Multivector Fields and Connections. Setting Lagrangian Equations in Field Theories

TL;DR

This paper explores the integrability of multivector fields and their relation to connections in differentiable manifolds, applying these concepts to formulate and analyze Lagrangian equations in classical field theories.

Contribution

It establishes the equivalence between integrable multivector fields and connections in jet bundles, providing a geometric framework for Lagrangian field equations.

Findings

Characterizes integrable multivector fields and connections with canonical integral manifolds.

Sets Lagrangian evolution equations in three equivalent geometric forms.

Analyzes solution existence, non-uniqueness, and Noether's theorem in field theories.

Abstract

The integrability of multivector fields in a differentiable manifold is studied. Then, given a jet bundle $J^1E\to E\to M$, it is shown that integrable multivector fields in $E$ are equivalent to integrable connections in the bundle $E\to M$ (that is, integrable jet fields in $J^1E$). This result is applied to the particular case of multivector fields in the manifold $J^1E$ and connections in the bundle $J^1E\to M$ (that is, jet fields in the repeated jet bundle $J^1J^1E$), in order to characterize integrable multivector fields and connections whose integral manifolds are canonical lifting of sections. These results allow us to set the Lagrangian evolution equations for first-order classical field theories in three equivalent geometrical ways (in a form similar to that in which the Lagrangian dynamical equations of non-autonomous mechanical systems are usually given). Then, using multivector fields; we discuss several aspects of these evolution equations (both for the regular and singular cases); namely: the existence and non-uniqueness of solutions, the integrability problem and Noether's theorem; giving insights into the differences between mechanics and field theories.