Table Of ContentSingularities of Caustics and Wave Fronts
Mathematics and Its Applications (Soviet Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. A. KIRILLOV, MGU, Moscow, U.S.S.R.
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.S.S.R.
S. P. NOVIKOV, Landau Institute o/Theoretical Physics, Moscow, U.S.S.R.
M. C. POLY VANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.
Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.
Volume 62
Singularities of Caustics
and Wave Fronts
by
V. 1. Amold
Steklov Institute, Moscow, V.S.S.R.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging in Publication Data
Arnol 'do V. 1. <Vladimir Igorevich). 1937-
Singularities of caustics and wave fronts I by V.I. Arnold.
p. cm. -- (Mathematics and its appllcations. Sovlet serles
v. 62)
Translated from the Russian.
Includes bibllographical references and index.
ISBN 978-1-4020-0333-2 ISBN 978-94-011-3330-2 (eBook)
DOI 10.1007/978-94-011-3330-2
1. Geometry. Differential. 2. Slngularities <Mathematics)
1. Title. II. Serles, Mathematlcs and its applications <S pringer
Science+BusinessMedia, B.v.l. Soviet series ; 62.
QA649.A75 1990
516.3'6--dc20 90-19243
ISBN 978-1-4020-0333-2
Printed an acid-free paper
This book has been typeset with LATEX.
All Rights Reserved
© 1990 by Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1990
Softcover reprint ofthe hardcover Ist edition 1990
No part of the material protected by this copyright notice may be reproduced ar
utilized in any form or by any means. electronic or mechanical.
including photocopying. recording or by any information storage and
retrieval system. without written permission from the copyright owner,
SERIES EDITOR'S PREFACE
'Et moi, ... ) si j'avait su comment en revenir, One service mathematics has rendered the
je n'y serais point aile.' human race. It has put common sense back
Jules Verne where it belongs, on the topmost shelf next
to the dusty canister labelled 'discarded non
The series is divergent; therefore we may be sense'.
able to do something with it. ErieT. Bell
O. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Lie algebras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu
lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more)
all come together. And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via
finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And,
accordingly, the applied mathematician needs to be aware of much more. Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
extra mathematical sophistication that this requires. For that is where the rewards are. Linear
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci-
vi SERIES EDITOR'S PREFACE
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com
puters; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre
quently proved the shortest path between 'real' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no telling where all this is leading -
fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis
cipline which are used in others. Thus the series still aims at books dealing with:
- a central concept which plays an important role in several different mathematical and/or
scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- infiuences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
This book is about caustics -and about a large part of everything else in mathematics. Let me start
by quoting from the article on caustics in the (Russian) Encyclopaedia of Mathematics: 'A caustic is
the envelope of rays reflected or refracted by a given curve'. The editorial comments to the anno
tated expanded and revised English language edition (Kluwer Academic Publishers) add the follow
ing. 'In terms of purely geometric optics these [caustics] are curves of light of infinite brightness
consisting of points through which infinitely many reflected or refracted light rays pass. In reality
they can often be observed as a pattern of pieces of very bright curves; e.g. on a sunny day at the
seashore on the bottom beneath a bit of wavy water: or at the bottom of a cup of tea into which
light is shining.
That sounds like geometric optics and, quickly, singularities and bifurcation theory but still a far
cry from, say, cobordism, characteristic classes, Dynkin diagrams and Weyl groups.
However, in 1972, Arnold himself, in a seminal article, discovered that the A,D,E Dynkin
diagrams appear naturally in the discussion of (certain types of) singularities. He also observed that
the A,D,E diagrams occur in many different parts of mathematics and posed (in 1974) the problem
of finding the common origin of all these A - D - E classification results. As far as singularities are
concerned the B,C,F,H diagrams turned up in the first half of the eighties and by now caustics,
wave fronts, Legendre transforms, singularities of ray systems; and groups of reflections and Weyl
groups are firmly linked: although not a few mysteries remain, and though Arnolds problem as to
other A - D - E etc. classification results (the list of these has meanwhile also grown) is far from
solved.
Arnold and his students have been the main initiators and developers in all this. Arnold has also
found another employ for the word 'perestroika' in this context and it is fair to say that he has
caused and overseen a restructuring of singularity theory. This book is his own account of all this in
his well known, classic, lucid expository syle.
