Calculating Lagrangian of electromagnetism

Question

I know that the interaction terms of the Lagrangian of electromagnetism are given by

$$L_{int}=-q\phi (\mathbf{x},t)+q\mathbf{v}(t)\cdot \mathbf{A}(\mathbf{x},t).$$

The above equation is replaced by terms involving a continuous charge density $\rho$ and current density $\mathbf{j}$. The resulting Lagrangian density for the electromagnetic field is: $$\mathcal{L}=-\rho \phi +\mathbf{j}\cdot \mathbf{A}＋\frac{\epsilon _{0}}{2}E^{2}-\frac{1}{2\mu _{0}}B^{2} .$$

The first problem is that I know where the first two terms $-\rho \phi +\mathbf{j}\cdot \mathbf{A}$ come from but I don't know where the last two terms $\frac{\epsilon _{0}}{2}E^{2}-\frac{1}{2\mu _{0}}B^{2}$ come from.

Next varying the Lagrangian density with respect to $\phi$ and $\mathbf{A}$, we get Gauss' law $$0=-\rho +\epsilon _{0}\triangledown \cdot \mathbf{E}$$ and Ampère's law $$0=\mathbf{j}+\epsilon _{0}\frac{\partial \mathbf{E}}{\partial t}-\frac{1}{\mu _{0}}\triangledown \times \mathbf{B},$$ respectively. The second problem is that I don't know how to calculate these variations clearly.

Answer 1

Working from the original (obviously scalar) Langrangian

You started with this term $$-{\frac {1}{4\mu _{0}}}F_{\mu \nu }F_{\rho \sigma }\eta ^{\mu \rho }\eta ^{\nu \sigma }$$

Which reduces to

$$=-{\frac {1}{4\mu _{0}}}F_{\mu \nu }F^{\mu \nu }$$

You need to take account of the fact that the E and B fields can be written in terms of the Faraday tensor $${\displaystyle F_{\mu \nu }}$$ We define this tensor as:

$${\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}$$

Which in turn reduces the above to:

$$={\displaystyle {\epsilon _{0} \over 2}{E}^{2}-{1 \over {2\mu _{0}}}{B}^{2}}$$

The derivation of the two inhomogeneous Maxwell equations also derives from

$${\displaystyle \partial _{\mu }F^{\mu \nu }=\mu _{0}J^{\nu }}$$

Now you can write  Gauss's law and Ampère's law using the following replacements :

$${\displaystyle {\begin{aligned}{\frac {1}{c}}E^{i}&=-F^{0i}\\\epsilon ^{ijk}B_{k}&=-F^{ij}\end{aligned}}}$$

where i, j, k range through 1, 2, and 3