Fundamental Lemma of Variational Calculus

[arhanbezbora]

I was just going through the derivation of the euler - lagrange equation that rests on the proof of the fundamental lemma of the calculus of variations which states the following:

If a function g(x) vanishes at the endpoints i.e g(a) = 0, g(b) = 0 and is continuously differentiable within the interval, and a second function f(x) is smooth within the interval and if
integral[ f(x)g(x) ] from a to b = 0,

then f(x) is identically zero within the interal (a,b).

This result is simple and intuitive if g(x) is positive within (a,b) but what happens if we have f(x) = k and g(x) = sin(x). Then g(x) = 0 at the endpoints 0 and 2pi but

integral[ k * sin(x) ] from a to b = 0 for k not equal to zero.

I would appreciate it if someone explained this to me and cleared my doubts as regarding the lemma. thanks a lot :)

 

[HallsofIvy]

That "lemma" is clearly not true. Are you sure it doesn't say "\int f(x)g(x)dx= 0 for every such funcition g"? Or perhaps "g(x)= 0 only at the endpoints"?

 

[dhris]

That "lemma" is clearly not true. Are you sure it doesn't say "\int f(x)g(x)dx= 0 for every such funcition g"?

Yes, that seems to be the correct way to state it:
http://en.wikipedia.org/wiki/Fundamental_lemma_of_calculus_of_variations

It's for every function g not any function g.

 

[quasar987]

To say "for every function g" and "for any function g" is the same thing. But "for every function g", and "for a function g" is not the same thing.

 

[dhris]

Oh man. I must have meant to say "some", and it came out as "any".

 

[arhanbezbora]

oh i see...i made a mistake in reading it and interpreted it as being true for some function g rather than for every function belonging to C(infinity). thanks a lot you guys.