Approximating perfection: a mathematician's journey into the world of mechanics

234 Pages·2004·6.903 MB·English

by  Cloud, Michael J.;Lebedev, Leonid P

Approximating Perfection

Approximating Perfection
A Mathematician’s Journey into the World of
Mechanics
Leonid P. Lebedev
Michael J. Cloud
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright (cid:2)c 2004 by Princeton University Press
PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,New
Jersey 08540
IntheUnitedKingdom: PrincetonUniversityPress,3MarketPlace,Wood-
stock, Oxfordshire OX20 1SY
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Lebedev, L. P.
Approximating perfection: a mathematician’s journey into the
world of mechanics / Leonid P. Lebedev, Michael J. Cloud.
p. cm.
Includes biographical references and index.
ISBN 0-691-11726-8(acid-free paper)
1. Mechanics, Analytic. I. Cloud, Michael J. II. Title.
QA805.L38 2004
531 — dc22 2003062201
British Library Catalog-in-PublicationData is available
The publisher would like to acknowledge the authors of this volume for
providing the camera-ready copy from which this book was printed.
This book has been composed in Computer Modern
Printed on acid-free paper. ∞
www.pupress.princeton.edu
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Contents
Preface vii
Chapter1. The Tools of Calculus 1
1.1 IsMathematical Proof Necessary? 2
1.2 Abstraction, Understanding,Infinity 6
1.3 Irrational Numbers 8
1.4 What Is a Limit? 11
1.5 Series 15
1.6 Function Continuity 19
1.7 How to Measure Length 21
1.8 Antiderivatives 33
1.9 DefiniteIntegral 35
1.10 The Length of a Curve 42
1.11 Multidimensional Integrals 44
1.12 ApproximateIntegration 47
1.13 On theNotion of a Function 52
1.14 Differential Equations 53
1.15 Optimization 59
1.16 Petroleum Exploration and Recovery 61
1.17 Complex Variables 63
1.18 Moving On 65
Chapter2. The Mechanics of Continua 67
2.1 WhyDo ShipsFloat? 67
2.2 The Main Notions of Classical Mechanics 71
2.3 Forces, Vectors, and Objectivity 74
2.4 More on Forces; Statics 76
2.5 Hooke’s Law 80
2.6 Bending of a Beam 84
2.7 Stress Tensor 94
2.8 Principal Axesand Invariantsof the Stress Tensor 100
2.9 On theContinuum Model and Limit Passages 102
2.10 Equilibrium Equations 104
2.11 The Strain Tensor 108
2.12 Generalized Hooke’s Law 113
2.13 Constitutive Laws 114
2.14 Boundary ValueProblems 115
2.15 Setupof Boundary Value Problems of Elasticity 118
vi CONTENTS
2.16 Existence and Uniquenessof Solution 120
2.17 Energy; Minimal Principle for a Spring 126
2.18 Energy in Linear Elasticity 128
2.19 DynamicProblems of Elasticity 132
2.20 Oscillations of a String 134
2.21 Lagrangian and Eulerian Descriptions of Continuum Media 137
2.22 TheEquations of Hydrodynamics 140
2.23 D’Alembert–Euler Equation of Continuity 142
2.24 SomeOtherModels of Hydrodynamics 144
2.25 Equilibrium of an Ideal Incompressible Liquid 145
2.26 Force on an Obstacle 148
Chapter3. Elements of theStrength of Materials 151
3.1 What Arethe Problems of theStrength of Materials? 151
3.2 Hooke’s Law Revisited 152
3.3 Objectiveness of Quantities in Mechanics Revisited 157
3.4 Plane Elasticity 159
3.5 Saint-Venant’sPrinciple 161
3.6 Stress Concentration 163
3.7 Linearity vs. Nonlinearity 165
3.8 Dislocations, Plasticity, Creep, and Fatigue 166
3.9 Heat Transfer 172
3.10 Thermoelasticity 175
3.11 Thermal Expansion 177
3.12 A Few Wordson theHistory of Thermodynamics 178
3.13 Thermodynamics of an Ideal Gas 180
3.14 Thermodynamics of a Linearly Elastic Rod 182
3.15 Stability 186
3.16 StaticStability of a Straight Beam 188
3.17 Dynamical Tools for StudyingStability 193
3.18 AdditionalRemarks on Stability 195
3.19 Leak Prevention 198
Chapter4. Some Questions of Modeling in theNatural Sciences 201
4.1 Modeling and Simulation 201
4.2 Computerization and Modeling 203
4.3 Numerical Methods and Modeling in Mechanics 206
4.4 Complexity in theReal World 208
4.5 TheRole of theCosine in EverydayMeasurements 209
4.6 Accuracy and Precision 211
4.7 How Trees Stand Up against the Wind 213
4.8 WhyKing Kong Cannot Be as Terrible as in the Movies 216
Afterword 219
Recommended Reading 221
Index 223
Preface
Althoughengineeringtextbooksonceprovidedmorebreadththantheydo
today, few ever took the time to offer the reader a true perspective. We all
know that myriad formulas are essential to engineering practice. However,
modern textbooks have begun to allow formulas and procedural recipes to
preoccupy the mind of the student. We have already reached a stage where
proofs once deemed essential receive no mention whatsoever. The situation
willundoubtedlyworsenascomputationalmethods demandanevengreater
share of the engineering curriculum. In some areas, we are rapidly nearing
thepointwhereevenapassingfamiliaritywithcomputational“recipes”will
be deemed unnecessary: engineers will simply feed data into “canned” rou-
tines and receive immediate output. (It is likely that students will continue
to welcome this prospect with open arms, until it finally dawns on them
