Problem with Maxwell Lagrangian Density

[Strangelet]

Homework Statement

I have to expand the following term:

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

to get in the end this form:

$$\dfrac{1}{2}\left(\partial_{\mu}A_{\nu}\right) \left(\partial^{\mu}A^{\nu}\right) - \dfrac{1}{2}\left(\partial_{\mu}A^{\mu}\right)^2$$

Homework Equations

I really don't know how to make the calculation. I tried to multiply terms but I think I din't get some rule about index.. sigh!

The Attempt at a Solution

 

[Physicist97]

Here's a hint, ${\partial}_{\mu}A_{\nu}{\partial}^{\nu}A^{\mu}={\partial}_{\nu}A_{\mu}{\partial}^{\mu}A^{\nu}=({\partial}_{\mu}A^{\mu})^{2}$ . I'm not near a computer and have a hard time writing TeX from my phone, but foiling it out and using this relation should get you the right answer.

 

[TSny]

I have to expand the following term:

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

to get in the end this form:

$$\dfrac{1}{2}\left(\partial_{\mu}A_{\nu}\right) \left(\partial^{\mu}A^{\nu}\right) - \dfrac{1}{2}\left(\partial_{\mu}A^{\mu}\right)^2$$

I don't believe the expression $\dfrac{1}{4} F_{\mu\nu}F^{\mu\nu}$ is equal to the form you want to get. However, recall that two Lagrangian densities lead to the same equations of motion if they differ by the divergence of some expression. So, try to show that the initial and final forms differ only by a divergence of some expression.