A new Lagrangian dynamic reduction in field theory

TL;DR

This paper introduces a novel Lagrangian reduction method in classical field theory that unifies covariant and dynamic approaches by constructing a gauge-invariant Lagrangian on an infinite-dimensional space, demonstrating their equivalence.

Contribution

It presents a new Lagrangian reduction technique that bridges covariant and dynamic methods for symmetric classical field theories on principal bundles.

Findings

Covariant and dynamic reductions yield equivalent equations of motion.

A new gauge-invariant Lagrangian on an infinite-dimensional space is constructed.

The method unifies two classical symmetry reduction approaches.

Abstract

For symmetric classical field theories on principal bundles there are two methods of symmetry reduction: covariant and dynamic. Assume that the classical field theory is given by a symmetric covariant Lagrangian density defined on the first jet bundle of a principal bundle. It is shown that covariant and dynamic reduction lead to equivalent equations of motion. This is achieved by constructing a new Lagrangian defined on an infinite dimensional space which turns out to be gauge group invariant.