What's the proof of the Euler-Lagrange requirement for minimising an integral?

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What's the proof of the Euler-Lagrange requirement for minimising an integral?

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http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation Open the section labeled "Derivation of one-dimensional Euler-Lagrange equation." All you're really doing is re-doing the episilon/delta thing you did when you first derived derivatives of functions, however. It's just that you're working with functionals. That actually makes it easier, however, as functionals are usually much better behaved than functions.
http://upload.wikimedia.org/math/f/6/e/f6ee48b778d912689720cd77e1f4a0b3.png This is true??
\[F_{x} =x^2\] \[\frac{1}{dx}\int\limits_0^3x^2dt=\int\limits_0^32xdt\] The left side will be 0 and the right side not necessarily so.

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And can you explain how the fundamental Lemming applies here specifically?
Also: where is the proof that the minimum occurs when epsilon is 0? I think it is assumed in this proof- could you direct me to it?
In the Wikipedia derivation you are taking the derivative of a *parameter* of the integrand, while in your example above you are switching the order of differentiation and integration with respect to the variable of integration -- which is a definite no-no. They are very different things. I think if you are not clear on when you can and can't differentiate under the integral, you need to get clear on that before you can tackle the calculus of variations.

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