that the same taskcouldbe performedby someonewho lacksa hard-earned
engineering degree.) Unfortunately, all of this points to a diminishing grasp
of just how complex real systems (industrial or otherwise) really are. One
could argue that this is part of normal social progress: that a major goalof
science should be our freedom from having to think too much. Why should
the averagepersonnotbe abletosolveproblemsthatsurpassedtheabilities
of everytrue genius a century ago? But the argumentquickly wearsthin —
anyone who engages in researchand development activity, for instance, will
certainly require a realunderstanding of things. Training in the use of rigid
recipes may be appropriate for a fast-food cook, but not for the chef who
will be expected to develop new dishes for persons having special dietary
needs. The latter will have to learn a few things about chemistry, biology,
even medicine, in order to function in a truly professional capacity.
This book is not a textbook on engineering mechanics, although it does
contain topics from mechanics, the strength of materials, and elasticity. It
considers the background behind mechanics, some aspects of calculus, and
other portions ofmathematics that play key roles in applications. The logic
that underlies modeling in mechanics is its real emphasis. The book is,
however,intendedtobeusefultoanyonewhomustdealwithmodelingissues
— even in such areas as biology. Students and experts alike may discover
explanationsthat serveto justify routine actions,orthatoffer a better view
of particular problems in their areas of interest.
TheauthorsaregratefultoYu.P.StepanenkooftheResearchInstituteof
Mechanics and Applied Mathematics at Rostov State University. Professor
Stepanenko is an engineer and designer of measurement devices, and our
viii PREFACE
fruitful discussions with him have resulted in many of the examples cited in
this book. Edward Rothwell and Leonard Moriconi read large portions of
the manuscript and provided valuable feedback. We are deeply indebted to
oureditor,VickieKearn,forseveralyears’worthofhelpandencouragement.
Thanks are also due to editorial assistant Alison Kalett, production editor
Gail Schmitt, designer Lorraine Doneker, and copyeditor Anne Reifsnyder.
Finally, Natasha Lebedeva and BethLannon-Clouddeservethanks for their
patience and support.
Chapter One
The Tools of Calculus
The complexity of Nature has led to the existence of various sciences that
consider the same natural objects using different tools and approaches. An
expert in the physics of solids may find it hard to communicate with an
expert in the mechanics of solids; even between these closely related sub-
jects we find significant differences in mathematical tools, terms, and view-
pointstakentowardtheobjectsofinvestigation. Thephysicistandengineer,
however, do share a bit of common background: the tools of mathematical
physics. These have evolved during the long history of our civilization.
The heartof any physicaltheory — say,mechanics or electrodynamics—
is a collection of main ideas expressed in terms of some particular wording.
The next layer consists of mathematical formulation of these main ideas. It
is interesting to note that mathematical formulations can be both broader
and narrower than word statements. Wording, especially if left somewhat
“fuzzy,”isoftencapableofawiderrangeofapplicationbecauseitskirtspar-
ticularcaseswhereadditionalrestrictionswouldbeimperative. Ontheother
hand, mathematicalstudies often yield results and ideas that areimportant
in practice — for example, system traits such as energy and entropy.
Among the most mathematical of the disciplines within physics is me-
chanics. At first glance other branches such as quantum mechanics or field
theorymightseemmoresophisticated,buttheinfluenceofmechanicsonthe
rest of physics has been profound. Its main ideas, although they reflect our
everyday experience, are deep and complicated. The models of mechanics
and the mathematical tools that have been developed for the solution of
mechanical problems find application in many other mathematical sciences.
The tools, in particular, are now regarded as an important part of mathe-
maticsaswell. Indeed,therelationshipbetweenmechanicsandmathematics
has become so tight that it is possible to consider mechanics as a branch of
mathematics (although much of mechanics lacks the formalizedstructure of
pure mathematics).
Inthisbookweshallexploretheroleofmathematicsinthedevelopmentof
mechanics. The historical pattern of interaction between these two sciences
may yield a glimpse into the future development of certain other fields of
knowledge (e.g., biology) in which mathematical rigor currently plays a less
fundamental role.
Our use of the term “mechanics” will include both “classical mechanics”
and “continuum mechanics.” The former treats problems in the statics and
dynamics of rigid bodies, while the latter treats the motions, deformations,