Skip to main content
SpringerLink
Log in

Representation of the Variational Sequence by Differential Forms

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

In the paper the representation of the finite order variational sequence on fibered manifolds in field theory is studied. The representation problem is completely solved by a generalization of the integration by parts procedure using the concept of the Lie derivative of forms with respect to vector fields along canonical maps of prolongations of fibered manifolds. A close relationship between the obtained coordinate invariant representation of the variational sequence and some familiar objects of physics, such as Lagrangians, dynamical forms, Euler–Lagrange mapping and Helmholtz–Sonin mapping is pointed out and illustrated by examples.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, I. and Duchamp, T.: On the existence of global variational principles, Amer. J. Math. 102 (1980), 781–867.

    Google Scholar 

  2. Anderson, I. M.: Introduction to the variational bicomplex, Contemporary Math. 132 (1992), 51–73.

    Google Scholar 

  3. Bocharov, A. V., Chetverikov, V. N., Duzhin, S. V., Khor’kova, N. G., Krasil’schik, I. S., Samokhin, A. V., Thorkhov, Yu. N., Verbovetsky, A. M. and Vinogradov, A. M.: Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, I. S. Krasil’schik and A. M. Vinogradov (eds), Amer. Math. Soc., 1999.

  4. Dedecker, P. and Tylczyjew, W. M.: Spectral sequences and the inverse problem of calculus of variations, In: Differential Geometric Methods in Mathematical Physics, Proc. Internat. Coll. Aix-en Provence, France, 1979, Lecture Notes in Math. 836, Springer, Berlin, 1980, pp. 498–503.

    Google Scholar 

  5. Grigore, D. R.: The variational sequence on finite jet bundle extensions and the Lagrange formalism, Diff. Geom. Appl. 10 (1999), 43–77.

    Article  Google Scholar 

  6. Kolář, I., Michor, P. and Slovák, J.: Natural Operations in Differential Geometry, Springer, Berlin, 1993.

    Google Scholar 

  7. Krasilschik, I. S., Lychagin, V. V. and Vinogradov, A. M.: Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Advanced Studies in Contemporary Mathematics 1, Gordon & Breach, New York, 1986.

    Google Scholar 

  8. Krbek, M.: The representation of the variational sequence by forms, Ph.D. Thesis, Masaryk University, Brno, 2002.

  9. Krbek, M., Musilová, J. and Kašparová, J.: The variational sequence: Local and global properties, In: Proceedings of the Seminar on Differential Geometry, Math. Publications 2, Silesian University Opava, Opava, 2000, pp. 15–38.

    Google Scholar 

  10. Krbek, M., Musilová, J. and Kašparová, J.: The representation of the variational sequence in field theory, In: L. Kozma, P. T. Nagy and L. Tamássy (eds), Steps in Differential Geometry, Proc. Colloq. Differential Geometry, Debrecen, Hungary, 2000, Debrecen, 2001, pp. 147–160.

  11. Krbek, M. and Musilová, J.: Representation of the variational sequence, Rep. Math. Phys. 51 (2003), 251–258.

    Article  Google Scholar 

  12. Krupka, D.: Variational sequences on finite order jet spaces, In: J. Janyška and D. Krupka (eds), Differential Geometry and its Applications, Proc. Conf. Brno, Czechoslovakia, 1989, World Scientific, Singapore, 1990, pp. 236–254.

    Google Scholar 

  13. Krupka, D.: The geometry of Lagrange structures, Preprint Series on Global Analysis GA 7/1997, Mathematical Institute, Silesian University, Opava, 1997.

  14. Krupka, D. and Janyška, J.: Lectures on Differential Invariants, J. E. Purkyně University, Brno, 1990.

    Google Scholar 

  15. Krupka, D. and Musilová, J.: Trivial Lagrangians in field theory, Diff. Geom. Appl. 9 (1998), 293–305.

    Article  Google Scholar 

  16. Krupka, D. and Musilová, J.: Recent results in variational sequence theory, In: L. Kozma, P. T. Nagy and L. Tamássy (eds), Steps in Differential Geometry, Proc. Colloq. Differential Geometry, Debrecen, Hungary, 2000, Debrecen, 2001, pp. 161–186.

