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I Equations of motion from Born-Infeld Lagrangian

  1. Jan 30, 2017 #1
    We can write the Born-Infeld Lagrangian as:

    [tex]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}}[/tex]

    with [itex]G^{\mu\nu}=\frac{\partial L}{\partial F_{\mu\nu}}[/itex] how can we show that in empty space the equations of motion take the form [itex]\partial_{\mu}G^{\mu\nu}=0[/itex]
    We should start with an Euler-Lagrange equation, but how can i write it for this Lagrangian?
     
  2. jcsd
  3. Jan 30, 2017 #2

    Demystifier

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    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.
     
    Last edited: Jan 30, 2017
  4. Jan 30, 2017 #3
    Thank you, Demystifier.
    I have never seen [itex]F_{\mu\nu}[/itex] written like that, but using that:
    [itex]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}[/itex]
    [itex](F_{\mu\nu}\widetilde{F}^{\mu\nu})^{2}=((A_{\nu\mu}-A_{\mu\nu})\widetilde{F}^{\mu\nu})^{2}[/itex]

    [itex] \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}}}[/itex]
    What do i do with the [itex]\partial_{\nu}[/itex] now?
     
  5. Jan 30, 2017 #4

    Demystifier

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    Than you should first learn ordinary electrodynamics. See the Jackson's textbook.
     
  6. Jan 30, 2017 #5

    PeterDonis

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    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.
     
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