . . . . . . . . . . . . 304
4.17 Closed Operators . . . . . . . . . . . . . . . . . . . . . . . 311
4.18 On the Sobolev Imbedding Theorem . . . . . . . . . . . . 315
4.19 Some Energy Spaces in Mechanics . . . . . . . . . . . . . 320
4.20 Introduction to Spectral Concepts . . . . . . . . . . . . . 337
4.21 The Fredholm Theory in Hilbert Spaces . . . . . . . . . . 343
4.22 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
5. Applications of Functional Analysis in Mechanics 359
5.1 Some Mechanics Problems from the Standpoint of the Cal-
culus of Variations; the Virtual Work Principle . . . . . . 359
5.2 Generalized Solution of the Equilibrium Problem for a
Clamped Rod with Springs . . . . . . . . . . . . . . . . . 364
5.3 Equilibrium Problem for a Clamped Membrane and its
Generalized Solution . . . . . . . . . . . . . . . . . . . . . 367
5.4 Equilibrium of a Free Membrane . . . . . . . . . . . . . . 369
5.5 Some Other Equilibrium Problems of Linear Mechanics . 371
5.6 The Ritz and Bubnov–Galerkin Methods . . . . . . . . . . 379
5.7 The Hamilton–Ostrogradski Principle and Generalized
Setup of Dynamical Problems in Classical Mechanics . . . 381
5.8 Generalized Setup of Dynamic Problem for Membrane . . 383
5.9 Other Dynamic Problems of Linear Mechanics . . . . . . 397
5.10 The Fourier Method . . . . . . . . . . . . . . . . . . . . . 399
5.11 An Eigenfrequency Boundary Value Problem Arising in
Linear Mechanics . . . . . . . . . . . . . . . . . . . . . . . 400
5.12 The Spectral Theorem . . . . . . . . . . . . . . . . . . . . 404
5.13 The Fourier Method, Continued . . . . . . . . . . . . . . . 410
5.14 Equilibrium of a von K´arma´n Plate . . . . . . . . . . . . . 415
5.15 A Unilateral Problem . . . . . . . . . . . . . . . . . . . . 425
5.16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
Appendix A Hints for Selected Exercises 433