Constraints in Euler-Poincaré Reduction of Field Theories

Abstract

The goal of this short note is to show the geometric structure of the Euler-Poincaré reduction procedure in Field Theories with special emphasis on the nature of the set of variations and the set of admissible sections. The method of Lagrange multipliers is also applied for a deeper study of these constraints.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Abraham, R., Marsden, J.E.: Foundations of Mechanics. Benjamin/Cummings Publishing, Advanced Book Program, Reading (1978)

    MATH  Google Scholar 

  2. 2.

    Anderson, I.M., Fels, M.E., Torre, C.: Group invariant solutions without transversality and the principle of symmetric criticality. In: Bäcklund and Darboux transformations. The Geometry of Solitons, Halifax, NS, 1999. CRM Proc. Lecture Notes, vol. 29, pp. 95–108. Am. Math. Soc., Providence (2001)

    Google Scholar 

  3. 3.

    Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Graduate Texts in Mathematics, vol. 60. Springer, New York (1989)

    Google Scholar 

  4. 4.

    Castrillón López, M., García, P.L., Ratiu, T.: Euler-Poincaré reduction on principal bundles. Lett. Math. Phys. 58(2), 167–180 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Castrillón, M., García, P.L., Rodrigo, C.: Euler-Poincaré reduction in principal fibre bundles and the problem of Lagrange. Differ. Geom. Appl. 25(6), 585–593 (2007)

    MATH  Article  Google Scholar 

  6. 6.

    Castrillón López, M., Marsden, J.: Some remarks on Lagrangian and Poisson reduction for field theories. J. Geom. Phys. 48(1), 52–83 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Castrillón López, M., Muñoz Masqué, J.: The geometry of the bundle of connections. Math. Z. 236(4), 797–811 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    García, P.L.: The Poincaré-Cartan invariant in the calculus of variations. Symp. Math. 14, 219–246 (1974)

    Google Scholar 

  9. 9.

    Giachetta, G., Mangiarotti, L., Sarnanashvily, G.: New Lagrangian and Hamiltonian Methods in Field Theory. World Scientific, Singapore (1997)

    MATH  Google Scholar 

  10. 10.

    Goldschmidt, H., Sternberg, S.: The Hamiltonian-Cartan formalism in the calculus of variations. Ann. Inst. Fourier 23(1), 203–267 (1973)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Kobayashi, S., Numizu, K.: Foundations of Differential Geometry, vol. I. Wiley-Interscience, New York (1963)

    MATH  Google Scholar 

  12. 12.

    Kobayashi, S., Numizu, K.: Foundations of Differential Geometry, vol. II. Wiley-Interscience, New York (1969)

    MATH  Google Scholar 

  13. 13.

    Marsden, J.E., Patrick, G., Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199, 351–395 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, 2nd edn. Texts in Applied Mathematics, vol. 17. Springer, New York (1999)

    MATH  Google Scholar 

  15. 15.

    Moser, J., Veselov, A.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139(2), 217–243 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Saunders, D.J.: The Geometry of Jet Manifolds. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  17. 17.

    Vankerschaver, J.: Euler-Poincaré reduction for discrete field theories. J. Math. Phys. 48(3) (2007)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to M. Castrillón López.

Additional information

The author was partially supported by Ministerio de Ciencia e Innovacion (Spain) under grants MTM2010-19111 and MTM2011-22528.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Castrillón López, M. Constraints in Euler-Poincaré Reduction of Field Theories. Acta Appl Math 120, 87–99 (2012). https://doi.org/10.1007/s10440-012-9695-1

Download citation

Keywords

  • Euler-Poincaré equations
  • Lagrange multipliers
  • Reduction
  • Symmetries
  • Variational calculus

Mathematics Subject Classification (2010)

  • 58A15
  • 70S05
  • 70S15