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) 
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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://www.numdam.org/articles/10.5802/aif.2549/

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