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-Poincare 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.

Highlights

  • The Lagrangian equations of a first-order field theory are a set of second-order partial differential equations in the unknown fields φA(t), depending on k parameters tα

  • One may think of k-symplectic field theory as the model that resembles the closest the standard symplectic formalism of both Lagrangian and Hamiltonian mechanics on a tangent and a cotangent bundle, respectively

  • In the presence of a Lagrangian with a symmetry group G, we identify in Section 5 the action under which the Lagrangian kvector fields are invariant, and we show that they can be reduced to k-vector fields on the reduced space (Tk1Q)/G

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Summary

Introduction

The Lagrangian equations of a first-order field theory are a set of second-order partial differential equations in the unknown fields φA(t), depending on k parameters tα. This part of the problem involves the introduction of a new concept, that of a principal k-connection on the principal bundle M → M/G. Since it resembles the so-called mechanical connection which appears in the context of a Lagrangian system whose kinetic energy is associated to a Riemannian metric We end the paper with an application of our results to the context of harmonic maps

Integrability of a k-vector field
Connections and curvature
The tangent bundle of k1-velocities
The connection associated to a k-vector field
Invariant k-vector fields
Integrability and curvature
Decomposition by making use of a principal connection
Lagrangian k-symplectic field theory
Canonical operations on Tk1Q
Second-order partial differential equations
Lagrangian sopdes
Symmetry reduction of a Lagrangian k-vector field
Invariant Lagrangian sopdes
Local frames of vector fields on Tk1Q
The reduced Lagrangian sopde
The integrability of an invariant sopde
Reconstruction
Reconstruction method
The mechanical k-connection
An application on harmonic maps

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