Skip to main content

On the generalization of symplectic geometry to multiple integrals in the Calculus of Variations

  • Chapter IV. Symplectic Structures — Mechanics
  • Conference paper
  • First Online:
Book cover Differential Geometrical Methods in Mathematical Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 570))

A table of contents is placed on page 456

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Ph. Antoine, Etude de la structure de certains espaces fonctionnels; applications. Thèse, Université de Paris-Sud (Orsay), 1972.

    Google Scholar 

  2. G.A. Bliss, Lectures on the Calculus of Variations, Chicago University Press, 1946.

    Google Scholar 

  3. E. Cartan, Leçons sur les invariants intégraux. Paris, Hermann, 1921.

    MATH  Google Scholar 

  4. —, Les systèmes différentiels extérieurs et leurs applications géométriques. Paris, Hermann, A.S.I. 1945.

    MATH  Google Scholar 

  5. H. Cartan, Les systèmes d'équations extérieures. Notes minéographiées, Séminaire Julia, Fac. des Sciences de Paris, 1937.

    Google Scholar 

  6. C. Caratheodory, Uber die Variationsrechnung bei mehrfachen Integralen. Acta Szeged, 4 (1929), 193–216.

    MATH  Google Scholar 

  7. R. Courant and D. Hilbert, Methods of Mathematical Physics, I, II. Interscience Publ., New York 1953, 1962.

    MATH  Google Scholar 

  8. P. Dedecker, Sur les intégrales multiples du Calcul des Variations. Comptes rendus du IIIe congrès Nat. des Sci., Bruxelles, 1950.

    Google Scholar 

  9. —, Les systémes d'équations extérieures. Equations différentielles extérieures et Calcul des Variations. Séminaire de Topologie de Strasbourg, 1951 (deux articles).

    Google Scholar 

  10. —, Calcul des variations, formes différentielles et champs géodésiques. Colloques internat. du C.N.R.S., Strasbourg, 1953.

    MATH  Google Scholar 

  11. —, Systèmes différentiels extérieurs, invariants intégraux et suites spectrales. Convegno internazionale di Geometria Differenziale, Venezia, Padova, Bologna e Pisa, 1953.

    Google Scholar 

  12. —, Calcul des Variations et topologie algébrique. Mém. Soc. Roy. Sci. Liège, XIX (1957), 1–216.

    MathSciNet  MATH  Google Scholar 

  13. —, à paraître, Accademia Nazionale dei Lincei, Roma, Contributi del Centro Interdisciplinare di Science Matematiche e. 1. Applicazioni.

    Google Scholar 

  14. Th. de Donder, Théorie invariantive du Calcul des Variations. Paris, Gauthier-Villars, 1935.

    MATH  Google Scholar 

  15. P. Funk, Variationsrechnung und ihre Anwendung in Physik und Technik. Springer Verl., Grundlehren Math. Wiss. Bd. 94, 1962.

    Google Scholar 

  16. P.L. García, Geometría simpléctica en la teoría clásica de campos. Coll. Math. 19 (1968).

    Google Scholar 

  17. —, The Poincaré-Cartan invariant in the Calculus of Variations. Symposia Mathematica, vol. 14, Istituto Nazionale di Alta Matematica, Roma, 1974.

    MATH  Google Scholar 

  18. P.L. García and A. Pérez-Rendón, Symplectic approach to the theory of quantized fields. I, Comm. Math. Phys. 13 (1969), 22–44; II, Archiv for Rat. Mechanics and Analysis 43 (1971), 101–124.

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the Calculus of Variations. Ann. Inst. Fourier, 23 (1973), 203–267.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Hadamard, Leçons sur la propagation des ondes et des équations de l'hydrodynamique. Paris, Hermann, 1903.

    MATH  Google Scholar 

  21. R. Hermann, E. Cartan's Geometric theory of partial differential equations. Advances in Math. 1 (1965), 265–317.

    Article  MathSciNet  MATH  Google Scholar 

  22. —, A differential geometric formalism for multiple integrals in C.V. A.M.S. Summer Institute in Global Analysis, Berkeley, Ca., July 1–26, 1968. See chapt. V in: Lie Algebras and Quantum Mechanics, Benjamin Lect. Notes.

    Google Scholar 

  23. J. Kijowski, A finite-dimensional canonical formalism in the classical field theory. Commun. Math. Phys., 30 (1973), 99–128.

    Article  MathSciNet  Google Scholar 

  24. J. Kijowski and W. Szcyrba, A canonical structure for classical field theories. Commun. Math. Phys. 46 (1976), 183–206.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Klötzler, Mehrdimensionale Variationsrechnung. Birkhäuser Verl., Basel und Stuttgart, 1970.

    Book  MATH  Google Scholar 

  26. C. Lanczos, The Variational principles of Mechanics. University of Toronto Press, 1949.

    Google Scholar 

  27. Th. Lepage, Champs stationnaires, champs géodésiques et formes intégrables, I, II. Bull. Acad. Roy. Belg., Classe des Sciences, 28 (1942), 73–92, 247–268.

    MathSciNet  MATH  Google Scholar 

  28. J. Leray, L'anneau spectral et l'anneau filtré d'homologie d'un espace localement compact et d'une application continue. J. Math. Pures et App. (9) 29 (1950), 1–139.

    MathSciNet  MATH  Google Scholar 

  29. A. Liesen, Feldtheorie in der Variationsrechnung mehrfacher Integralen, I, II. Math. Ann. 171 (1967), 194–218, 273–292.

    Article  MathSciNet  MATH  Google Scholar 

  30. C.B. Morrey, Multiple integrals in the Calculus of Variations. Springer Verl., Grundlehren Math. Wiss. Bd. 130, 1966.

    Google Scholar 

  31. M. Morse, The Calculus of Variations in the large. Amer. Math. Soc. Colloquium Pub. vol. 14, 1934.

    Google Scholar 

  32. R. Palais and S. Smale, A generalized Morse theory. Bull. Amer. Math. Soc., 70 (1964), 165–172.

    Article  MathSciNet  MATH  Google Scholar 

  33. H. Poincaré, Leçons sur les méthodes nouvelles de la Mécanique céleste. Paris, Gauthier Villars, 1892–1899.

    MATH  Google Scholar 

  34. J.P. Serre, Homologie singulière des espaces fibrés; applications. Ann. of Math. 54 (1951), 425–505.

    Article  MathSciNet  MATH  Google Scholar 

  35. E. Vessiot, Sur la théorie des multiplicités et le Calcul des Variations. Bull. Soc. Math. de France, 40 (1912), 68–139.

    MathSciNet  MATH  Google Scholar 

  36. H. Weyl, Geodesic fields. Ann. of Math., 36 (1935), 607–629.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Konrad Bleuler Axel Reetz

Rights and permissions

Reprints and permissions

Copyright information

© 1977 Springer-Verlag

About this paper

Cite this paper

Dedecker, P. (1977). On the generalization of symplectic geometry to multiple integrals in the Calculus of Variations. In: Bleuler, K., Reetz, A. (eds) Differential Geometrical Methods in Mathematical Physics. Lecture Notes in Mathematics, vol 570. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087794

Download citation

  • DOI: https://doi.org/10.1007/BFb0087794

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08068-8

  • Online ISBN: 978-3-540-37498-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics