Applying the Euler-Lagrange equations to Maxwell's Theory

Question

In Prof. David Tong's notes, specifically on page 10, he gives the Lagrangian of Maxwell's theory to be

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

and then he computes the following

$$ \frac{\partial\mathcal{L}}{\partial(\partial_\mu A_\nu)} = -\partial_\mu A_\nu + (\partial_\rho A^\rho)\eta^{\mu\nu}. $$

I can see how the first term in the derivative is computed but am having problems with the second term. Any help is appreciated!

Answer 1

We have $\frac12 (\partial_{\mu}A^{\mu})^2 = \frac12 (\partial_{\alpha} A^{\alpha})(\partial_{\beta}A^{\beta})= \frac12 (\partial_{\alpha} A_{\sigma}) \eta^{\sigma\alpha}(\partial_{\beta}A_{\rho}) \eta^{\rho\beta}$ so the derivative w.r.t. $\partial_{\mu} A_{\nu}$ is

$$\frac12\delta_{\alpha}^{\mu} \delta_{\sigma}^{\nu} \eta^{\sigma\alpha}(\partial_{\beta}A_{\rho}) \eta^{\rho\beta}+\frac12(\partial_{\alpha} A_{\sigma}) \eta^{\sigma\alpha}\delta_{\beta}^{\mu} \delta_{\rho}^{\nu} \eta^{\rho\beta}= \frac12 \eta^{\mu\nu} (\partial_{\beta} A^{\beta})+\frac12 (\partial_{\alpha}A^{\alpha}) \eta^{\mu\nu} = (\partial_{\rho}A^{\rho})\eta^{\mu\nu} $$

where I've freely labeled and relabeled dummy indices.