# Solving the Euler-Lagrange Equation for a given Lagrange Density [duplicate]

Assume that our Lagrangian density is of the form $$\mathcal L = -\frac{1}{2} \left( \partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\partial_{\nu} A_{\mu}\partial^{\mu}A^{\nu}\right) - J^{\mu}A_{\mu}.$$

(If you wonder where it comes from: https://en.wikipedia.org/wiki/Electromagnetic_tensor#Lagrangian_formulation_of_classical_electromagnetism, section "Lagrangian formulation of classical electromagnetism".)

Euler-Lagrange equation (according to Wikipedia): $$\partial_{\mu}\left(\frac{\partial \mathcal L}{\partial \left(\partial_{\mu}A_{\nu}\right)}\right) - \frac{\partial \mathcal L}{\partial A_{\nu}} = 0.$$

However, I cannot follow yet why Wikipedia then obtains what they do. (i) What is $$\frac{\partial \mathcal L}{\partial \left(\partial_{\mu}A_{\nu}\right)}$$ supposed to mean?

(ii) When I derive $$\mathcal L$$ with regard to $$\partial_{\mu}A_{\nu}$$, how can I derive $$\partial_{\nu}A_{\mu}\partial^{\mu}A^{\nu}$$ with regards to $$\partial_{\mu}A_{\nu}$$? I would have said that this is not possible, since once, the indices are up, and then they are down ...

(iii) When deriving $$J^{\nu}$$ with regards to $$A_{\nu}$$, how is this possible?

(I am really sorry for asking essentially three times the same, I realized that I don't know how to derive in SRT ...)

em