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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

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