Equations of motion from Born-Infeld Lagrangian

[OhNoYaDidn't]

We can write the Born-Infeld Lagrangian as:

$$L_{BI}=1 - \sqrt{ 1+\frac{1}{2}F_{\mu\nu }F^{\mu\nu}-\frac{1}{16}\left(F_{\mu\nu}\widetilde{F}^{\mu\nu} \right)^{2}}$$

with $G^{\mu\nu}=\frac{\partial L}{\partial F_{\mu\nu}}$ how can we show that in empty space the equations of motion take the form $\partial_{\mu}G^{\mu\nu}=0$
We should start with an Euler-Lagrange equation, but how can i write it for this Lagrangian?

 

[Demystifier]

The EL equation for this case is
$$\partial_{\nu}\frac{\partial L}{\partial A_{\mu,\nu}}=0$$
where $A_{\mu,\nu}=\partial_{\nu}A_{\mu}$. Using $F_{\mu\nu}=A_{\nu,\mu}-A_{\mu,\nu}$, the rest should be straightforward. See also Jackson to see how covariant Maxwell equations are derived for ordinary $F_{\mu\nu}F^{\mu\nu}$ action. For other details about Born Infeld see Zwiebach - A First Course in String Theory.

 

[OhNoYaDidn't]

Thank you, Demystifier.
I have never seen $F_{\mu\nu}$ written like that, but using that:
$F_{\mu\nu}F^{\mu\nu}=(A_{\nu\mu}-A_{\mu\nu})(A^{\nu\mu}-A^{\mu\nu})=A_{\nu\mu}A^{\nu\mu}-A_{\nu\mu}A^{\mu\nu}-A_{\mu\nu}A^{\nu\mu}+A_{\mu\nu}A^{\mu\nu}$
$(F_{\mu\nu}\widetilde{F}^{\mu\nu})^{2}=((A_{\nu\mu}-A_{\mu\nu})\widetilde{F}^{\mu\nu})^{2}$

$\frac{\partial L}{\partial A_{\mu\nu}} = \frac{-\frac{1}{4}({-A^{\nu\mu}+A^{\mu\nu}})+\frac{1}{16}A_{\mu\nu}F_{\mu\nu}(\widetilde{F}^{\mu\nu})^{2}}{\sqrt{ 1+\frac{1}{2}F_{\mu\nu }F^{\mu\nu}-\frac{1}{16}\left(F_{\mu\nu}\widetilde{F}^{\mu\nu} \right)^{2}}}$
What do i do with the $\partial_{\nu}$ now?

 

[Demystifier]

I have never seen $F_{\mu\nu}$ written like that

Than you should first learn ordinary electrodynamics. See the Jackson's textbook.

 

[PeterDonis]

using that

You left out the commas. Look closely at what Demystifier posted; there are commas, so it's $F_{\mu \nu} = A_{\nu , \mu} - A_{\mu , \nu}$. The commas are partial derivatives, so what he wrote is the same as $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$.

As Demystifier said, you need a good background in ordinary electrodynamics for the topic under discussion.