Is it possible to write the Lagrangian density for the EM field and charges $$L=-\frac{1}{4\mu_0}F^{\mu \nu}F_{\mu \nu}+j^{\mu}A_{\mu}$$ only in terms of the Electromagnetic Tensor and current vector? Can you also solve the Euler-Lagrange equation without reference to the potential? For example, witht the Lagrangian density for empty space: $$L=-\frac{1}{4\mu_0}F^{\mu \nu}F_{\mu \nu}$$ There are no derivatives of $F^{\mu \nu}$, so I don't see how you could derive the field equations from it.
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Qmechanic
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Phineas Nicolson
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Possible duplicate: https://physics.stackexchange.com/q/53018/2451 and links therein. – Qmechanic Feb 28 '21 at 06:56
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By going through the answers to the linked question it is clear that Occam's razor must be applied. This is a dead end street. – my2cts Feb 28 '21 at 10:50
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The field strength (Faraday Tensor) is defined as:
$$F_{μν}= \partial_{μ}Α_{ν} - \partial_{ν} Α_{μ}$$
You have to vary with respect to the electromagnetic $U(1)$ field $A_{μ}$ everywhere it appears.
Noone
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I think it does. The field one varies with respect to is the $A_{\mu}$ which is incorporated in the electromagnetic tensor, which is what i think the OP forgot. – Noone Feb 28 '21 at 09:25
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