QFT Index Question Homework: Solving Euler-Lagrange Equations w/ F_{mu,nu}

[spacelike]

Homework Statement

I'm learning QFT and trying to do a basic problem finding the equations of motion from the Euler-Lagrange equation given a lagrangian.

The lagrangian is in terms of:
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}

so then my issue comes in with this part of the Euler-Lagrange equation:
\frac{\partial\mathcal{L}}{\partial (\partial_{\mu}\phi)}

The Attempt at a Solution

Now, I'm not sure if I am supposed to treat this as two separate fields or not. My first attempt to solve this I made a change of index from \mu\rightarrow\lambda and \nu\rightarrow\gamma in the Euler-Lagrange equation so that I got terms that look something similar to:
(there's more terms and factors but I'm just showing the relevant part)
\frac{\partial (\partial_{\mu}A_{\nu})}{\partial (\partial_{\lambda}A_{\gamma})}-\frac{\partial (\partial_{\nu}A_{\mu})}{\partial (\partial_{\lambda}A_{\gamma})}
This then results in delta functions which multiply the other factors in the equations and I get the final answer.

OR

Am I supposed to only change ONE index and treat A_{\mu} and A_{\nu} as separate fields, so that I would only do \mu\rightarrow\lambda (again, only for the euler-lagrange equation)
and get two equations with terms similar to:
\frac{\partial (\partial_{\mu}A_{\nu})}{\partial (\partial_{\lambda}A_{\mu})}-\frac{\partial (\partial_{\nu}A_{\mu})}{\partial (\partial_{\lambda}A_{\mu})}
\frac{\partial (\partial_{\mu}A_{\nu})}{\partial (\partial_{\lambda}A_{\nu})}-\frac{\partial (\partial_{\nu}A_{\mu})}{\partial (\partial_{\lambda}A_{\nu})}


NOTE: I did it the first way and the answer looks reasonable to me, but I just want to make sure my technique was correct.

 

[dextercioby]

You should never have the same index in the differentiations, you need to adapt all indices so that the equations respect the correct covariance requirement

\left[\partial_{\mu}\left(\frac{\partial_{\sigma}A_{\lambda}-\partial_{\lambda}A_{\sigma}}{\partial\left(\partial_{\mu}A_{\nu}\right)}\right)\right] F^{\sigma\lambda}

So you can see that:
* the free index is \nu
* "fractions" corresponding to differentiations do not mix indices
* no index appears more than twice, twice iff summed over.

There's something wrong with the LaTex code...Hmmmmm...

 

[Oxvillian]

\left[\partial_{\mu} \left(\frac{\partial_{\sigma}A_{\lambda}-\partial_{\lambda}A_{\sigma}}{\partial \left(\partial_{\mu}A_{\nu}\right)}\right)\right] F^{\sigma\lambda}

 

[dextercioby]

\left[\partial_{\mu} \left(\frac{\partial_{\sigma}A_{\lambda}-\partial_{\lambda}A_{\sigma}}{\partial \left(\partial_{\mu}A_{\nu}\right)}\right)\right] F^{\sigma\lambda}

I missed an operator.

\left[\partial_{\mu} \left(\frac{\partial\left(\partial_{\sigma}A_{\lambda}-\partial_{\lambda}A_{\sigma}\right)}{\partial \left(\partial_{\mu}A_{\nu}\right)}\right)\right] F^{\sigma\lambda}

Now the \lambda is not correctly parsed...

 

[Oxvillian]

\left[\partial_{\mu} \left(\frac{\partial \left( \partial_{\sigma}A_{\lambda} -\partial_{\lambda}A_{\sigma}\right)}{\partial \left(\partial_{\mu}A_{\nu}\right)}\right)\right] F^{\sigma\lambda}

Problem seems to be that the bb software is fond of inserting spaces in inappropriate places in order to break up long space-less lines.