Talk:Fundamental lemma of the calculus of variations

message: Functional J is a functional of Lagrangian NOT the dependent variable y !
--mcyp 11:35, 24 January 2007 (UTC)
thread_title: Definition of Functional

message: 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)
thread_title: External Links

message: 71.165.190.99 just changed the hypothesis from f smooth to f continuous. Now the given proof doesn't work, because r f is not necessarily smooth. The proof can be edited to work as long as f is C^k and for all h in C^k the integral of f h is 0. That is, the smoothness on f and h must match. The given proof is nice for Wikipedia because it's so simple. The lemma can be stated more generally (see PlanetMath's version), but then the proof is more demanding. What version should we state and prove? Joshua R. Davis (talk) 20:35, 8 May 2008 (UTC)
replies: text: But then (if f is smooth, say, as it often is) the result is much weaker, since in your version our hypothesis is that the integral of fh is zero for all locally integrable h, while in the current version the hypothesis only involves smooth h. Algebraist 23:14, 22 March 2009 (UTC)
thread_title:  Continuity assumption 

message: Putting this in discussion part as I am not sure whether this requirement is already included by h(x)\in C^k. But I think it is neccessary to require \exists x\in[a,b]:h(x)\neq 0  i.e. h(x) cannot be constant zero for the lemma to hold.  —Preceding unsigned comment added by 87.122.44.44 (talk) 01:03, 16 January 2011 (UTC)
replies: text: The statement of the lemma is correct as it stands. The conclusion of the lemma holds if a certain property is satisfied for every h, including the h that is identically 0 and many other hs. The lemma makes no claim about f, if the property holds only for the h that is identically 0. Mgnbar (talk) 01:57, 16 January 2011 (UTC)
thread_title:  Small correction for correctness 

message: I got bold and rewrote it, after some discussion on Wikipedia talk:WikiProject Mathematics#Fundamental lemma of calculus of variations.
Hope it is better. You are invited to improve it further. (Really, I intend to do some more soon.)
About the proof: if you want to restore it, do so. But we should not restrict the formulations to the very weak case according to the very simple proof. Just formulate the relevant simple case and prove it, if you like.
About Section "Applications": I doubt it is correct, but for the moment I am tired; let it stay as it was, and then we'll see.
About the old refs: restore them if you like (but please keep my refs). Boris Tsirelson (talk) 20:10, 2 June 2015 (UTC)
replies: text: Part of the "Basic version" is misleading. It says that replacing Cinfinity with C0 as the notion of smoothness is okay and may be strong enough for a given task. But, because smoothness is used in the assumptions of the lemma, replacing Cinfinity with C0 makes the assumptions stronger, not weaker. Do you see my problem? How should we fix it? Mgnbar (talk) 11:40, 4 April 2019 (UTC)
thread_title:  Rewritten