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vlad imir i. ar nold

Collected Works
vo lume ii
Hydrodynamics, Bifurcation Theory, and Algebraic Geometr y 1965­1972

123


VLADIMIR I. ARNOLD

Collected Works

123


Vladimir I. Arnold, drawing, 1968.
Photo by Dmitry Arnold


VLADIMIR I. ARNOLD
Collected Works


VLADIMIR I. ARNOLD

Collected Works

VOLUME II

Hydrodynamics, Bifurcation Theory, and Algebraic Geometry 1965-1972


VLADIMIR I. ARNOLD
Collected Works

VOLUME II
Hydrodynamics, Bifurcation Theory, and Algebraic Geometry 1965-1972

Edited by
Alexander B. Givental Boris A. Khesin Alexander N. Varchenko Victor A. Vassiliev Oleg Ya. Viro

123


Vladimir I. Arnold
June 12, 1937 ­ June 3, 2010

Editors Alexander B. Givental Department of Mathematics University of California Berkeley, CA, USA Boris A. Khesin Department of Mathematics University of Toronto Toronto, ON, Canada Alexander N. Varchenko Department of Mathematics University of North Carolina Chapel Hill, NC, USA Victor A. Vassiliev Steklov Mathematical Institute Russian Academy of Sciences Moscow, Russia Oleg Ya. Viro Institute for Mathematical Sciences Stony Brook University Stony Brook, NY, USA

ISBN 978-3-642-31030-0 ISBN 978-3-642-31031-7 (ebook) DOI 10.1007/978-3-642-31031-7 Library of Congress Control Number: 2013937321 Springer-Verlag Berlin Heidelberg 2014 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)


Preface

This volume of the Collected Works appears in print after Vladimir Arnold's untimely death in June 2010. His passing was a terrible loss for mathematics and science in general. We hope that this project of Collected Works, which is needed now more than ever, will contribute to establishing the tremendous legacy of V.I. Arnold, a remarkable mathematician and human being. Some memories of V.I. Arnold can be found in the recent March and April 2012 issues of the Notices of the AMS. Our Editorial team has also suffered an unprecedented blow since Volume I was published in 2009. Jerry Marsden passed away in September 2010, and Vladimir Zakalyukin passed away in December 2011. We dedicate this volume to their memory. This Volume II of the Collected Works includes papers written by V.I. Arnold mostly during the period from 1965 to 1972. This was an amazingly productive period, starting with a year Arnold spent in Paris. During this period he made fundamental contributions to the fields of hydrodynamics, algebraic geometry, singularity and bifurcation theories, and dynamical systems. We have also added later papers by Arnold on related topics so as to make this volume more complete and comprehensive. Several papers were translated from Russian specifically for this volume. Unfortunately it was not possible to replace certain previously existing translations, completed in the era of the Iron Curtain, when it could not be expected that translators understood the subject. As an alarming example, we refer the reader to the paper "Local problems of analysis", and to the editors' comment therein. As a counter-example, see the paper "On the arrangements of ovals...", which has been translated once again. For the same reason some of the titles in the reprinted translations were incorrect. Since it was not possible to correct them in the reprinted articles, we have given their correct versions in the Contents along with the versions used in the reprinted contributions. November 2012 Alexander Givental Boris Khesin Alexander Varchenko Victor Vassiliev Oleg Viro

VII


Acknowledgements

The Editors thank the GÆttingen State and University Library for providing the original articles for this edition, as well as D. Auroux, A. Chenciner, G. Gould and R. Montgomery for the translation and editing of several papers in this volume. They also thank the Springer office in Heidelberg, and in particular Ruth Allewelt and Martin Peters, for their extensive help and tireless support with this project.

