# Equations of motion from Born-Infeld Lagrangian

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In summary, the Born-Infeld Lagrangian can be written as a function of the electromagnetic field strength tensor and its dual. Using the Euler-Lagrange equation and the covariant form of Maxwell's equations, we can show that in empty space, the equations of motion take the form of \partial_{\mu}G^{\mu\nu}=0. This can be derived by considering the EL equation for the Lagrangian and using the relationship between the field strength tensor and the vector potential. It is recommended to have a good understanding of ordinary electrodynamics before studying this topic.
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?

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.

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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?

I have never seen $F_{\mu\nu}$ written like that
Than you should first learn ordinary electrodynamics. See the Jackson's textbook.

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.

## 1. What is the Born-Infeld Lagrangian?

The Born-Infeld Lagrangian is a mathematical expression that describes the dynamics of electromagnetic fields in a nonlinear way. It was first proposed by physicists Max Born and Leopold Infeld in 1934 as an attempt to resolve the singularity issue in classical electrodynamics.

## 2. How are equations of motion derived from the Born-Infeld Lagrangian?

The equations of motion can be derived from the Born-Infeld Lagrangian using the Euler-Lagrange equations, which relate the Lagrangian to the physical quantities of the system. By varying the Lagrangian with respect to the fields, we can obtain the equations of motion for the electromagnetic field.

## 3. What are some applications of the Born-Infeld Lagrangian?

The Born-Infeld Lagrangian has been used in various fields of physics, including particle physics, cosmology, and string theory. It has also found applications in engineering, such as in the design of nonlinear optical devices and in the study of fluid dynamics.

## 4. How does the Born-Infeld Lagrangian differ from Maxwell's equations?

Unlike Maxwell's equations, which describe the dynamics of electromagnetic fields in a linear way, the Born-Infeld Lagrangian takes into account the nonlinear behavior of these fields. This allows for the avoidance of singularities and the incorporation of quantum effects.

## 5. Are there any open questions or controversies surrounding the Born-Infeld Lagrangian?

While the Born-Infeld Lagrangian has been successful in resolving certain issues in classical electrodynamics, there is ongoing debate and research regarding its consistency with other fundamental theories, such as quantum mechanics and general relativity. Additionally, there are still open questions about the physical interpretation of some of its terms.

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