Table Of ContentAsymptotology
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science. Amsterdam, The Netherlands
Volume 551
Asymptotology
Ideas, Methods, and Applications
by
Igor V. Andrianov
Pridneprovie State Academy of Civil Engineering and Architecture ( PGASA),
Dnepropetrovsk, Ukraine
Leonid I. Manevitch
N.N. Semenov Institute for Chemical Physics of Russian Academy of Sciences,
Moscow, Russia
with help from
Michiel Hazewinkel
CWI (Center for Mathematics and Computer Science),
Amsterdam, The Netherlands
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4613-4816-0 ISBN 978-1-4419-9162-1 (eBook)
DOI 10.1007/978-1-4419-9162-1
Printed on acid-free paper
All Rights Reserved
© 2002 Springer Science+ Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2002
Softcover reprint of the hardcover 1st edition 2002
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming, recording
or otherwise, without written permission from the Publisher, with the exception
of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
We dedicate this book
to our parents.
Contents
Foreword xi
Preface Xlll
Acknowledgments XV
Synopsis XVll
1. INTRODUCTION 1
1. General Remarks 1
2. What does "Popular Science Book" Mean? 2
3. Why "Asymptotology"? 4
2. WHAT ARE ASYMPTOTIC METHODS? 7
1. Reduction of the System's Dimension 8
2. Regular Asymptotics and Boundary Layers 9
3. Asymptotic Series 11
4. Averaging and Homogenization 16
5. Continuous limits 19
6. Local and Nonlocal Linearization 22
7. Estimation of Asymptotic Solution Errors 23
8. Summation Procedures 25
9. "Padeons" 26
10. How to Make Both Ends Meet 27
11. Renormalization 32
12. Asymptotics and Computers 34
13. Are Asymptotic Methods a Panacea? 36
3. A LITTLE MATHEMATICS 41
1. Basic Formalism 42
2. A Simple Example 46
3. Regular and Singular Asymptotics 47
4. Asymptotic Decomposition 56
5. Supplementary Asymptotics 58
Vll
vm ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
6. Continuous Approximation of a Chain of Masses 59
7. Search for Small Parameters 62
8. Newton Polyhedron (i[j
9. Catastrophe Theory 68
4. HOW ASYMPTOTIC METHODS WORK 81
1. Celestial Mechanics 82
2. Theory of Plates and Shells 83
3. Polymer Physics 86
4. Asymptotics and Engineering 89
5. Theory of Composite Materials 90
6. Biology 9:3
7. Et Cetera 94
8. Asymptotics and Art 95
9. Asymptotics in Pictures 97
10. Formation of New Concepts 98
11. Is Understanding an Asymptotic Process? 99
5. ASYMPTOTIC METHODS AND PHYSICAL THEORIES 103
l. Asymptotic Correspondence of Physical Theories 103
2. Mechanics by Aristotle and Galileo--Newton 104
3. Newton Mechanics and Special Relativity Theory 107
4. Geometrical and Wave Optics 109
5. Classical and Quantum Mechanics 110
6. "Simple Theories" in Physics 111
7. ·'The Cube of Theories'' 112
8. Asymptotic Ways of Thinking for Beginners 114
6. PHENOMENOLOGY AND FIRST PRINCIPLES 119
1. Basic Relations of Shell Theory 121
2. How to Construct Consistent Phenomenological Theories 12:3
3. Some Conclusions 126
7. A LITTLE HISTORY 129
1. Method of Averaging 129
2. Triumphs of Perturbation Methods 133
:3. Galileo and the Principle of Idealization 13S
8. FATHERS OF ASYMPTOTIC METHODS 137
1. Leonhard Euler 138
2. Alexis-Claude Clairaut 142
:3. Jean Le Rond d'Alembert 145
4. Joseph-Louis Lagrange 148
5. Pierre-Simon Laplace 1GO
6. Carl Friedrich Gauss 1S5
Contents IX
7. Jules-Henri Poincare 157
8. Alexander M. Lyapunov 161
9. Henri Eugene Pade 164
10. Ludwig Prandtl 169
11. Balthasar Vander Pol 172
12. Nickolay M. Krylov 174
13. Nickolay N. Bogoliubov 176
9. CONCLUSION 179
Appendices 183
A- Linear and Nonlinear Mathematical Physics: 183
from Harmonic Waves to Solitons
1. The Quasi-Linear World 184
2. On the Way to Nonlinear Physics 187
3. How Solitons Work 193
B- Certain Mathematical Notions 197
of Catastrophe Theory
1. Representation of Functions by Jets 197
2. Equivalency of a Function and its k-th Jet 199
3. Representation of Functions by Jets in Ordinary Points 201
4. Jets at Non-Degenerate Critical Points 202
5. Jets at Degenerate Critical Points 204
6. Control Parameters 205
C- Asymptotics and Scaling Transformations 207
1. Estimation of Variables 208
2. Subsequent Approximations 209
D- Asymptotic Approaches: Attempt at a Definition 215
1. Asymptotic Methods or a New Mathematics? 215
2. Uncertainty-Complementarity-Compatibility 219
E- Some Web-Pages 223
References 225
About the Authors 239
Author Index 241
Topic Index 24 7
Foreword
Asymptotic methods belong to the, perhaps, most romantic area of
modern mathematics. They are widely known and have been used in me
chanics, physics and other exact sciences for many, many decades. But
more than this, asymptotic ideas are found in all branches of human
knowledge, indeed in all areas of life. In this broader context they have
not and perhaps cannot be fully formalized. However, they are mar
velous, they leave room for fantasy, guesses and intuition; they bring us
very near to the border of the realm of art.
Many books have been written and published about asymptotic meth
ods. Most of them presume a mathematically sophisticated reader. The
authors here attempt to describe asymptotic methods on a more accessi
ble level, hoping to address a wider range of readers. They have avoided
the extreme of banishing formulae entirely, as done in some popular
science books that attempt to describe mathematical methods with no
mathematics. This is impossible (and not wise). Rather, the authors
have tried to keep the mathematics at a moderate level. At the same
time, using simple examples, they think they have been able to illustrate
all the key ideas of asymptotic methods and approaches, to depict in de
tail the results of their application to various branches of knowledge -
from astronomy, mechanics, and physics to biology, psychology and art.
The book is supplemented by several appendices, one of which con
tains the profound ideas of R.G. Barantsev, "Asymptotic Approaches:
Attempt at a definition."
V.M. Alexandrov
Xl
