Extrema of the Lagrangian density

i. Overview

In this module we define what it means for an electromagnetic potential to be an extremum of the Lagrangian density in presence of a Lorentz current density.

This is equivalent to the electromagnetic potential satisfying Maxwell's equations with sources, i.e. Gauss's law and Ampère's law.

ii. Key results

IsExtrema : The condition on an electromagnetic potential to be an extrema of the lagrangian.

iii. Table of contents

A. Is Extema condition in the distributional case

A.1. IsExtrema and Gauss's law and Ampère's law

A.2. IsExtrema in terms of Vector Potentials

A.3. The exterma condition in terms of tensors

A.4. The invariance of the exterma condition under Lorentz transformations

iv. References

A. Is Extema condition in the distributional case

The above results looked at the extrema condition for electromagnetic potentials that are functions. We now look at the case where the electromagnetic potential is a distribution.

The proposition on an electromagnetic potential, corresponding to the statement that it is an extrema of the lagrangian.

A.1. IsExtrema and Gauss's law and Ampère's law

We show that A is an extrema of the lagrangian if and only if Gauss's law and Ampère's law hold. In other words,

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$$ and $$\mu_0 \varepsilon_0 \frac{\partial \mathbf{E}i}{\partial t} - \sum_j \partial_j \mathbf{B}{j i} + \mu_0 \mathbf{J}_i = 0.$$ Here $\mathbf{B}$ is the magnetic field matrix.