Demystifying the Lagrangian formalism for field theories

TL;DR

This paper clarifies the derivation and motivation of the Lagrangian formalism in field theories, illustrating its application with Electrodynamics and Maxwell's equations.

Contribution

It systematically derives the Lagrangian formulation for field theories, emphasizing the independence of Euler-Lagrange equations under coordinate transformations.

Findings

Euler-Lagrange equations are independent under coordinate changes

Electrodynamics Lagrangian leads to Maxwell's equations

Provides a clear derivation of the action principle for fields

Abstract

This paper expands on previous work to derive and motivate the Lagrangian formulation of field theories. In the process, we take three deliberate steps. First, we give the definition of the action and derive Euler-Lagrange equations for field theories. Second, we prove the Euler-Lagrange equations are independent under arbitrary coordinate transformations and motivate that this independence is desirable for field theories in physics. We then use the Lagrangian for Electrodynamics as an example field Lagrangian and prove that the related Euler-Lagrange equations lead to Maxwell's equations.