# Fundemental lemma of the calculus of variations

TooFastTim

## Homework Statement

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

## Homework Equations

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

## 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.

## Answers and Replies

Staff Emeritus
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Isn't ##dx/d\varepsilon =0##?

TooFastTim
I think I have it. The description in wiki is a little limited (alternatively my imagination is limited ) so by a little mixing and matching of proofs I think I have found it.

Thanks anyway.

Staff Emeritus
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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.