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Euler-Lagrange Simplification

  1. Jan 25, 2009 #1
    1. The problem statement, all variables and given/known data

    If the integrand f(y, y', x) does not depend explicitly on x, that is, f = f(y, y') then
    [tex]\frac{df}{dx} = \frac{\partial f}{\partial y}y' + \frac{ \partial f } {\partial y' } y''[/tex]


    Use the Euler-Lagrange equation to replace [tex]\partial f / \partial y[/tex] on the right and hence show that [tex]\frac{df}{dx} = \frac{d}{dx} ( y' \frac{\partial f}{\partial y'} ) [/tex]
    2. Relevant equations

    [tex]\frac{\partial f }{\partial y} = \frac{d}{dx} \frac{\partial f}{\partial y'}[/tex]


    3. The attempt at a solution

    By substituting in for df/df, I get an extra term that I can't seem to make go away.

    [tex]\frac{df}{dx} = \frac{d}{dx} y' \frac{ \partial f }{\partial y'} + \frac{\partial f}{\partial y'} y'' [/tex]

    I can't seem to get rid of that extra term, it seems like it should be straight forward but...
     
  2. jcsd
  3. Jan 26, 2009 #2

    CompuChip

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    Science Advisor
    Homework Helper

    Maybe it helps if you work backwards... what is
    [tex]
    \frac{df}{dx} = \frac{d}{dx} ( y' \frac{\partial f}{\partial y'} ) [/tex]
    ?
     
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