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.
@article{AIF_2010__60_3_1125_0,
author = {Gay-Balmaz, Fran\c cois and Ratiu, Tudor S.},
title = {A new Lagrangian dynamic reduction in field theory},
journal = {Annales de l'Institut Fourier},
volume = {60},
year = {2010},
pages = {1125-1160},
doi = {10.5802/aif.2549},
zbl = {pre05763362},
mrnumber = {2680826},
language = {en},
url = {http://dml.mathdoc.fr/item/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) pp. 1125-1160. doi : 10.5802/aif.2549. http://gdmltest.u-ga.fr/item/AIF_2010__60_3_1125_0/
[1] Euler-Poincaré reduction on principal bundles, Lett. Math. Phys., Tome 58 (2001) no. 2, pp. 167-180 | Article | MR 1876252 | Zbl 1020.58018
[2] Covariant and dynamical reduction for principal bundle field theories, Ann. Glob. Anal. Geom., Tome 34 (2008) no. 3, pp. 263-285 | Article | MR 2434857 | Zbl 1162.83347
[3] Lagrangian reduction by stages, Mem. Amer. Math. Soc., Tome 152 (2001) no. 722, pp. x+108 | MR 1840979 | Zbl 1193.37072 | Zbl pre01636378
[4] Reduction in principal bundles: covariant Lagrange-Poincaré equations, Comm. Math. Phys., Tome 236 (2003) no. 2, pp. 223-250 | Article | MR 1981991 | Zbl 1037.53056
[5] Reduction in principal fiber bundles: covariant Euler-Poincaré equations, Proc. Amer. Math. Soc., Tome 128 (2000), pp. 2155-2164 | Article | MR 1662269 | Zbl 0967.53019
[6] 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, Tome 115 (1980), pp. 204-224 | MR 583572
[7] Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids, J. Symp. Geom., Tome 6 (2008) no. 2, pp. 189-237 | MR 2434440 | Zbl 1149.70015
[8] The geometric structure of complex fluids, Adv. in Appl. Math., Tome 42 (2008) no. 2, pp. 176-275 | Article | MR 2493976 | Zbl 1161.37052
[9] Momentum Maps and Classical Relativistic Fields. Part II: Canonical Analysis of Field Theories, preprint, arXiv:math-ph/0411032v1 (2004)
[10] Momentum Maps and Classical Relativistic Fields. Part I: Covariant Field Theory, preprint, arXiv:physics/9801019v2 (2004)
[11] On dynamics of various magnetically ordered structures, The Physics of Metals and Metallography, Tome 77 (1994) no. 4, pp. 342-347
[12] Backward electromagnetic waves in a magnetically disordered dielectric, Low. Temp. Phys., Tome 26 (2000) no. 6, pp. 422-424 | Article