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

L.  Búa; T.  Mestdag; M.  Salgado

doi:10.3934/jgm.2015.7.395

Article Contents

2015, Volume 7, Issue 4: 395-429. Doi: 10.3934/jgm.2015.7.395

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

1.
Departamento de Xeometría e Topoloxía, Universidade de Santiago de Compostela
2.
Department of Mathematics, Ghent University, Krijgslaan 281, B-9000 Gent

Received: March 31, 2015

Revised: July 31, 2015

Published: November 2015

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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.

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Mathematics Subject Classification: 37J15, 53Z05, 70S05, 70S10.

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