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Euler-Lagrange 2nd derivative

  1. Nov 29, 2007 #1


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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:


    where the electromagnetic field tensor is:


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


    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


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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


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    Thank you, that's very helpful.
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