- #1

spaghetti3451

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## Homework Statement

Given the Maxwell Lagrangian ##\mathcal{L} = -\frac{1}{2} (\partial_{\mu}A_{\nu})(\partial^{\mu}A^{\nu}) + \frac{1}{2} (\partial_{\mu}A^{\mu})^{2}##,

show that

(a) ##\frac{\partial \mathcal{L}}{\partial (\partial_{\mu}A_{\nu})} = - \partial^{\mu}A^{\nu}+(\partial_{\rho}A^{\rho})\eta^{\mu\nu}## and hence obtain the equations of motion ##\partial_{\mu}F^{\mu\nu}=0##.

(b) we may rewrite the Maxwell Lagrangian (up to an integration by parts) in the compact form ##\mathcal{L} = - \frac{1}{4} F_{\mu\nu}F^{\mu\nu}##.

## Homework Equations

## The Attempt at a Solution

(a) ##\mathcal{L} = -\frac{1}{2} (\partial_{\mu}A_{\nu})(\partial^{\mu}A^{\nu}) + \frac{1}{2} (\partial_{\mu}A^{\mu})(\partial_{\mu}A^{\mu})##

##= -\frac{1}{2} (\partial_{\mu}A_{\nu})(\partial_{\rho}A_{\sigma})(\eta^{\rho\mu}\eta^{\sigma\nu}) + \frac{1}{2} (\partial_{\mu}A_{\rho})(\partial_{\mu}A_{\sigma})(\eta^{\rho\mu}\eta^{\sigma\mu})##

Am I on the right track? Do I now differentiate each of the terms using the product rule?