VIII


Contents

1

A variational principle for three-dimensional steady flows of an ideal fluid
Published as "Variational Principle for three-dimensional steady-state flows of an ideal fluid" in J. Appl. Math. Mech. 29:5, 1002-1008, 1965. Translation of Prikl. Mat. Mekh. 29:5, 846-851, 1965

........ 1

2 3 4

On the Riemann curvature of diffeomorphism groups
Translation of C.R. Acad. Sc. Paris 260, 5668-5671, 1965. Translated by Denis Auroux

....................... 9

C. R. Acad. Sc. Paris 261, 17-20, 1965

Sur la topologie des Ècoulements stationnaires des fluides parfaits (French) ................................................................................... 15

Conditions for non-linear stability of stationary plane curvilinear flows of an ideal fluid
Sov. Math. Dokl. 162, No. 5, 773-777, 1965. Translation of Dokl. Akad. Nauk SSSR, 162:5, 975-978, 1965

................................................................................................................. 19 .......... 25

5 6

On the topology of three-dimensional steady flows of an ideal fluid
J. Appl. Math. Mech. 30:1, 223-226 1966. Translation of Prikl. Mat. Mekh. 30:1, 183-185, 1966

On an a priori estimate in the theory of hydrodynamical stability
Am. Math. Soc. Transl. (2) 79, 267-269, 1969. Translation of Izv. Vyssh. Uchebn. Zaved. Mat. 5:54, 3-5, 1966

........................................................................................................................ 29

7

On the differential geometry of infinite-dimensional Lie groups and its applications to the hydrodynamics of perfect fluids
Translation of Annales de L'Institut Fourier, Vol. 16, No. 1, 319-361, 1966. Translated by Alain Chenciner

............................................................................................................... 33

8

On a variational principle for the steady flows of perfect fluids and its applications to problems of non-linear stability
Translation of Journal de MÈcanique, Vol. 5, No. 1, 29-43, 1966. Translated by Alain Chenciner

......... 71

9

On a characteristic class arising in quantization conditions
Published as "Characteristic class entering in quantization conditions" in Funct. Anal. Appl. 1, 1-13, 1967. Translation of Funkts. Anal. Prilozh. 1:1, 1-14, 1967

................................................. 85

10

A note on the Weierstrass preparation theorem
Published as "A note on Weierstrass' auxiliary theorem" in Funct. Anal. Appl. 1, 173-179, 1967. Translation of Funkts. Anal. Prilozh. 1:3, 1-8, 1967

.................................................................. 99

11

The stability problem and ergodic properties for classical dynamical systems
Am. Math. Soc. Transl. (2) 70, 5-11, 1969. Translation of Proc. Internat. Congr. Math., Moscow 1966, 387-392. 1968

........................................................................................................ 107


12

A remark on the ramification of hyperelliptic integrals as functions of parameters
Published as "Remark on the branching of hyperelliptic integrals as functions of the parameters" in Funct. Anal. Appl. 2, 187-189, 1968. Translation of Funkts. Anal. Prilozh. 2:3, 1-3, 1968. Translated by Victor Vassiliev

.......................................................................................................... 115 .......................... 119

13 14

Singularities of smooth mappings
Russ. Math. Surv. 23, 1-43, 1968. Translation of Usp. Mat. Nauk 23:1, 3-44, 1968

Remarks on singularities of finite codimension in complex dynamical systems
Funct. Anal. Appl. 3, 1-5, 1969. Translation of Funkts. Anal. Prilozh. 3:1, 1-6, 1969. Translated by Victor Vassiliev

.............................................................................................................. 163

15 16

Translation of Usp. Mat. Nauk 23:4, 247-248, 1968. Translated by Gerald Gould

Braids of algebraic functions and the cohomology of swallowtails ........................... 171

Hamiltonian nature of the Euler equations in the dynamics of a rigid body and of an ideal fluid Translation of Usp. Mat. Nauk 24:3, 225-226, 1969. Translated by Gerald Gould ........................... 175 On the one-dimensional cohomology of the Lie algebra of divergence-free vector fields and rotation numbers of dynamical systems
Published as "One-dimensional cohomologies of the Lie algebras of nondivergent vector fields and rotation numbers of dynamic systems" in Funct. Anal. Appl. 3, 319-321, 1969. Translation of Funkts. Anal. Prilozh. 3:4, 77-78, 1969. Translated by Victor Vassiliev

17

........................................ 179

18

The cohomology ring of the colored braid group
Math. Notes 5, 138-140, 1969. Translation of Mat. Zametki 5:2, 227-231, 1969. Translated by Victor Vassiliev

