# [Physics] Deriving Euler-Lagrange for Electrodynamics Lagrangian

electromagnetismhomework-and-exerciseslagrangian-formalismtensor-calculusvariational-calculus

For $\mathcal L = -\frac14 F_{\mu\nu}F^{\mu\nu}$ I would appreciate some help evaluating

$$\frac{\partial \mathcal L}{\partial(\partial_{\mu}A_{\nu})}.$$

I've found

$$\frac{\partial \mathcal L}{\partial(\partial_{\mu}A_{\nu})} = -\frac14\frac{\partial}{\partial(\partial_{\mu}A_{\nu})}(\partial_{\lambda}A_{\sigma} – \partial_{\sigma} A_{\lambda})(\partial_{\lambda} A_{\sigma} – \partial_{\sigma} A_{\lambda})$$

$$= -\frac14 \frac{\partial}{\partial(\partial_{\mu}A_{\nu})} \left( 2\partial_{\lambda}A_{\sigma}-2\partial_{\lambda}A_{\sigma}\partial_{\sigma}A_{\lambda}\right)$$
$$= -\frac14 4\left(\partial_{\nu} A_{\mu} – \partial_{\mu} A_{\nu}\right) \qquad(*)$$
$$= F_{\mu\nu}$$

but I cannot understand the step taken to get to $(*)$. In my attempt, I take the chain rule to obtain

$$\frac{\partial \mathcal L}{\partial(\partial_{\mu}A_{\nu})} = -\frac12 F_{\mu\nu} \frac{\partial(\partial_{\alpha}A_{\beta} – \partial_{\beta} A_{\lambda})}{\partial(\partial_{\mu} A_{\nu})}$$

$$= -\frac12 F_{\mu\nu}(1) – \frac12 F_{\mu\nu} \frac{\partial(\partial_{\beta}A_{\lambda})}{\partial(\partial_{\mu}A_{\nu})}$$
$$= -\frac12 F_{\mu\nu} – \frac12F_{\mu\nu}\delta_{\beta\mu} \delta_{\lambda\nu} = -\frac12 F_{\mu\nu} – \frac12 F_{\beta\lambda}$$

which seems to give me almost what I want, but I must have run into an error because this expression no longer makes sense, as each term should be indexed by $\mu,\nu$.

So my 2 questions are: how do we get to $(*)$ and where did my attempt go wrong?

To begin, the antisymmetry of $F_{\mu\nu}$ implies $\mathcal{L}=-\frac{1}{2}\partial_\mu A_\nu F^{\mu\nu}$. We now differentiate with respect to $\partial_\rho A_\sigma$. By the product rule, the result is $$-\frac{1}{2}(\delta_\mu^\rho \delta_\nu^\sigma F^{\mu\nu} + \partial_\mu A_\nu (g^{\mu\rho}g^{\nu\sigma}-g^{\nu\rho}g^{\mu\sigma}))=-\frac{1}{2}(F^{\rho\sigma}+\partial^\rho A^\sigma - \partial^\sigma A^\rho)=-F^{\rho\sigma}.$$