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

TL;DR

This paper explores symmetry reduction and solution reconstruction in k-symplectic field theory, providing explicit reduced equations and a new concept of k-connection for solution reconstruction.

Contribution

It introduces explicit coordinate expressions for reduced equations and a novel k-connection concept for reconstructing solutions in invariant k-symplectic field theories.

Findings

Derived explicit coordinate forms of reduced PDEs

Interpreted integrability conditions via connection curvatures

Proposed a k-connection-based reconstruction method

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.