V.I. Arnold (1937–2010)

  • Leonid Polterovich
  • Inna ScherbakEmail author
Historischer Beitrag


This article is devoted to V.I. Arnold, a famous mathematician who passed away in June 2010. We discuss life and times of Arnold, and review some of his seminal contributions to symplectic geometry and singularities theory which were among Arnold’s favorite subjects.


Symplectic manifold Lagrangian submanifold Hamiltonian system Isolated singularity Reflection group 

Mathematics Subject Classification (2000)

01A70 57R17 70H08 14B05 11F22 


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  1. 1.
    Arnold, V.I.: On Liouville’s theorem concerning integrable problems of dynamics. Transl. Am. Math. Soc. 61, 292–296 (1967) [Russian original: 1963] Google Scholar
  2. 2.
    Arnold, V.I.: Proof of a theorem by A.N. Kolmogorov on the persistence of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv. 18, 9–36 (1963) CrossRefGoogle Scholar
  3. 3.
    Arnold, V.I.: Sur une propriété topologique des applications globalement canoniques de la mécanique classique. C. R. Acad. Sci. Paris 261, 3719–3722 (1965) MathSciNetGoogle Scholar
  4. 4.
    Arnold, V.I.: Instability of dynamical systems with several degrees of freedom. Sov. Math. 5, 581–585 (1964) Google Scholar
  5. 5.
    Arnold, V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16, 319–361 (1966) CrossRefGoogle Scholar
  6. 6.
    Arnold, V.I.: On a characteristic class entering into conditions of quantization (Russian). Funct. Anal. Appl. 1, 1–13 (1967) CrossRefGoogle Scholar
  7. 7.
    Arnold, V.I.: A stability problem and ergodic properties of classical dynamical systems. In: Proc. Internat. Congr. Math. (Moscow), pp. 387–392 (1966) Google Scholar
  8. 8.
    Arnold, V.I.: The cohomology ring of the colored braid group. Math. Notes 5, 138–140 (1969) CrossRefGoogle Scholar
  9. 9.
    Arnold, V.I.: On the arrangement of the ovals of real plane curves, involutions of 4-dimensional smooth manifolds, and the arithmetic of integral quadratic forms. Funct. Anal. Appl. 5, 169–176 (1971) Google Scholar
  10. 10.
    Arnold, V.I.: Modes and quasimodes. Funct. Anal. Appl. 6, 94–101 (1972) CrossRefGoogle Scholar
  11. 11.
    Arnold, V.I.: Normal forms for functions near degenerate critical points, the Weyl groups A k, D k, E k and Lagrangian singularities. Funct. Anal. Appl. 6, 235–272 (1972) Google Scholar
  12. 12.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60. Springer, New York (1989). [Russian original: 1974] Google Scholar
  13. 13.
    Arnold, V.I.: Critical points of functions on a manifold with boundary, the simple Lie groups B k, C k, F 4 and singularities of evolutes. Russ. Math. Surv. 33, 99–116 (1978) CrossRefGoogle Scholar
  14. 14.
    Arnold, V.I.: Indices of singular points of 1-forms on a manifold with boundary, convolution of invariants of reflection groups, and singular projections of smooth surfaces. Russ. Math. Surv. 34, 1–42 (1979) CrossRefGoogle Scholar
  15. 15.
    Arnold, V.I.: First steps in symplectic topology. Russ. Math. Surv. 41, 1–21 (1986) CrossRefGoogle Scholar
  16. 16.
    Arnold, V.I.: Catastrophe Theory. In: Dynamical Systems V. Encyclopaedia of Mathematical Sciences, vol. 5. Springer, Berlin (1994) [Russian original: 1986] Google Scholar
  17. 17.
    Arnold, V.I.: Huygens & Barrow, Newton & Hooke. Birkhäuser, Basel (1990) CrossRefGoogle Scholar
  18. 18.
    Arnold, V.I.: Singularities of caustics and wave fronts. In: Mathematic and Its Applications (Soviet Series), vol. 62. Kluwer Academic, Dordrecht (1990) Google Scholar
  19. 19.
    Arnold, V.I.: On the teaching of mathematics. Russ. Math. Surv. 53, 229–236 (1998) zbMATHCrossRefGoogle Scholar
  20. 20.
    Arnold, V.I.: Yesterday and Long Ago. Springer, Berlin (2007). [Russian original: “Istorii davnie i nedavnie” 3rd edn, PHASIS, Moscow, 2006] zbMATHGoogle Scholar
  21. 21.
    Demidovich, V.B.: Interview with V.I. Arnold. In: Mekhmatiane Vspominaut [Employees of the University Recollect]: 2 [Russian], pp. 25–58. Moscow State University, Moscow (2009) Google Scholar
  22. 22.
    Arnold, V.I., Avez, A.: Ergodic Problems of Classical Mechanics. Benjamin, New York (1968) Google Scholar
  23. 23.
    Arnold, V.I., Goryunov, V.V., Lyashko, O.V., Vasil’ev, V.A.: Singularity theory I: local and global theory. In: Dynamical Systems VI. Encyclopaedia of Mathematical Sciences, vol. 6. Springer, Berlin (1993). Singularity Theory II: Classification and Applications. In: Dynamical systems VIII. Encyclopaedia of Mathematical Sciences, vol. 8. Springer, Berlin (1993) Google Scholar
  24. 24.
    Arnold, V.I., Goryunov, V.V., Lyashko, O.V., Vasil’ev, V.A.: Singularity theory II: classification and applications. In: Dynamical Systems VIII. Encyclopaedia of Mathematical Sciences, vol. 8. Springer, Berlin (1993) Google Scholar
  25. 25.
    Arnold, V.I.: Some remarks on symplectic monodromy of Milnor fibrations. In: Hofer, H., Taubes, C., Weinstein, A., Zehnder, E. (eds.) The Floer Memorial Volume. Progr. Math., vol. 133, pp. 99–104. Birkhäuser, Boston (1995) CrossRefGoogle Scholar
