K-symplectic formalism on Lie algebroids

TL;DR

This paper develops a geometric framework for classical field theories on Lie algebroids using k-symplectic geometry, generalizing standard theories and exploring systems with symmetry and Poisson sigma models.

Contribution

It introduces a novel geometric description of Lagrangian and Hamiltonian field theories on Lie algebroids within the k-symplectic framework, including a Legendre transformation.

Findings

Generalizes classical field theory to Lie algebroids

Establishes a relation between Lagrangian and Hamiltonian formalisms

Provides examples with symmetry and Poisson sigma models

Abstract

In this paper we introduce a geometric description of Lagrangian and Hamiltonian classical field theories on Lie algebroids in the framework of k-symplectic geometry. We discuss the relation between Lagrangian and Hamiltonian descriptions through a convenient notion of Legendre transformation. The theory is a natural generalization of the standard one; in addition, other interesting examples are studied, in particular, systems with symmetry and Poisson sigma models.