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Fundemental lemma of the calculus of variations

  1. Aug 29, 2012 #1
    1. The problem statement, all variables and given/known data

    Hi, I've been revising the calculus of variations and using the wiki entry on the euler lagrange equation (http://en.wikipedia.org/wiki/Euler-Lagrange_equation) as a reference. Scroll down and you'll see: Derivation of one-dimensional Euler–Lagrange equation. Expand this. In it you'll see the statement: "It follows from the total derivative that" and:

    dF/dε= dx/dε*∂F/∂x + dgε/dε*∂F/∂gε + dg'ε/dε*∂Fε/∂g'ε

    2. Relevant equations

    What happened to the first term (dx/dε*∂F/∂x)?

    3. The attempt at a solution

    I understand that the first term has gone to zero. But how? If π(a) and π(b) both = 0 surely f(x) is a line with f(x) = 0? In which case it is clear that that term will go to zero.
     
  2. jcsd
  3. Aug 29, 2012 #2

    vela

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    Isn't ##dx/d\varepsilon =0##?
     
  4. Aug 29, 2012 #3
    I think I have it. The description in wiki is a little limited (alternatively my imagination is limited :smile:) so by a little mixing and matching of proofs I think I have found it.

    Thanks anyway.
     
  5. Aug 29, 2012 #4

    vela

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    I don't see any obvious problem with the derivation on Wikipedia other than it throws in the unnecessary term that seems to have confused you.
     
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