  26. 26.
    Arnold, V.I., Khesin, B.: Topological Methods in Hydrodynamics. Applied Mathematical Sciences, vol. 125. Springer, New York (1998) zbMATHGoogle Scholar
  27. 27.
    Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical aspects of classical and celestial mechanics. In: Dynamical Systems III. Encyclopaedia of Mathematical Sciences, vol. 3. Springer, Berlin (2006) Google Scholar
  28. 28.
    Buhovsky, L.: The Maslov class of Lagrangian tori and quantum products in Floer cohomology. J. Topol. Anal. 2, 57–75 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Bourbaki, N.: Lie Groups and Lie Algebras. Elements of Mathematics. Springer, Berlin (2002). Chaps. IV–VI [French original: 1968] zbMATHCrossRefGoogle Scholar
  30. 30.
    Chekanov, Y.: Critical points of quasifunctions, and generating families of Legendrian manifolds. Funct. Anal. Appl. 30, 118–128 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Chekanov, Y.: Lagrangian tori in a symplectic vector space and global symplectomorphisms. Math. Z. 223, 547–559 (1996) MathSciNetzbMATHGoogle Scholar
  32. 32.
    Conley, C.C., Zehnder, E.: The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold. Invent. Math. 73, 33–49 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Eliashberg, Y.: Symplectic topology in the nineties. Differ. Geom. Appl. 9, 59–88 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Floer, A.: Morse theory for Lagrangian intersections. J. Differ. Geom. 28, 513–547 (1988) MathSciNetzbMATHGoogle Scholar
  35. 35.
    Floer, A.: Cuplength estimates on Lagrangian intersections. Commun. Pure Appl. Math. 42, 335–356 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Givental, A.: Singularity theory and symplectic topology. In: The Arnoldfest, Toronto, ON, 1997. Fields Inst. Commun., vol. 24, pp. 201–207. Amer. Math. Soc., Providence (1999) Google Scholar
  37. 37.
  38. 38.
    Gusein-Zade, S., Varchenko, A.: Vladimir Arnold. In: European Mathematical Society Newsletter No. 78, pp. 28–29 (December 2010) Google Scholar
  39. 39.
    Hofer, H.: Lagrangian embeddings and critical point theory. Ann. Inst. H. Poincaré, Anal. Non Lineaire 2, 407–462 (1985) MathSciNetzbMATHGoogle Scholar
  40. 40.
    Hofer, H., Zehnder, E.: Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser, Basel (1994) zbMATHCrossRefGoogle Scholar
  41. 41.
    Kaloshin, V., Levi, M.: An example of Arnold diffusion for near-integrable Hamiltonians. Bull. Am. Math. Soc. 45, 409–427 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Kolmogorov, A.N.: On the persistence of conditionally periodic motions under a small change of the Hamilton function. Dokl. Akad. Nauk SSSR 98, 527–530 (1954). (in Russian). English translation: A.N. Kolmogorov. In: Casati, G., Ford J. (eds.) Stochastic Behavior in Classical and Quantum Hamiltonian Systems. Lecture Notes in Physics, vol. 93, pp. 51–56. Springer, Berlin (1979) MathSciNetzbMATHGoogle Scholar
  43. 43.
    Laudenbach, F., Sikorav, J.-C.: Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibre cotangent. Invent. Math. 82, 349–357 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Maslov, V.P.: Théorie des perturbations et méthodes asymptotiques. Dunod, Paris (1972). [Russian original: 1965] zbMATHGoogle Scholar
  45. 45.
    McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Oxford Mathematical Monographs. Clarendon Press, New York (1998) zbMATHGoogle Scholar
  46. 46.
    McDuff, D., Salamon, D.: J-Holomorphic Curves and Symplectic Topology. Amer. Math. Soc., Providence (2004) zbMATHGoogle Scholar
  47. 47.
    Moser, J.: On invariant curves of area preserving mappings of an annulus. Nachr. Akad. Wiss. Gött., Math. Phys. Kl. 1–20 (1962) Google Scholar
  48. 48.
  49. 49.
    Schwarz, M.: Morse Homology. Progress in Mathematics, vol. 111. Birkhäuser, Basel (1993) zbMATHCrossRefGoogle Scholar
  50. 50.
    Seidel, P.: Lagrangian two-spheres can be symplectically knotted. J. Differ. Geom. 52(1), 145–171 (1999) MathSciNetzbMATHGoogle Scholar
  51. 51.
    Shcherbak, O.: Wave fronts and reflection groups. Russ. Math. Surv. 43, 149–194 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Sikorav, J.-C.: Problèmes d’intersections et de points fixes en géométrie hamiltonienne. Comment. Math. Helv. 62, 62–73 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Weinstein, A.: Lectures on symplectic manifolds. In: Expository Lectures from the CBMS Regional Conference Held at the University of North Carolina, 8–12 March, 1976. Regional Conference Series in Mathematics, vol. 29. Am. Math. Soc., Providence (1977) Google Scholar
  54. 54.
  55. 55.
    Zehnder, E.: Generalized implicit function theorems with applications to some small divisor problems I, II. Commun. Pure Appl. Math. 28, 91–140 (1975); 29, 49–111 (1976) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Vieweg+Teubner und Deutsche Mathematiker-Vereinigung 2011

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA

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