- #1
dEdt
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If you don't have any charges or currents, the electromagnetic Lagrangian becomes ##\mathcal{L}=-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}##. The standard way to derive Maxwell's equations in free space is to replace ##F_{\alpha\beta}## by ##\partial_\alpha A_\beta -\partial_\beta A_\alpha## and apply the Euler-Lagrange equations. But why can't you just apply the Euler-Lagrange equations right away? I see no reason why this shouldn't work, except that it gives the wrong answer:
[tex]\frac{\partial\mathcal{L}}{\partial F_{\beta\gamma}} -\partial_\alpha\left(\frac{\partial\mathcal{L}}{\partial (\partial_\alpha F_{\beta\gamma})}\right)=-\frac{1}{4}F^{\beta\gamma}=0.[/tex]
What's wrong with my math?
[tex]\frac{\partial\mathcal{L}}{\partial F_{\beta\gamma}} -\partial_\alpha\left(\frac{\partial\mathcal{L}}{\partial (\partial_\alpha F_{\beta\gamma})}\right)=-\frac{1}{4}F^{\beta\gamma}=0.[/tex]
What's wrong with my math?