A new Lagrangian dynamic reduction in field theory

François Gay-Balmaz[1]; Tudor S. Ratiu[1]

  • [1] École Polytechnique Fédérale de Lausanne Section de Mathématiques Station 8 1015 Lausanne (Switzerland)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 3, page 1125-1160
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Gay-Balmaz, François, and Ratiu, Tudor S.. "A new Lagrangian dynamic reduction in field theory." Annales de l’institut Fourier 60.3 (2010): 1125-1160. <http://eudml.org/doc/116291>.

@article{Gay2010,
abstract = {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.},
affiliation = {École Polytechnique Fédérale de Lausanne Section de Mathématiques Station 8 1015 Lausanne (Switzerland); École Polytechnique Fédérale de Lausanne Section de Mathématiques Station 8 1015 Lausanne (Switzerland)},
author = {Gay-Balmaz, François, Ratiu, Tudor S.},
journal = {Annales de l’institut Fourier},
keywords = {Covariant reduction; dynamic reduction; affine Euler-Poincaré equation; covariant Euler-Poincaré equation; Lagrangian; principal bundle field theory; covariant reduction; affine Euler-Poincaré equation; covariant Euler-Poincaré equation},
language = {eng},
number = {3},
pages = {1125-1160},
publisher = {Association des Annales de l’institut Fourier},
title = {A new Lagrangian dynamic reduction in field theory},
url = {http://eudml.org/doc/116291},
volume = {60},
year = {2010},
}

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
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 3
SP - 1125
EP - 1160
AB - 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.
LA - eng
KW - Covariant reduction; dynamic reduction; affine Euler-Poincaré equation; covariant Euler-Poincaré equation; Lagrangian; principal bundle field theory; covariant reduction; affine Euler-Poincaré equation; covariant Euler-Poincaré equation
UR - http://eudml.org/doc/116291
ER -

References

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  1. M. Castrillón-López, P. L. García Pérez, T. S. Ratiu, Euler-Poincaré reduction on principal bundles, Lett. Math. Phys. 58 (2001), 167-180 Zbl1020.58018MR1876252
  2. M. Castrillón-López, J. E. Marsden, Covariant and dynamical reduction for principal bundle field theories, Ann. Glob. Anal. Geom. 34 (2008), 263-285 Zbl1162.83347MR2434857
  3. M. Castrillón-López, J. E. Marsden, T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc. 152 (2001) Zbl1193.37072MR1840979
  4. M. Castrillón-López, T. S. Ratiu, Reduction in principal bundles: covariant Lagrange-Poincaré equations, Comm. Math. Phys. 236 (2003), 223-250 Zbl1037.53056MR1981991
  5. M. Castrillón-López, T. S. Ratiu, S. Shkoller, Reduction in principal fiber bundles: covariant Euler-Poincaré equations, Proc. Amer. Math. Soc. 128 (2000), 2155-2164 Zbl0967.53019MR1662269
  6. I. E. Dzyaloshinskiĭ, 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 115 (1980), 204-224 MR583572
  7. F. Gay-Balmaz, T. S. Ratiu, Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids, J. Symp. Geom. 6 (2008), 189-237 Zbl1149.70015MR2434440
  8. F. Gay-Balmaz, T. S. Ratiu, The geometric structure of complex fluids, Adv. in Appl. Math. 42 (2008), 176-275 Zbl1161.37052MR2493976
  9. M. Gotay, J. Isenberg, J. E. Marsden, Momentum Maps and Classical Relativistic Fields. Part II: Canonical Analysis of Field Theories, preprint, arXiv:math-ph/0411032v1 (2004) 
  10. M. Gotay, J. Isenberg, J. E. Marsden, R. Montgomery, Momentum Maps and Classical Relativistic Fields. Part I: Covariant Field Theory, preprint, arXiv:physics/9801019v2 (2004) 
  11. A. A. Isaev, M. Yu. Kovalevskii, S. V. Peletminskii, On dynamics of various magnetically ordered structures, The Physics of Metals and Metallography 77 (1994), 342-347 
  12. E. A. Ivanchenko, Backward electromagnetic waves in a magnetically disordered dielectric, Low. Temp. Phys. 26 (2000), 422-424 

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