no. 3
A new Lagrangian dynamic reduction in field theory
[Une nouvelle réduction Lagrangienne dynamique en théorie des champs]
Gay-Balmaz, François1 ; Ratiu, Tudor S.1
1 École Polytechnique Fédérale de Lausanne Section de Mathématiques Station 8 1015 Lausanne (Switzerland)
Annales de l'Institut Fourier, Tome 60 (2010) no. 3, pp. 1125-1160.

Considérons une théorie des champs décrite par une densité lagrangienne définie sur le fibré des jets d’ordre 1 d’un fibré principal. Si la densité est invariante sous l’action du groupe de structure du fibré il y a deux approches possibles pour réduire le système : l’approche covariante et l’approche dynamique. Dans cet article nous montrons que ces deux approches produisent les mêmes équations réduites. Afin d’obtenir ce résultat, nous construisons, à partir de la densité lagrangienne, un nouveau lagrangien défini sur un espace de dimension infinie et invariant sous l’action du groupe des transformations de jauge.

For symmetric classical field theories on principal bundles there are two methods of symmetry reduction: covariant and dynamic. Assume that the classical field theory is given by a symmetric covariant Lagrangian density defined on the first jet bundle of a principal bundle. It is shown that covariant and dynamic reduction lead to equivalent equations of motion. This is achieved by constructing a new Lagrangian defined on an infinite dimensional space which turns out to be gauge group invariant.

DOI : 10.5802/aif.2549
Classification : 70S05, 70S15, 70H03, 70H30, 70H99, 58E30, 58E40
Keywords: Covariant reduction, dynamic reduction, affine Euler-Poincaré equation, covariant Euler-Poincaré equation, Lagrangian, principal bundle field theory
Mot clés : réduction covariante, équations d’Euler-Poincaré affines, équations d’Euler-Poincaré covariantes, Lagrangien, théorie des champs sur un fibré principal
Gay-Balmaz, François 1 ; Ratiu, Tudor S. 1

1 École Polytechnique Fédérale de Lausanne Section de Mathématiques Station 8 1015 Lausanne (Switzerland) 
@article{AIF_2010__60_3_1125_0,
     author = {Gay-Balmaz, Fran\c{c}ois and Ratiu, Tudor S.},
     title = {A new {Lagrangian} dynamic reduction in field theory},
     journal = {Annales de l'Institut Fourier},
     pages = {1125--1160},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {60},
     number = {3},
     year = {2010},
     doi = {10.5802/aif.2549},
     mrnumber = {2680826},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2549/}
}
TY  - JOUR
AU  - Gay-Balmaz, François
AU  - Ratiu, Tudor S.
TI  - A new Lagrangian dynamic reduction in field theory
JO  - Annales de l'Institut Fourier
PY  - 2010
SP  - 1125
EP  - 1160
VL  - 60
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2549/
DO  - 10.5802/aif.2549
LA  - en
ID  - AIF_2010__60_3_1125_0
ER  - 
%0 Journal Article
%A Gay-Balmaz, François
%A Ratiu, Tudor S.
%T A new Lagrangian dynamic reduction in field theory
%J Annales de l'Institut Fourier
%D 2010
%P 1125-1160
%V 60
%N 3
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2549/
%R 10.5802/aif.2549
%G en
%F AIF_2010__60_3_1125_0
Gay-Balmaz, François; Ratiu, Tudor S. A new Lagrangian dynamic reduction in field theory. Annales de l'Institut Fourier, Tome 60 (2010) no. 3, pp. 1125-1160. doi : 10.5802/aif.2549. http://archive.numdam.org/articles/10.5802/aif.2549/

[1] Castrillón-López, M.; García Pérez, P. L.; Ratiu, T. S. Euler-Poincaré reduction on principal bundles, Lett. Math. Phys., Volume 58 (2001) no. 2, pp. 167-180 | DOI | MR | Zbl

[2] Castrillón-López, M.; Marsden, J. E. Covariant and dynamical reduction for principal bundle field theories, Ann. Glob. Anal. Geom., Volume 34 (2008) no. 3, pp. 263-285 | DOI | MR | Zbl

[3] Castrillón-López, M.; Marsden, J. E.; Ratiu, T. S. Lagrangian reduction by stages, Mem. Amer. Math. Soc., Volume 152 (2001) no. 722, pp. x+108 | MR | Zbl

[4] Castrillón-López, M.; Ratiu, T. S. Reduction in principal bundles: covariant Lagrange-Poincaré equations, Comm. Math. Phys., Volume 236 (2003) no. 2, pp. 223-250 | DOI | MR | Zbl

[5] Castrillón-López, M.; Ratiu, T. S.; Shkoller, S. Reduction in principal fiber bundles: covariant Euler-Poincaré equations, Proc. Amer. Math. Soc., Volume 128 (2000), pp. 2155-2164 | DOI | MR | Zbl

[6] Dzyaloshinskiĭ, I. E. Macroscopic description of spin glasses., Modern trends in the theory of condensed matter (Proc. Sixteenth Karpacz Winter School Theoret. Phys., Karpacz, 1979), Lecture Notes in Physics, Volume 115 (1980), pp. 204-224 | MR

[7] Gay-Balmaz, F.; Ratiu, T. S. Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids, J. Symp. Geom., Volume 6 (2008) no. 2, pp. 189-237 | MR | Zbl

[8] Gay-Balmaz, F.; Ratiu, T. S. The geometric structure of complex fluids, Adv. in Appl. Math., Volume 42 (2008) no. 2, pp. 176-275 | DOI | MR | Zbl

[9] Gotay, M.; Isenberg, J.; Marsden, J. E. Momentum Maps and Classical Relativistic Fields. Part II: Canonical Analysis of Field Theories, preprint, arXiv:math-ph/0411032v1, 2004

[10] Gotay, M.; Isenberg, J.; Marsden, J. E.; Montgomery, R. Momentum Maps and Classical Relativistic Fields. Part I: Covariant Field Theory, preprint, arXiv:physics/9801019v2, 2004

[11] Isaev, A. A.; Yu. Kovalevskii, M.; Peletminskii, S. V. On dynamics of various magnetically ordered structures, The Physics of Metals and Metallography, Volume 77 (1994) no. 4, pp. 342-347

[12] Ivanchenko, E. A. Backward electromagnetic waves in a magnetically disordered dielectric, Low. Temp. Phys., Volume 26 (2000) no. 6, pp. 422-424 | DOI

Cité par Sources :