.............................................................................................................. 183

19

On cohomology classes of algebraic functions invariant under Tschirnhausen transformations
Funct. Anal. Appl. 4, 74-75, 1970. Translation of Funkts. Anal. Prilozh. 4:1, 84-85, 1970. Translated by Victor Vassiliev

.......................................................................................................... 187

20

Trivial problems
Translation of Proc. 5th Int. Conf. on Nonlinear Oscillations, Kiev 1969. Vol. 1, 630-631, Ukrain. Acad. Sciences, Kiev 1970. Translated by Gerald Gould

........................................................... 191

21

Local problems of analysis
Moscow Univ. Math. Bull. 25 (1970), 77-80, 1970. Translation of Vestn. Mosk. Univ. Ser. I, Mat. Mekh. 25:2, 52-56, 1970

............................................................................................ 193

22

Algebraic unsolvability of the problem of stability and the problem of topological classification of singular points of analytic systems of differential equations (Russian) Usp. Mat. Nauk 25:2, 265-266, 1970 ................................................................................... 197 On some topological invariants of algebraic functions
Transact. Math. Moscow Soc. 21, 30-52, 1970. Translation of Tr. Mosk. Mat. Obsc., 27-46, 1970

23

....... 199


24 25

Topological invariants of algebraic functions II
Funct. Anal. Appl. 4, 91-98, 1970. Translation of Funkts. Anal. Prilozh. 4:2, 1-9, 1970

................... 223

Algebraic unsolvability of the problem of Lyapunov stability and the problem of topological classification of singular points of an analytic system of differential equations Funct. Anal. Appl. 4, 173-180, 1970. Translation of Funkts. Anal. Prilozh. 4:3, 1-9, 1970 ................. 231 On the arrangement of ovals of real plane algebraic curves, involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms
Published as "Distribution of ovals of the real plane of algebraic curves, of involutions of fourdimensional smooth manifolds, and the arithmetic of integer-valued quadratic forms" in Funct. Anal. Appl. 5, 169-176, 1971. Translation of Funkts. Anal. Prilozh. 5:3, 1-9, 1971. Translated by Oleg Viro

26

..................................................................................................................... 239

27

The topology of real algebraic curves (works of I.G. Petrovsky and their development) Translation of Usp. Mat. Nauk 28:5, 260-262, 1973. Translated by Oleg Viro ................................ 251 On matrices depending on parameters
Russ. Math. Surv. 22, 29-43, 1971. Translation of Usp. Mat. Nauk 26:2, 101-114, 1971

28 29 30 31

................... 255 .................. 271

Lectures on bifurcations in versal families
Russ. Math. Surv. 27, 54-123, 1972. Translation of Usp. Mat. Nauk 27:5, 119-184, 1972

Versal families and bifurcations of differential equations (Russian) Izd. Inst. Akad. Nauk Ukrain. SSR, Kiev, 42-49, 1972 ............................................................... 341 Remarks on the behaviour of a flow of a three-dimensional ideal fluid in the presence of a small perturbation of the initial vector field
Published as "Notes on the three-dimensional flow pattern of a perfect fluid in the presence of a small perturbation of the initial velocity field" in J. Appl. Math. Mech. 36:2, 236-242, 1972. Translation of Prikl. Mat. Mekh. 36:2, 255-262, 1972

.............................................................................. 349

32

The asymptotic Hopf invariant and its applications
Selecta Math. Sov. 5:4, 327-345, 1986. Translation of Proc. All-Union School in Diff. Eq. with Infinite Number of Variables and in Dyn. Syst. with Infinite Number of Degrees of Freedom, Dilhan 1973, 229-256, 1974

.............................................................................................. 357

33

A magnetic field in a moving conducting fluid (with Ya.B. Zeldovich, A.A.Ruzmaikin, and D.D. Sokolov) Translation of Usp. Mat. Nauk 36:5, 220-221, 1981. Translated by Gerald Gould ........................... 377 A magnetic field in a stationary flow with stretching in a Riemannian manifold (with Ya.B. Zeldovich, A.A.Ruzmaikin, and D.D. Sokolov)
Published as "A magnetic field in a stationary flow with stretching in Riemannian space" in Sov. Phys. JETP 54:6, 1083-1086, 1982. Translation of Zh. Eksp. Teor. Fiz. 81:6, 2052-2056, 1981

