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Vladimir Igorevich Arnold and the Invention of Symplectic Topology

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Contact and Symplectic Topology

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 26))

Abstract

In 1965, with a Comptes rendus note of Vladimir Arnold, a new discipline, symplectic topology, was born. In 1986, its (remarkable) first steps were reported by Vladimir Arnold himself. In the meantime…

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Notes

  1. 1.

    This one was Jean-Claude Sikorav.

  2. 2.

    If A is not a diffeomorphism, counter-examples can be constructed with n=1. Note of V.I. Arnold.

  3. 3.

    All of this was very fast, including the mail from Paris to Palermo (recall that there was no air-mail and that Palermo was already on an island). All the dates given here can be found on the printed journal. For some reason (which I was unable to understand), they were cut out in Poincaré’s complete works, even the date he probably wrote himself at the end of his paper.

  4. 4.

    I copied this example from [54].

  5. 5.

    “Dernier”, which means last, was not in the American title. Also, the translation kept the original phrasing “théorème de géométrie” rather than “théorème géométrique”, as in English.

  6. 6.

    Note that, in the preface Marston Morse wrote for the 1966 edition of this 1927 book, he insisted on the relationship between Birkhoff’s work on periodic orbits and “the work of Moser, Arnold and others on stability”.

  7. 7.

    See my paper, An extension of Poincaré’s last geometric theorem, Acta Mathematica, vol. 47 (1926). Note of G.D. Birkhoff.

  8. 8.

    Soon translated in English as [17].

  9. 9.

    This was also very fast: the translation in English in Russian mathematical surveys would arrive in the libraries less than one year after the publication of the Russian original.

  10. 10.

    This one was Jürgen Moser.

  11. 11.

    Note that Figure 3 contains a 5-fold covering and a 3-fold covering of the map in Figure 2.

  12. 12.

    A stability problem and ergodic properties of classical dynamical systems.

  13. 13.

    Let me mention here the beautiful little book [13] he wrote on this subject for a general audience in the eighties.

  14. 14.

    If you don’t know, look at [59].

  15. 15.

    Again, you should read [59].

  16. 16.

    In any case, you should read [59].

  17. 17.

    Let me quote what I wrote at the very moment I learned his death in a short online paper [23]: he was charming, provocative, brilliant, cultured, funny, caustic sometimes even wicked, adorable, quick, lively, incisive, yes, all this together.

  18. 18.

    Nothing is perfect. One thing I never understood and never dared to ask, is why there is a Lagrangian but no Hamiltonian treatment of the spinning top in this book.

  19. 19.

    It seems that the idea was Jerry Marsden’s. The translation was made by Karen Vogtmann and edited by Alan Weinstein, who knew the domain and its lexicon better.

  20. 20.

    This one was [5–7], before becoming [15].

  21. 21.

    Mikhail Gromov’s paper [49] (at icm Nice 1970), where the h-principle for Lagrangian immersions was announced, should also be mentioned.

  22. 22.

    Math. Reviews waited until May 1979 to publish a review of the 1974 Russian edition. The reviewer was very enthusiastic, so enthusiastic that he added a very elegant remark:

    The reader should be aware that the reviewer participated in the English translation of the work under review, and so has been prejudiced in favor of the book by the pleasure which that project provided.

    This one was Alan Weinstein.

  23. 23.

    The French translation has no parenthesis, only a comma.

  24. 24.

    and Gromov.

  25. 25.

    Not taking Moser’s homotopy method [55] (see also [62]) into account.

  26. 26.

    Replacing Fourier series by a broken geodesics idea, Chaperon himself soon gave a more elementary proof in [39], which is the basis of the proof given in [54].

  27. 27.

    And this became one of the most expensive Springer series in the 1990’s.

  28. 28.

    The contents of [2] would deserve a whole paper… Note that the adjectives Lagrangian, Legendrian, in the sense used in symplectic geometry, were invented in [2].

  29. 29.

    Allow me to mention that this was the way I entered symplectic geometry. See [44] and [20–22].

  30. 30.

    See [56].

  31. 31.

    Note that Nikishin’s article [57] quoted in [35] more or less disappeared from the literature. The statement and a (different) proof were given in [37] without any reference. A few years later the conjecture for CP n was announced by Fortune and Weinstein [47] then published by Fortune [46] with no mention that the CP 1-case was already known. Even in [14] the S 2-case is mentioned as an analogous of Poincaré’s geometric theorem, but not in connection with the proof of the conjecture for surfaces (attributed both to Eliashberg [43] and Floer [45]).

  32. 32.

    By “we” here, I mean the community. I could also mention that some of us (and here, by “us”, I mean the two authors of [24]) wrote a textbook to explain all this (a translation to English will be available soon).

  33. 33.

    See Review 52#5337 on Math. Reviews. Already in 1972, it was possible to publish double translations without checking the signification. The title of our favorite Poincaré paper [58] became there “A certain theorem of geometry”.

  34. 34.

    Joined by a one-parameter family of symplectomorphisms with single valued (but time-dependent) Hamiltonians. Note of V.I. Arnold.

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Acknowledgement

I thank Bob Stanton and Marcus Slupinski for their help with the translation of the adjectives in footnote 17.

Many thanks to Alan Weinstein and Karen Vogtmann, who were so kind to send me recollections and information and also to Alan, for allowing me to publish an excerpt of a letter Arnold had sent to him.

I am very grateful to Mihai Damian, Leonid Polterovich and Marc Chaperon, who kindly agreed to read preliminary versions of this paper, for their friendly comments and suggestions.

The last sentence in this paper was inspired by [61].

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  1. Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084, Strasbourg cedex, France

    Michèle Audin

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  1. Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, Orsay, France

    Frédéric Bourgeois

  2. Lab. de Mathématiques Jean Leray UMR6629, University of Nantes, Nantes, France

    Vincent Colin

  3. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    András Stipsicz

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Audin, M. (2014). Vladimir Igorevich Arnold and the Invention of Symplectic Topology. In: Bourgeois, F., Colin, V., Stipsicz, A. (eds) Contact and Symplectic Topology. Bolyai Society Mathematical Studies, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-02036-5_1

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