  17. Krupka, D.: Variational sequences and bicomplexes, In: O. Kowalski, D. Krupka and J. Slovák (eds), Diff. Geom. Appl., Proc. Conf. Brno, Czech Republic, 1998, Masaryk University, Brno, 1999, pp. 525–532.

    Google Scholar 

  18. Krupková, O.: The Geometry of Ordinary Variational Equations, Lecture Notes in Math. 1678, Springer, Berlin, 1997.

    Google Scholar 

  19. Mangiarotti, L. and Modugno, M.: New operators on jet spaces, Ann. Fac. Sci. Toulouse 2(5) (1983), 171–198.

    Google Scholar 

  20. Musilová, J.: Variational sequence in higher order mechanics, In: J. Janyška, I. Kolář and J. Slovák (eds), Differential Geometry and its Applications, Proc. Conf. Brno, Czech Republic, 1995, Masaryk University, Brno, 1996, pp. 611–624.

    Google Scholar 

  21. Musilová, J. and Krbek, M.: A note to the representation of the variational sequence in mechanics, In: I. Kolář, O. Kowalski, D. Krupka and J. Slovák (eds), Differential Geometry and its Applications, Proc. Conf. Brno, Czech Republic, 1998, Masaryk University, Brno, 1999, pp. 511–523.

    Google Scholar 

  22. Saunders, D.: The Geometry of Jet Bundles, London Math. Soc. Lecture Note Series, Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  23. Štefánek, J.: A representation of the variational sequence in higher order mechanics, In: J. Janyška, I. Kolář and J. Slovák (eds), Differential Geometry and its Applications, Proc. Conf. Brno, Czech Republic, 1995, Masaryk University, Brno, 1996, pp. 469–478.

    Google Scholar 

  24. Šeděnková, J.: On the invariant variational sequences in mechanics, In: Rend. Cont. Mat. Palermo, Proc. 22-nd Winter School Geometry and Physics, Srní, Czech Republic, 2002, pp. 185–190.

  25. Takens, F.: A global version of the inverse problem of calculus of variations, J. Diff. Geom. 14 (1979), 543–562.

    Google Scholar 

  26. Tulczyjew, W. M.: The Euler–Lagrange resolution, In: Differential Geometric Methods in Mathematical Physics, Internat. Coll., Aix en Provence, 1979, Lecture Notes in Math. 836, Springer, Berlin, 1980, pp. 22–48.

    Google Scholar 

  27. Vinogradov, A. M.: On the algebro-geometric foundations of Lagrangian field theory, Soviet Math. Dokl. 18 (1977), 1200–1204.

    Google Scholar 

  28. Vinogradov, A. M.: A spectral sequence associated with a non-linear differential equation, and algebro-geometric foundations of Lagrangian field theory with constraints, Soviet Math. Dokl. 19 (1978), 144–148.

    Google Scholar 

  29. Vinogradov, A. M.: The \({\mathcal{C}}\) -spectral sequence, Lagrangian formalism and conservation laws I and II, J. Math. Anal. Appl. 100(1) (1984), 1–129.

    Article  Google Scholar 

  30. Vitolo, R.: Finite order Lagrangian bicomplexes, Math. Proc. Cambridge Philos. Soc. 125 (1999), 321–333.

    Article  Google Scholar 

  31. Vitolo, R.: On different geometric formulations of Lagrangian formalism, Diff. Geom. Appl. (1999), 225–255.

  32. Vitolo, R.: Finite order formulation of Vinogradov’s C-spectral sequence, Acta Appl. Math. (2002), 133–154.

Download references

Author information

Authors and Affiliations

  1. Institute of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Brno, Czech Republic

    Michael Krbek & Jana Musilová

Authors
  1. Michael Krbek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jana Musilová
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Krbek.

Additional information

Mathematics Subject Classifications (2000)

58E99, 49F99.

Jana Musilová: Research of both authors supported by grants MSM 0021622409 and 201/03/0512.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Krbek, M., Musilová, J. Representation of the Variational Sequence by Differential Forms. Acta Appl Math 88, 177–199 (2005). https://doi.org/10.1007/s10440-005-4980-x

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-005-4980-x

Keywords

Search

Navigation