k-cosymplectic classical field theories: Tulczyjew, Skinner--Rusk and Lie-algebroid formulations

TL;DR

This paper reviews and extends the k-cosymplectic formalism for first-order classical field theories, introducing new formulations based on Tulczyjew, Skinner-Rusk, and Lie algebroids, applicable to singular and regular systems.

Contribution

It develops new geometric formulations for k-cosymplectic field theories, unifying Lagrangian and Hamiltonian approaches and extending them to Lie algebroids.

Findings

New interpretation of classical field equations via submanifolds of tangent bundles.

Unified Lagrangian-Hamiltonian formalism using Skinner-Rusk approach.

Formulations applicable to singular and almost-regular systems.

Abstract

The k-cosymplectic Lagrangian and Hamiltonian formalisms of first-order field theories are reviewed and completed. In particular, they are stated for singular and almost-regular systems. Subsequently, several alternative formulations for k-cosymplectic first-order field theories are developed: First, generalizing the construction of Tulczyjew for mechanics, we give a new interpretation of the classical field equations in terms of certain submanifolds of the tangent bundle of the $k^1$-velocities of a manifold. Second, the Lagrangian and Hamiltonian formalisms are unified by giving an extension of the Skinner-Rusk formulation on classical mechanics. Finally, both formalisms are formulated in terms of Lie algebroids.