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henpen
 3 years ago
What's the proof of the EulerLagrange requirement for minimising an integral?
henpen
 3 years ago
What's the proof of the EulerLagrange requirement for minimising an integral?

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Carl_Pham
 3 years ago
Best ResponseYou've already chosen the best response.1http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation Open the section labeled "Derivation of onedimensional EulerLagrange equation." All you're really doing is redoing 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.

henpen
 3 years ago
Best ResponseYou've already chosen the best response.0http://upload.wikimedia.org/math/f/6/e/f6ee48b778d912689720cd77e1f4a0b3.png This is true??

henpen
 3 years ago
Best ResponseYou've already chosen the best response.0\[F_{x} =x^2\] \[\frac{1}{dx}\int\limits_0^3x^2dt=\int\limits_0^32xdt\] The left side will be 0 and the right side not necessarily so.

henpen
 3 years ago
Best ResponseYou've already chosen the best response.0And can you explain how the fundamental Lemming applies here specifically?

henpen
 3 years ago
Best ResponseYou've already chosen the best response.0Also: 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?

Carl_Pham
 3 years ago
Best ResponseYou've already chosen the best response.1In 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 nono. 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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