On the k-Symplectic, k-Cosymplectic and Multisymplectic Formalisms of Classical Field Theories

TL;DR

This paper explores the relationships between k-symplectic, k-cosymplectic, and multisymplectic formalisms in classical field theories, establishing equivalences and clarifying geometric aspects of solutions.

Contribution

It demonstrates the equivalence between k-symplectic and autonomous k-cosymplectic formalisms and clarifies the geometric nature of solutions in these frameworks.

Findings

Proves the equivalence between k-symplectic and autonomous k-cosymplectic theories.

Establishes the relation between k-cosymplectic and certain multisymplectic theories.

Clarifies the geometric interpretation of Hamilton-de Donder-Weyl and Euler-Lagrange solutions.

Abstract

The objective of this work is twofold: First, we analyze the relation between the k-cosymplectic and the k-symplectic Hamiltonian and Lagrangian formalisms in classical field theories. In particular, we prove the equivalence between k-symplectic field theories and the so-called autonomous k-cosymplectic field theories, extending in this way the description of the symplectic formalism of autonomous systems as a particular case of the cosymplectic formalism in non-autonomous mechanics. Furthermore, we clarify some aspects of the geometric character of the solutions to the Hamilton-de Donder-Weyl and the Euler-Lagrange equations in these formalisms. Second, we study the equivalence between k-cosymplectic and a particular kind of multisymplectic Hamiltonian and Lagrangian field theories (those where the configuration bundle of the theory is trivial).