Approximating Perfection: A Mathematician’s Journey into the World of Mechanics

323 Pages·2004·6.35 MB·English

by  L. P. Lebedev

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 © 2004 by Princeton University Press Published by Princeton University Press, 41
William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, 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 Catalogin-Publication Data 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
Chapter 1. The Tools of Calculus 1
1.1 Is Mathematical 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 Definite Integral 35
1.10 The Length of a Curve 42
1.11 Multidimensional Integrals 44
1.12 Approximate Integration 47
1.13 On the Notion 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
Chapter 2. The Mechanics of Continua 67
2.1 Why Do Ships Float? 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 Axes and Invariants of the Stress Tensor 100
2.9 On the Continuum 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 Value Problems 115
2.15 Setup of Boundary Value Problems of Elasticity 118
2.16 Existence and Uniqueness of Solution 120
2.17 Energy; Minimal Principle for a Spring 126
2.18 Energy in Linear Elasticity 128
2.19 Dynamic Problems of Elasticity 132
2.20 Oscillations of a String 134
2.21 Lagrangian and Eulerian Descriptions of Continuum Media 137
2.22 The Equations of Hydrodynamics 140
2.23 D’Alembert–Euler Equation of Continuity 142
2.24 Some Other Models of Hydrodynamics 144
2.25 Equilibrium of an Ideal Incompressible Liquid 145
2.26 Force on an Obstacle 148
Chapter 3. Elements of the Strength of Materials 151
3.1 What Are the Problems of the Strength 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’s Principle 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 Words on the History 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 Static Stability of a Straight Beam 188
3.17 Dynamical Tools for Studying Stability 193
3.18 Additional Remarks on Stability 195
3.19 Leak Prevention 198
Chapter 4. Some Questions of Modeling in the Natural 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 the Real World 208
4.5 The Role of the Cosine in Everyday Measurements 209
4.6 Accuracy and Precision 211
4.7 How Trees Stand Up against the Wind 213
4.8 Why King Kong Cannot Be as Terrible as in the Movies 216
Afterword 219
Recommended Reading 221
Index 223
Preface
Although engineering textbooks once provided more breadth than
they do 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 will undoubtedly worsen
as computational methods demand an even greater share of the
engineering curriculum. In some areas, we are rapidly nearing the point
where even a passing familiarity with computational “recipes” will be
deemed unnecessary: engineers will simply feed data into “canned”
routines 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 task could be performed by someone who lacks a
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 goal of science should be our freedom from
having to think too much. Why should the average person not be able to
solve problems that surpassed the abilities of every true genius a century
ago? But the argument quickly wears thin — anyone who engages in
research and development activity, for instance, will certainly require a
real understanding 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