Symmetries and conservation laws in the Gunther k-symplectic formalism of field theory

TL;DR

This paper explores symmetries and conservation laws in the k-symplectic formalism for classical field theories, extending Noether's theorem and analyzing gauge symmetries within Hamiltonian and Lagrangian frameworks.

Contribution

It introduces a comprehensive study of symmetries, Cartan symmetries, and conservation laws in the k-symplectic formalism, including new results on gauge symmetries and their relation to Cartan symmetries.

Findings

Established Noether's theorem in various k-symplectic contexts

Characterized equivalent Lagrangians and gauge symmetries

Linked gauge symmetries with Cartan symmetries

Abstract

This paper is devoted to studying symmetries of k-symplectic Hamiltonian and Lagrangian first-order classical field theories. In particular, we define symmetries and Cartan symmetries and study the problem of associating conservation laws to these symmetries, stating and proving Noether's theorem in different situations for the Hamiltonian and Lagrangian cases. We also characterize equivalent Lagrangians, which lead to an introduction of Lagrangian gauge symmetries, as well as analyzing their relation with Cartan symmetries.