Obtaining field equations from an action

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Homework Statement


Provided an action:
[tex]S[A_\nu] = \int\left(\frac{1}{4}(A_{\gamma,\mu}-A_{\mu,\gamma})(A_{\zeta,\alpha}-A_{\alpha,\zeta})\eta^{\gamma\zeta}\eta^{\mu\alpha}+\frac{1}{2}\nu^2A_\mu A_\gamma -\beta A_\mu J^\mu\right)\sqrt{-\eta}~d^4x[/tex]

How would one go about finding the field equations for the same? I do understand that using the Euler-Lagrange method is how one should start out.

Does it tell us what kind of field we're looking at, at a glance?

Homework Equations





The Attempt at a Solution



I know that the terms in the parentheses are just ##F_{\mu\nu}F^{\mu\nu}##, but am unsure of how to proceed.

Additionally:
Does anyone know any good resources online or books that derive field equations and extract physics from a given action in steps, giving a good detailed process from which one can observe and learn how the Euler Lagrange method is being applied, in terms of the 4-vector notation. My struggle is essentially in decoding the physics, largely due to problems with understanding the notation.
 
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Orodruin
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I do understand that using the Euler-Lagrange method is how one should start out.
Right, so what do you get when you do the EL equation for a single component ##A_\lambda## of the field?
 

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