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Homework Help: Euler-Lagrange 2nd derivative

  1. Nov 29, 2007 #1

    neu

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    1. The problem statement, all variables and given/known data
    I'm asked to get Maxwell's equations using the Euler-lagrange equation:

    [tex]\partial\left(\frac{\partial L}{\partial\left\partial_{\mu}A_{\nu}\right)}\right)-\frac{\partial L}{\partial A_{\nu}}=0[/tex]

    with the EM Langrangian density:

    [tex]L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-j_{\mu}A^{\mu}[/tex]

    where the electromagnetic field tensor is:

    [tex]F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}[/tex]

    3. The attempt at a solution
    I'm able to multiply out the density with the full form of the tensor F to get:

    [tex]\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}\partial_{\mu}A_{\nu}\left(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}\right)=\frac{1}{2}\left(\partial_{\mu}A_{\nu}F^{\mu\nu}\right)[/tex]

    My problem is that I know that the derivative w.r.t the scalar potential A for the 1st term in the density is zero as it only contains derivatives. i.e

    [tex]\frac{\partial }{\partial A_{\mu}}\left(\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}\right)=0[/tex]

    But I'm unable to show it explicity
     
  2. jcsd
  3. Nov 29, 2007 #2

    nrqed

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    I am not sure what you mean by "explicitly". As in other applications of the Lagrange formulation, you must treat [tex] A_\mu [/tex] and [tex] \partial_\nu A_\mu [/tex] independent quantities. So the derivative you wrote is trivially zero.
     
  4. Nov 29, 2007 #3

    neu

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    Thank you, that's very helpful.
     
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