I'm trying to calculate the euler-lagrange equations of the following Lagrangian density.

$$\mathcal{L} = -\frac{1}{2}(\partial_\alpha A_\beta)(\partial^\alpha A^\beta)+\frac{1}{2}(\partial_\alpha A^\alpha)^2+\frac{\mu^2}{2} A_\alpha A^\alpha $$

The Euler-Lagrange formula: $\frac{\partial \mathcal{L}}{\partial A^\alpha} =\partial_\beta (\frac{\partial \mathcal{L}}{\partial(\partial_\beta A^\alpha)})$

I have calculated $\frac{\partial\mathcal{L}}{\partial A^\alpha} = \mu^2 A_\alpha$ but I can't figure out how to calculate $\partial_\beta (\frac{\partial \mathcal{L}}{\partial(\partial_\beta A^\alpha)})$. Mainly my problem is the index notation here. Although I understand the basic rules I'm kinda lost here. The equations of motion are given as

$$[g_{\alpha\beta}(\partial^2+\mu^2)-\partial_\alpha\partial_\beta]A^\beta = 0$$

so I figured I should insert the metric somewhere though I don't understand how and why we would do that.

Note: $g_{\alpha\beta}$ is the minkowski metric.

  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/3005/2451 , physics.stackexchange.com/q/64272/2451 and links therein. $\endgroup$ – Qmechanic Oct 12 '19 at 18:59
  • $\begingroup$ Thank you, the answers there are helpful. My problem is that it looks like I have an extra term in my Lagrangian. Is it possible that you could show how my lagrangian is equal to $\mathcal{L} = J^\mu A_\mu + \frac{1}{{4\mu _0 }}F^{\mu \nu } F_{\mu \nu } $ ? $\endgroup$ – grand_unifier Oct 12 '19 at 20:28
  • $\begingroup$ Once you understand the other terms the extra mass term will be easy. $\endgroup$ – Qmechanic Oct 12 '19 at 20:34