Talk:Fundamental lemma of calculus of variations

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I think the proof is important - some students use wikipedia to help them understand what they learn in lectures better. Plenty of other pages have proofs: for example Noether's theorem, (and it is one of the main reasons I use wikipedia). This is a quite a short proof. Oh yeah, and I missed out the details of h(x) because they were written in the statement of the lemma (h ∈ C2[a,b] with h(a) = h(b) = 0)

[edit] Proof of the principle

The proof is by contradiction:

\int_a^b f(x)h(x) dx = 0, \forall \, h(x): h(a)=h(b)=0.

Assume that for some c in the interior of the interval one has f(c) = 2e > 0.

By continuity, and the intermediate value theorem, there exists a neighbourhood [c0,c1] of c within [x0,x1] on which f(x) > e. Then,

\int_{c_0}^{c_1} f(x)h(x) \, dt > e\int_{c_0}^{c_1} h(x) > 0.

That gives a contradiction: therefore the only way for the integral to be zero in general is if

f(x)=0 \forall x \in [x_0, x_1].


Your proof is still wrong. How do you know that
\int_{c_0}^{c_1} h(x) > 0.
also, how do you know that
\int_{c_0}^{c_1} f(x)h(x) \, dt > e\int_{c_0}^{c_1} h(x)?
The function h needs to be chosen such that h(a) = h(b) = 0 but more is needed. It must be non-negative, and positive only in the small interval [c0,c1]. Why does such a function exist? Things are a bit more complicated than what you wrote. Oleg Alexandrov (talk) 16:15, 29 March 2006 (UTC)


It is not hard to find such an h, but the proof above is certainly sloppy. The first inequality is just wrong, and there is no need for the Intermediate Value Theorem. -cj67

[edit] External Links

I have removed the link to a web page that only contains a proof of the Euler-Lagrange equation, but not of the lemma. I'd guess that some sort of Hilbert space basis is needed for a proof, but I haven't actually seen any proof of this lemma... Anyone have a good book handy? --Shastra 20:10, 15 July 2006 (UTC)

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