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The Lagrangian density for Electromagnetism is given by $$\mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu},$$ where $F_{\mu \nu} = \partial_{\mu}A_\nu - \partial_{\nu}A_\mu$ is the field strength tensor.

May I know how to show that $$\mathcal{L} = -\frac{1}{2}\partial^{\nu}A^\mu(\partial_{\mu}A_\nu - \partial_{\nu}A_\mu)~?$$ It seems like there's an easy way to do this, but I can't seem to realise it.

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Expand the original Lagrangian, into $$ \mathcal{L} = -\tfrac{1}{2}\left(\tfrac{1}{2}\partial^\mu A^\nu - \tfrac{1}{2}\partial^\nu A^\mu \right)F_{\mu\nu} = -\tfrac{1}{2}\left(\tfrac{1}{2}\partial^\mu A^\nu F_{\mu\nu} - \tfrac{1}{2}\partial^\nu A^\mu F_{\mu\nu}\right).$$ Then interchange the summed indices in the second term, use antisymmetry of $F_{\mu\nu}$, rename them and you'll get what you want.

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