I Coleman Lecture: Varying E-M Lagrangian - Problem 3.1 Explained

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The discussion focuses on Problem 3.1 from the Coleman Lectures on Relativity, specifically the derivation of the second equation using integration by parts. The simplification of the Lagrangian variation is presented, showing that the variation can be expressed in terms of the field strength tensor, F. By applying partial integration, the integral of the variation leads to the free Maxwell equations. The key takeaway is that this process confirms the relationship between the potential A and the field strength F, ultimately demonstrating that the divergence of F vanishes. This analysis clarifies the connection between the Lagrangian formulation and the resulting equations of motion in electromagnetism.
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This is from Coleman Lectures on Relativity, p.63. I understand that he uses integration by parts, but just can't see how he gets to the second equation. (In problem 3.1 he suggest to take a particular entry in 3.1 to make that more obvious, but that does not help me.)
 
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You can simplify the task a bit by writing
$$\delta \mathcal{L}=-\frac{1}{4} \delta (F_{\mu \nu} F^{\mu \nu}) = -\frac{1}{2} \delta F_{\mu \nu} F^{\mu \nu}=-\delta (\partial_{\mu} A_{\nu}) F^{\mu \nu}.$$
Then you have, indeed via partial integration)
$$\delta I = -\int \mathrm{d}^4 x \partial_{\mu} \delta A_{\nu} F^{\mu \nu} = + \int \mathrm{d}^4 x \delta A_{\nu} \partial_{\mu} F^{\mu \nu} \stackrel{!}{=}0,$$
and from this you get the free Maxwell equations
$$\partial_{\mu} F^{\mu \nu}=0, \quad F_{\mu \nu}=\partial_{\mu} A_{\nu} -\partial_{\nu} A_{\mu}$$
for the potential.
 
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