Explicit gauge covariant Euler-Lagrange equation

TL;DR

This paper introduces a gauge covariant Euler-Lagrange equation that simplifies deriving equations of motion for fields under external forces by incorporating the external force directly into the gauge covariant derivative.

Contribution

It presents a novel extension of the Euler-Lagrange equation using a gauge covariant derivative that includes external forces, streamlining calculations in gauge theories and general relativity.

Findings

Simplifies equations of motion for gauge and tensor fields.

Demonstrates the method with examples relevant to general relativity.

Provides a unified approach to external forces in gauge covariant frameworks.

Abstract

The application of a gauge covariant derivative to the Euler-Lagrange equation yields a shortcut to the equations of motion for a field subject to an external force. The gauge covariant derivative includes an external force as an intrinsic part of the derivative and hence simplifies Lagrangians containing tensor and gauge covariant fields. The gauge covariant derivative used in the covariant Euler-Lagrange equation is presented as an extension of the coordinate covariant derivative used in tensor analysis. Several examples provide useful demonstrations of the covariant derivative relevant to studies in general relativity and gauge theory.