The shortest path between two truths in the N ever lend books, for no one ever returns
real domain passes through the complex them; the only books I have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole France
La physique ne nous donne pas senlement The function of an expert is not to be more
I'o ccasion de resoudre des problemes ... eIle right than other people, but to be wrong for
nous fait pressentir la solution. more sophisticated reasons.
H. Poincare David Butler
Bussum, October 1990 Michiel Hazewinkel
Contents
Series Editor's Preface v
Introduction IX
1 Symplectic geometry 1
1.1 Symplectic manifolds. . . . . . . . . . . . . . . . . . . . .. . 1
1.2 Submanifolds of symplectic manifolds ............ . 7
1.3 Lagrangian manifolds, fibrations, mappings, and singularities 14
2 Applications of the theory of Lagrangian singularities 21
2.1 Oscillatory integrals .. 21
2.2 Lattice points ......... . 27
2.3 Perestroikas of caustics .... . 31
2.4 Perestroikas of optical caustics 35
2.5 Shock wave singularities and perestroikas of Maxwell sets 38
3 Contact geometry 43
3.1 Wave fronts .. 43
3.2 Singularities of fronts. 51
3.3 Perestroikas of fronts . 56
4 Convolution of invariants, and period maps 61
4.1 Vector fields tangent to fronts .... 61
4.2 Linearised convolution of invariants. 66
4.3 Period maps ............ . 73
4.4 Intersection forms of period maps. 78
4.5 Poisson structures 81
4.6 Principal period maps . . . . . . . 84
5 Lagrangian and Legendre topology 87
5.1 Lagrangian and Legendre cobordism ..... . 88
5.2 Lagrangian and Legendre characteristic classes 96
Vlll
5.3 Topology of complex discriminants 102
5.4 Functions with mild singularities . 108
5.5 Global properties of singularities . 113
5.6 Topology of Lagrangian inclusions 115
6 Projections of surfaces, and singularities of apparent contours 123
6.1 Singularities of projections from a surface to the plane 123
6.2 Singularities of projections of complete intersections 129
6.3 Geometry of bifurcation diagrams. . . . . . . . . . . 139
7 Obstacle problem 155
7.1 Asymptotic rays in symplectic geometry 157
7.2 Contact geometry of pairs of hypersurfaces 166
7.3 Unfurled swallowtails . 172
7.4 Symplectic triads. . . . . . . 186
7.5 Contact triads ........ 193
7.6 Hypericosahedral singularity. 199
7.7 Normal forms of singularities in the obstacle problem. 208
8 Transformation of waves defined by hyperbolic variational principles219
8.1 Hyperbolic systems and their light hypersurfaces . . . . . 220
8.2 Singularities of light hypersurfaces of variational systems. . . . . . . . 224
8.3 Contact normal forms of singularities of quadratic cones . . . . . . . . 228
8.4 Singularities of ray systems and wave fronts at nonstrict hyperbolic
points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Bibliography 241
Index 255
Introduction
Let us start with an example: consider the distance from a point in the euclidean plane
to a given curve; for instance, from an interior point of an elliptical domain to the
boundary ellipse (Fig. 1). The corresponding rays (the extremals of this variational
problem) are the normal lines to the ellipse. The minimal value ofthe functional (the
distance) satisfies, as a function of the initial point, the Hamilton-Jacobi equation
(\7u)2 = 1 (at the points where it is smooth). However, this function has singularities
(on the segment joining the focal points of the ellipse). The system of rays also has
singularities. They lie on the astroid, which is the envelope of the system of normals
of the ellipse. The envelope of a system of extremals is called the caustic of this
system. The caustic of our system has four cusps. These singularities are stable: any
curve sufficiently close to an ellipse has a caustic sufficiently close to an astroid and
having four cusps.
The level lines of a solution of a Hamilton-Jacobi equation are called fronts.
In our example the fronts are the equidistant curves of the ellipse. The equidistant
curves close to the ellipse are smooth curves. However, the equidistants which are
not so close to the ellipse have singularities. In Fig. 2 four cuspidal singularities on
an equidistant of an ellipse are depicted. These cusps are stable: curves equidistant
to any curve which is sufficiently close to the ellipse have similar singularities.
Of course, a level line of the shortest distance to the ellipse is only a part of an
equidistant curve. However, most of the properties of singularities of ray systems,
caustics and fronts are more transparent if we consider next to the minima also the
other extremal points of the functionals. In our example we start with the study of
the equidistant curves and then discern the parts we need.
For instance, consider the distance to the curve as a function of the initial point.
Figure 1: Rays and fronts of a perturbation, propagating inside an ellipse
IX
