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I have the Lagrange density for Maxwell field, which is $\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, where $F_{\mu\nu}=\partial_{\mu}A_\nu-\partial_{\nu}A_{\mu}$.

How can I obtain $\dfrac{\partial\mathcal{L}}{\partial(\partial_\mu A_\nu)}$ and $\partial_{\mu}\dfrac{\partial\mathcal{L}}{\partial(\partial_\mu A_\nu)}$ when trying to write down the Euler-Lagrange Equation?

Qmechanic
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Kane Young
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1 Answers1

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how about this,

$\begin{align}\mathcal{L}&=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\\ &=-\frac{1}{4}\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\partial^\mu A^\nu-\partial^\nu A^\mu\right)\\ &=-\frac{1}{4}\left[(\partial_\mu A_\nu)(\partial^\mu A^\nu)-(\partial_\mu A_\nu)(\partial^\nu A^\mu)-(\partial_\nu A_\mu)(\partial^\mu A^\nu)+(\partial_\nu A_\mu)(\partial^\nu A^\mu)\right]\\ &\left[We\;are\;exchanging\;\mu\;and\;\nu\; in\;last\;two\;terms\right]\\ &=-\frac{1}{2}\left[(\partial_\mu A_\nu)(\partial^\mu A^\nu)-(\partial_\mu A_\nu)(\partial^\nu A^\mu)\right]\end{align}$

Now differentiate using chain rule

mathguy
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