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Help with derivation of euler-lagrange equations

  1. Nov 21, 2009 #1

    I am trying to follow a derivation of the euler lagrange equations in one of my textbooks. It says that

    [tex]\int ( f\frac{dL}{dx} + f'\frac{dL}{dx'}) dt[/tex]


    [tex]f\frac{dL}{dx'} + \int f ( \frac{dL}{dx} - \frac{d}{dt}(\frac{dL}{dx'}) ) dt [/tex]

    where f is an arbitrary function and L is the Lagrangian.

    I'm not sure how to perform this step. I think it has something to do with integration by parts but can't work it out. Any help would be appreciated.
  2. jcsd
  3. Nov 21, 2009 #2


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    Homework Helper

    The integral over f (dL/dx) can be ignored, it simply sits in both expressions. So your question is, how do you get from

    [tex]\int \frac{df}{dt} \frac{dL}{dx'}[/tex]
    [tex]f \frac{dL}{dx'} - \int f \frac{d}{dt} \frac{dL}{dx'}[/tex]

    Because that is just partial integration in its purest form:
    [tex]\int f' g = f g - \int f g'[/tex]
    where g = dL/dx'.
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