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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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  • Hi Kane, and welcome to Physics Stack Exchange! What have you tried to work this out yourself, and where did you get stuck? – David Z Dec 28 '15 at 06:15
  • Possible duplicates: http://physics.stackexchange.com/q/3005/2451 and links therein. – Qmechanic Dec 28 '15 at 06:24
  • @DavidZ I am a novice at Einstein sum notation and I just can find derivatives to Einstein sum notation. I tried to applied the same pattern when trying to find the calculus of variation, yet fall to obtain the correct answer. So, could you please show me how to do it from the beginning? Thank you very much. – Kane Young Dec 28 '15 at 06:55

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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