Singularities 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