Central European Journal of Mathematics

, Volume 11, Issue 11, pp 1960–1981 | Cite as

The geometry of the space of Cauchy data of nonlinear PDEs

Research Article

Abstract

First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways — for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework for the space of Cauchy data for nonlinear PDEs. As an example, we derive a general notion of transversality conditions in the Calculus of Variations.

Keywords

Geometry of PDEs Cauchy data Lagrangian formalism Fiber bundles Jet spaces Flags 

MSC

14M15 35A99 53B15 53C80 57R99 58A20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor’kova N.G., Krasil’shchik I.S., Samokhin A.V., Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Transl. Math. Monogr., 182, American Mathematical Society, Providence, 1999Google Scholar
  2. [2]
    Bott R., Tu L.W., Differential Forms in Algebraic Topology, Grad. Texts in Math., 82, Springer, New York-Berlin, 1982MATHCrossRefGoogle Scholar
  3. [3]
    van Brunt B., The Calculus of Variations, Universitext, Springer, New York, 2004Google Scholar
  4. [4]
    Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior Differential Systems, Math. Sci. Res. Inst. Publ., 18, Springer, New York, 1991MATHCrossRefGoogle Scholar
  5. [5]
    Giaquinta M., Hildebrandt S., Calculus of Variations. I, Grundlehren Math. Wiss., 310, Springer, Berlin, 1996Google Scholar
  6. [6]
    Kijowski J., A simple derivation of canonical structure and quasi-local Hamiltonians in general relativity, Gen. Relativity Gravitation, 1997, 29(3), 307–343MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Krasil’shchik J., Verbovetsky A., Geometry of jet spaces and integrable systems, J. Geom. Phys., 2011, 61(9), 1633–1674MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Krupka D., Of the structure of the Euler mapping, Arch. Math. (Brno), 1974, 10(1), 55–61MathSciNetGoogle Scholar
  9. [9]
    Michor P.W., Manifolds of Differentiable Mappings, Shiva Mathematics Series, 3, Shiva Publishing, Nantwich, 1980MATHGoogle Scholar
  10. [10]
    Moreno G., A C-spectral sequence associated with free boundary variational problems, In: Geometry, Integrability and Quantization, Avangard Prima, Sofia, 2010, 146–156Google Scholar
  11. [11]
    Vinogradov A.M., Many-valued solutions, and a principle for the classification of nonlinear differential equations, Dokl. Akad. Nauk SSSR, 1973, 210, 11–14 (in Russian)MathSciNetGoogle Scholar
  12. [12]
    Vinogradov A.M., The C-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory, J. Math. Anal. Appl., 1984, 100(1), 1–40MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Vinogradov A.M., The C-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory, J. Math. Anal. Appl., 1984, 100(1), 41–129MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Vinogradov A.M., Geometric singularities of solutions of nonlinear partial differential equations, In: Differential Geometry and its Applications, Brno, 1986, Math. Appl. (East European Ser.), 27, Reidel, Dordrecht, 1987, 359–379Google Scholar
  15. [15]
    Vinogradov A.M., Cohomological Analysis of Partial Differential Equations and Secondary Calculus, Transl. Math. Monogr., 204, American Mathematical Society, Providence, 2001Google Scholar
  16. [16]
    Vinogradov A.M., Moreno G., Domains in infinite jet spaces: the C-spectral sequence, Dokl. Math., 2007, 75(2), 204–207MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Vitagliano L., Secondary calculus and the covariant phase space, J. Geom. Phys., 2009, 59(4), 426–447MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Vitagliano L., private communication, 2010Google Scholar

Copyright information

© Versita Warsaw and Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Mathematical Institute in OpavaSilesian University in OpavaOpavaCzech Republic

Personalised recommendations