How Can Maxwell's Equations Be Derived Using the Euler-Lagrange Equation?

[neu]

Homework Statement

I'm asked to get Maxwell's equations using the Euler-lagrange equation:

$$\partial\left(\frac{\partial L}{\partial\left\partial_{\mu}A_{\nu}\right)}\right)-\frac{\partial L}{\partial A_{\nu}}=0$$

with the EM Langrangian density:

$$L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-j_{\mu}A^{\mu}$$

where the electromagnetic field tensor is:

$$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$$

The Attempt at a Solution

I'm able to multiply out the density with the full form of the tensor F to get:

$$\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}\partial_{\mu}A_{\nu}\left(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}\right)=\frac{1}{2}\left(\partial_{\mu}A_{\nu}F^{\mu\nu}\right)$$

My problem is that I know that the derivative w.r.t the scalar potential A for the 1st term in the density is zero as it only contains derivatives. i.e

$$\frac{\partial }{\partial A_{\mu}}\left(\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}\right)=0$$

But I'm unable to show it explicity

 

[nrqed]

Homework Statement

I'm asked to get Maxwell's equations using the Euler-lagrange equation:

$$\partial\left(\frac{\partial L}{\partial\left\partial_{\mu}A_{\nu}\right)}\right)-\frac{\partial L}{\partial A_{\nu}}=0$$

with the EM Langrangian density:

$$L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-j_{\mu}A^{\mu}$$

where the electromagnetic field tensor is:

$$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$$

The Attempt at a Solution

I'm able to multiply out the density with the full form of the tensor F to get:

$$\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}\partial_{\mu}A_{\nu}\left(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}\right)=\frac{1}{2}\left(\partial_{\mu}A_{\nu}F^{\mu\nu}\right)$$

My problem is that I know that the derivative w.r.t the scalar potential A for the 1st term in the density is zero as it only contains derivatives. i.e

$$\frac{\partial }{\partial A_{\mu}}\left(\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}\right)=0$$

But I'm unable to show it explicity

I am not sure what you mean by "explicitly". As in other applications of the Lagrange formulation, you must treat $$A_\mu$$ and $$\partial_\nu A_\mu$$ independent quantities. So the derivative you wrote is trivially zero.

 

[neu]

Thank you, that's very helpful.