Geometry of Lagrangian First-order Classical Field Theories

TL;DR

This paper develops a geometric Lagrangian framework for first-order classical field theories using jet bundles, deriving Euler-Lagrange equations and Noether's theorem within this formalism.

Contribution

It introduces a comprehensive geometric formulation of first-order field theories using jet bundles, including derivations of equations and symmetries.

Findings

Derived Euler-Lagrange equations via variational and jet formalisms

Established Noether's theorem in the geometric setting

Applied the formalism to classical examples

Abstract

We construct a lagrangian geometric formulation for first-order field theories using the canonical structures of first-order jet bundles, which are taken as the phase spaces of the systems in consideration. First of all, we construct all the geometric structures associated with a first-order jet bundle and, using them, we develop the lagrangian formalism, defining the canonical forms associated with a lagrangian density and the density of lagrangian energy, obtaining the {\sl Euler-Lagrange equations} in two equivalent ways: as the result of a variational problem and developing the {\sl jet field formalism} (which is a formulation more similar to the case of mechanical systems). A statement and proof of Noether's theorem is also given, using the latter formalism. Finally, some classical examples are briefly studied.