Talk:Calculus of variations

Fermat's Principle section is ambiguous

It should be removed as right now it is unsourced and has sloppy notation. $n_{y}$ is undefined, and the process by which n is expanded is not explained, it is assumed and steps are missing. — Preceding unsigned comment added by Stwalczyk (talk • contribs) 00:28, 27 August 2013 (UTC)

Euler-Lagrange equation: loss of generality?

I don't claim to be an expert, but as far as I can see, there is no reason for L[x, y (x), y ′(x)] to be assumed twice continuously differentiable while proving the Euler-Lagrange equation; having continuous first partial derivatives is just enough, as the respective main article has it. Please, if there is anybody with special knowledge on the subject, review the proof in question.

--SiriusGR (talk) 23:01, 27 May 2015 (UTC)

The source for this article's derivation is Methods of Mathematical Physics. Vol. Vol. I (First English ed.). New York: Interscience Publishers, Inc. 1953. ISBN 978-0471504474. {{cite book}}: |volume= has extra text (help); Cite uses deprecated parameter |authors= (help) On p. 184 of this source is,

"The function F [L in our notation] is to be twice continuously differentiable with respect to its three arguments x, y, y' "

I didn't notice a source for the singly continuously differentiable requirement in the other article Euler-Lagrange equation.

FYI, the requirement for twice continuously differentiable probably comes from using the fundamental lemma of calculus of variations in the following step in the derivation, which requires that the part of the integrand in parentheses be a continuous function of x, according to Courant and Hilbert p. 185.

$\int _{x_{1}}^{x_{2}}\eta \left({\frac {\partial L}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L}{\partial f'}}\right)\,dx=0\,.$

According to the fundamental lemma of calculus of variations, the part of the integrand in parentheses is zero, ...

--Bob K31416 (talk) 00:48, 28 May 2015 (UTC)

Indeed, I apologise, $d/dx\left({\partial L}/{\partial f'}\right)$ should exist and be continuous in order to apply integration by parts, and that is the most obvious application of the twice differentiability assumption. Of course I cannot doubt the source of the proof... Now, should we do something about the other article? (I'm new to Wikipedia and I don't know what happens in such a case; I came across this problem while translating this article (Calculus of variations) for the Greek-language Wikipedia). Thank you!

--SiriusGR (talk) 14:34, 28 May 2015 (UTC)

I just now made the change in the corresponding derivation at the other article.[1] --Bob K31416 (talk) 16:16, 28 May 2015 (UTC)

But look again at "fundamental lemma of calculus of variations'; now differentiability is derived there, not assumed. Boris Tsirelson (talk) 16:34, 21 August 2015 (UTC)

Strong vs. weak

The article says:

"Both strong and weak extrema of functionals are for a space of continuous functions but weak extrema have the additional requirement that the first derivatives of the functions in the space be continuous. Thus a strong extremum is also a weak extremum, but the converse may not hold. "

Shouldn't this read, "**strong** extrema have the additional requirement..."? 74.92.140.243 (talk) 00:35, 22 May 2017 (UTC)

Our article uses the terminology of the given source. I think one can see how that terminology came about by noting that the weak extrema are for the case of functions that are continuous and have continuous first derivatives. This case is less general and thus weaker than the case of functions that are continuous and may or may not have continuous first derivatives. In other words, the set of functions that are continuous and have continuous first derivatives is contained in the set of functions that are continuous, so that weak extrema apply to fewer situations than strong extrema. --Bob K31416 (talk) 06:01, 12 August 2017 (UTC)

I think part of this section needs a small rewrite. From Gelfand & Fomin (p6, p13):

The functional $J[y]$ has a strong extremum for $y={\hat {y}}$ if there exists an $\varepsilon >0$ such that $J[y]-J[{\hat {y}}]$ has the same sign for all $y$ in the domain of definition of the functional which satisfy the condition: $\max |y-{\hat {y}}|<\varepsilon \,\,(1)$.

Whereas the functional $J[y]$ has a weak extremum for $y={\hat {y}}$ if there exists an $\varepsilon >0$ such that $J[y]-J[{\hat {y}}]$ has the same sign for all $y$ in the domain of definition of the functional which satisfy the condition: $\max |y-{\hat {y}}|+\max |y'-{\hat {y}}'|<\varepsilon \,\,(2)$.

When you write it like that, it's clear that every strong extremum is also a weak extremum, as if $J[y]-J[{\hat {y}}]$ has the same sign for all $y$ which satisfy condition (1) then of course it also has the same sign for all $y$ which satisfy condition (2), as the set of all $y$ which satisfy condition (2) is a subset of the set of all $y$ which satisfy condition (1).

103.102.228.151 (talk) 05:09, 3 April 2018 (UTC)

Merging Applications of the calculus of variations into Calculus of variations

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

Proceeding with merge for lack of objections.

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Applications of the calculus of variations is currently incorrectly listed as a disambiguation page, and in fact gives less detail than the Applications section here. I would suggest merging those bullet points that are not currently listed here from there, and then replacing Applications of the calculus of variations with a redirect to the Application section of Calculus of variations. Felix QW (talk) 16:49, 6 January 2022 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Nontrivial topology

What is meant by the warning that certain minimal surfaces may "have" nontrivial topology? Can't we assign it any topology we'd like? Why is this relevant here? 141.23.182.177 (talk) 19:48, 11 April 2022 (UTC)