34

.... 379


35

Stationary magnetic field in a periodic flow (with Ya.B. Zeldovich, A.A.Ruzmaikin, and D.D. Sokolov)
Published as "Steady-state magnetic field in a periodic flow" in Sov. Phys. Dok. 27:10, 814-816, 1982. Translation of Dokl. Akad. Nauk SSSR, 266:6, 1357-1361, 1982

......................................... 383

36

Some remarks on the antidynamo theorem
Moscow Univ. Math. Bull. 37, 57-66, 1982. Translation of Vestn. Mosk. Univ. Ser. I, Mat. Mekh. 6, 50-57, 1982

................................................................................................................... 387

37 38

Translation of Usp. Mat. Nauk 38:2, 226-227, 1983. Translated by Gerald Gould

Evolution of a magnetic field under the action of transfer and diffusion ........................... 397

The growth of a magnetic field in a three-dimensional steady incompressible flow (with E.I.Korkina)
Moscow Univ. Math. Bull, Ser. I, Math. Mech. 3, 50-54, 1983. Translation of Vestn. Mosk. Univ. Mat. 38:3, 43-46, 1983

..................................................................................................... 399

39

On evolution of a magnetic field under the action of drift and diffusion (Russian) Some Problems in Modern Analysis, 8-21, MGU, Moscow 1984 ................................................. 405 Exponential scattering of trajectories and its hydrodynamical applications
Translation of N.E. Kochin and the Development of Mechanics, 185-193, Nauka, Moscow 1984. Translated by Gerald Gould

40

.............................................................................................. 419

41 42 43 44

Proc. Royal Soc. London A 434:1890, 19-22, 1991

Kolmogorov's hydrodynamic attractors .................................................................. 429 Topological methods in hydrodynamics (with B.A. Khesin) ............................................................................ 433

Annu. Rev. Fluid Mech. 24, 145-166, 1992

MIR, Moscow 1965. Translated by Gerald Gould

Translator's preface to J. Milnor's book "Morse Theory" ................................................................... 455

Henri PoincarÈ: Selected Works in Three Volumes. Vol. I New Methods of Celestial Mechanics ­ Preface. From the editorial board. Comments Nauka, Moscow 1971, 747-752 . Translated by Gerald Gould .................................................... 457 Comments on the paper "On a geometric theorem" by Henri PoincarÈ
H. PoincarÈ: Selected Works in Three Volumes. Vol. II. Klassiki Nauki, Nauka, Moscow 1972, 987-989. Translated by Gerald Gould

45

.................................................................................. 463

Acknowledgements .......................................................................................................... 465


Books or book prefaces by V. Arnold written in 1965-1972, but not included in the "Collected Works" 1. (with A. Avez): "ProblÕmes ergodiques de la mÕcanique classique" (in French), Monographies Internationales de MathÕmatiques Modernes, No. 9, Gauthier-Villars, Paris, 1967, ii+243pp.; English translation: "Ergodic problems of classical mechanics", W. A. Benjamin, Inc., New York-Amsterdam, 1968, ix+286pp. 2. "Obyknovennye differentsialnye uravneniya" (in Russian) Nauka, Moscow, 1971, 239pp.; English translation: "Ordinary differential equations", The M.I.T. Press, Cambridge, 1973, ix+280pp.; Other translations/editions of this book are French (1984), Russian (2nd edition, unrevised, 1975), Russian (3rd edition, revised and extended, 1984), Polish (1975, 1983), Portuguese (1985), German (1980), etc. 3. Editor's preface to a collection of papers "Singularities of differentiable mappings", Mir, Moscow, 1968. 4. Translator's preface to F. Pham's book "Introduction to the topological study of Landau's singularities", Mir, Moscow, 1970. 5. Translator's preface to J. Milnor's "Singular points of complex hypersurfaces", Mir, Moscow, 1971.