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The Journal of Geometric Mechanics (JGM)
 

Symmetry reduction, integrability and reconstruction in $k$-symplectic field theory

Pages: 395 - 429, Volume 7, Issue 4, December 2015      doi:10.3934/jgm.2015.7.395

 
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L. Búa - Departamento de Xeometría e Topoloxía, Universidade de Santiago de Compostela, Spain (email)
T. Mestdag - Department of Mathematics, Ghent University, Krijgslaan 281, B-9000 Gent, Belgium (email)
M. Salgado - Departamento de Xeometría e Topoloxía, Universidade de Santiago de Compostela, Spain (email)

Abstract: We investigate the reduction process of a $k$-symplectic field theory whose Lagrangian is invariant under a symmetry group. We give explicit coordinate expressions of the resulting reduced partial differential equations, the so-called Lagrange-Poincaré field equations. We discuss two issues about reconstructing a solution from a given solution of the reduced equations. The first one is an interpretation of the integrability conditions, in terms of the curvatures of some connections. The second includes the introduction of the concept of a $k$-connection to provide a reconstruction method. We show that an invariant Lagrangian, under suitable regularity conditions, defines a `mechanical' $k$-connection.

Keywords:  Classical field theories, symmetry, reduction, integrability, reconstruction.
Mathematics Subject Classification:  37J15, 53Z05, 70S05, 70S10.

Received: April 2015;      Revised: August 2015;      Available Online: October 2015.

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