On the construction of K-operators in field theories as sections along Legendre maps

TL;DR

This paper generalizes the time-evolution operator from mechanics to covariant field theories using sections along Legendre maps, aiding in solving and relating Lagrangian and Hamiltonian field equations.

Contribution

It introduces a covariant construction of the K-operator for field theories using sections along maps, extending geometric tools from mechanics to field theories.

Findings

Provides a geometric framework for the K-operator in field theories.

Facilitates solutions to Lagrangian and Hamiltonian field equations.

Establishes equivalence between different formulations, especially for non-regular theories.

Abstract

The ``time-evolution operator'' in mechanics is a powerful tool which can be geometrically defined as a vector field along the Legendre map. It has been extensively used by several authors for studying the structure and properties of the dynamical systems (mainly the non-regular ones), such as the relation between the Lagrangian and Hamiltonian formalisms, constraints, and higher-order mechanics. This paper is devoted to defining a generalization of this operator for field theories, in a covariant formulation. In order to do this, we also use sections along maps, in particular multivector fields (skew-symmetric contravariant tensor fields of order greater than 1), jet fields and connection forms along the Legendre map. As a first relevant property, we use these geometrical objects to obtain the solutions of the Lagrangian and Hamiltonian field equations, and the equivalence among them (specially for non-regular field theories).