Skip to main content
SpringerLink
Log in
Book cover

Catastrophe Theory pp 67–75Cite as

Symplectic and Contact Geometry

  • Chapter

  • 394 Accesses

Abstract

Many questions in singularity theory (for instance, the classification of the singularities of caustics and wave fronts, and also the investigation of the various singularities in optimization and variational calculus problems) become understandable only within the framework of the geometry of symplectic and contact manifolds, which is refreshingly unlike the usual geometries of Euclid, Lobachevskij and Riemann.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographical Comments

The theory of singularities of Lagrangian mappings started in 1966. See:

  1. A Lagrangian equivalence between two Lagrangian singularities is a mapping of the first Lagrangian fibration to the second which maps fibres onto fibres, takes the first symplectic structure into the second and carries the first Lagrangian submanifold into the second.

    Google Scholar 

  2. V. I. Arnol’d: Characteristic class entering in quantization conditions. Funkts. Anal. Prilozh. 1:1 (1967), 1–14 (English translation: Funct. Anal. Appl. 1 (1967), 1-13).

    Article  Google Scholar 

  3. L. Hörmander: Fourier integral operators, I. Acta Math. 127 (1971), 79–183.

    Article  MATH  Google Scholar 

  4. V. I. Arnol’d: Integrals of rapidly oscillating functions and singularities of projections of Lagrangian manifolds. Funkts. Anal. Prilozh. 6:3 (1972), 61–62 (English translation: Funct. Anal. Appl. 6 (1972), 222-224).

    Google Scholar 

  5. V. I. Arnol’d: Normal forms for functions near degenerate critical points, the Weyl groups of A k, D k, E k and Lagrangian singularities. Funkts. Anal. Prilozh. 6:4 (1972), 3–25 (English translation: Funct. Anal. Appl. 6 (1972), 254-272).

    Google Scholar 

See also:

The Legendre singularities theory was published in 1974 in the Russian edition of:

  • V. I. Arnol’d: Mathematical Methods of Classical Mechanics. Nauka, Moscow 1974 (English translation: Graduate Texts in Mathematics 60, Springer-Verlag, New York-Heidelberg-Berlin 1978).

    Google Scholar 

  • V. I. Arnol’d: Critical points of smooth functions, in: Proceedings of the International Congress of Mathematicians (Vancouver, 1974) vol. 1. Canadian Mathematical Congress 1975, pp. 19–39.

    Google Scholar 

See also:

  • M. J. Sewell: On Legendre transformations and elementary catastrophes. Math. Proc. Camb. Philos. Soc. 82 (1977), 147–163.

    Article  MathSciNet  MATH  Google Scholar 

  • J.-G. Dubois, J.-P. Dufour: La théorie des catastrophes. V. Transformées de Legendre et thermodynamique. Ann. Inst. Henri Poincaré, Nouv. Ser., Sect. A 29 (1978), 1–50.

    MathSciNet  MATH  Google Scholar 

For the open swallowtail, see:

  • V. I. Arnol’d: Lagrangian manifolds with singularities, asymptotic rays, and the open swallowtail. Funkts. Anal. Prilozh. 15:4 (1981), 1–14 (English translation: Funct. Anal. Appl. 15 (1981), 235-246).

    Google Scholar 

  • V. I. Arnol’d: Singularities of Legendre varieties, of evolvents and of fronts at an obstacle, Ergodic Theory Dyn. Syst. 2 (1982), 301–309.

    Google Scholar 

  • A. B. Givental’: Lagrangian varieties with singularities and irreducible sl2-modules. Usp. Mat. Nauk 38:6 (1983), 109–110 (English translation: Russ. Math. Surv. 38:6 (1983), 121-122).

    MathSciNet  MATH  Google Scholar 

  • A. B. Givental’: Varieties of polynomials having a root of fixed co-multiplicity and the generalized Newton equation. Funkts. Anal. Prilozh. 16:1 (1982), 13–18 (English translation: Manifolds of polynomials having a root of fixed multiplicity, and the generalized Newton equation. Funct. Anal. Appl. 16 (1982), 10-14).

    MathSciNet  Google Scholar 

The Givental’ theorem on submanifolds of symplectic and contact spaces first appears in the 1981 Russian edition of the present booklet. It is a generalization of the Darboux-Weinstein theorem (the difference being that Givental' requires no information on transverse vectors). The Darboux-Weinstein theorem is proved in :

  • A. Weinstein: Lagrangian submanifolds and hamiltonian systems. Ann. Math., II. Ser. 98 (1973), 377–410.

    Article  MATH  Google Scholar 

  • V. I. Arnol’d, A. B. Givental’: Symplectic geometry (in Russian), in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat, Fundamental’nye na-pravleniya (Contemporary Problems of Mathematics, Fundamental directions) 4, Viniti, Moscow 1985 (to be translated by Springer-Verlag).

    Google Scholar 

  • V. I. Arnol’d: Singularities in variational calculus, in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. (Contemporary Problems of Mathematics) 22, Viniti, Moscow 1983, pp. 3–55 (English translation: J. Sov. Math. 27 (1984), 2679-2713).

    Google Scholar 

  • R. B. Melrose: Equivalence of glancing hypersurfaces. Invent. Math. 37 (1976), 165–191.

    Article  MathSciNet  MATH  Google Scholar 

  • R. B. Melrose: Equivalence of glancing hypersurfaces. II. Math. Ann. 255 (1981), 159–198.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Martinet: Sur les singularités des formes différentielles. Ann. Inst. Fourier, 20, 1 (1970), 95–178.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Roussarie: Modèles locaux de champs et de formes. Astérisque 30 (1975).

    Google Scholar 

  • M. Golubitsky, D. Tischler: An example of moduli for singular symplectic forms. Invent. Math. 38 (1977), 219–225.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Moscow, Moscow, 117234, USSR

    Vladimir Igorevich Arnold

Authors
  1. Vladimir Igorevich Arnold
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Arnold, V.I. (1986). Symplectic and Contact Geometry. In: Catastrophe Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-96937-9_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-96937-9_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16199-8

  • Online ISBN: 978-3-642-96937-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation