Help understanding the derivation of Maxwell's equations from Euler-Lagrange equations

Question

I am having trouble with the following points in the derivation of Maxwell's equations on page 10 of these notes.

1. Specifically, I am new to the co-/contra-variant notation used in relativistic forms and am somewhat confused as to why $$\mathcal{L} \sim \frac{1}{2}\sum_{i=1}^3\dot{A}_i^2\tag{1.18b}$$ follows from the Minkowski metric.

2. Also, could someone kindly provide a full worked derivation of equations (1.19) and (1.20)? For example, $\rho$ seems to have appeared from nowhere. I would really appreciate some help to improve my understanding of the notation.

Answer 1

In terms of $\mathcal{L}\sim\frac12\dot{A}_i^2$, with implicit summation over $i$, you have the point backwards: we choose a coefficient in $\mathcal{L}$ to get that result; we don't derive it. (The aim is to get a positive coefficient analogous to the $\frac12 m\sum_i\dot{x}_i^2$ in a mechanical Lagrangian.) Note that by Eq. (1.18) the $\sum_i\dot{A}_i^2$ coefficient in $\mathcal{L}$ is $\frac12$ because the Minkowski metric has $\partial_0A_i\partial^0A^i=-\sum_i\dot{A}_i^2$.

Since your subsequent confusions are due to the way dummy indices are juggled, I'll try preserve the indices in the differential operators by changing those in $\mathcal{L}$. To derive Eq. (1.18), note that $\mathcal{L}=-\frac12\partial_\alpha A_\beta\partial^\alpha A^\beta+\frac12\partial_\alpha A^\alpha\partial_\beta A^\beta$ implies$$\frac{\partial\mathcal{L}}{\partial(\partial_\mu A_\nu)}=-\frac12\delta_\alpha^\mu\delta_\beta^\nu\partial^\alpha A^\beta-\frac12\partial_\alpha A_\beta\eta^{\alpha\mu}\eta^{\beta\nu}+\frac12\delta_\alpha^\mu\eta^{\alpha\nu}\partial_\beta A^\beta+\frac12\partial_\alpha A^\alpha\delta_\beta^\mu \eta^{\beta\nu}$$(by repeated use of the product rule). We can tidy this up, viz.$$\frac{\partial\mathcal{L}}{\partial(\partial_\mu A_\nu)}=-\partial^\mu A^\nu+\eta^{\mu\nu}\partial_\alpha A^\alpha.$$We get Eq. (1.20) by differentiating again:$$\partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu A_\nu)}=-\partial_\mu\partial^\mu A^\nu+\eta^{\mu\nu}\partial_\mu\partial_\alpha A^\alpha.$$But this tidies up to$$\partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu A_\nu)}=-\partial^2A^\nu+\partial^\nu\partial_\alpha A^\alpha.$$But defining $F^{\mu\nu}:=\partial^\mu A^\nu-\partial^\nu A^\mu$,$$-\partial_\mu F^{\mu\nu}=-\partial^2 A^\nu+\partial_\mu\partial^\nu A^\mu=-\partial^2 A^\nu+\partial^\nu \partial_\alpha A^\alpha,$$as expected. (All I've done at the last $=$ is rename one index and permuted partial derivatives.)