Obtaining field equations from an action

[hepnoob92]

Homework Statement

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

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.

 

[Orodruin]

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?