Fundamental Lemma of Variational Calculus

In summary, the fundamental lemma of calculus of variations states that if a function vanishes at the endpoints and is continuously differentiable within the interval, then the function f(x) is identically zero within the interval.
  • #1
arhanbezbora
13
0
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 :)
 
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  • #2
That "lemma" is clearly not true. Are you sure it doesn't say "[itex]\int f(x)g(x)dx[/itex]= 0 for every such funcition g"? Or perhaps "g(x)= 0 only at the endpoints"?
 
  • #3
HallsofIvy said:
That "lemma" is clearly not true. Are you sure it doesn't say "[itex]\int f(x)g(x)dx[/itex]= 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.
 
  • #4
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.
 
  • #5
Oh man. I must have meant to say "some", and it came out as "any".
 
Last edited:
  • #6
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.
 

Related to Fundamental Lemma of Variational Calculus

What is the Fundamental Lemma of Variational Calculus?

The Fundamental Lemma of Variational Calculus is a theorem that states that if a functional is stationary at a point, then its first variation is also equal to zero at that point. This lemma is a crucial tool in variational calculus, which deals with finding the extreme values of functionals.

Why is the Fundamental Lemma of Variational Calculus important?

The lemma is important because it allows us to determine the extreme values of functionals by setting their first variations to zero. This simplifies the process of finding extreme values and is used in many areas of science, such as physics and engineering.

How is the Fundamental Lemma of Variational Calculus used in real-world applications?

The lemma is used in many real-world applications, such as in physics to determine the path of a particle that minimizes the action, or in engineering to optimize the shape of a structure to minimize stress. It is also used in economics to find the optimal production level that maximizes profit.

Is the Fundamental Lemma of Variational Calculus applicable to all types of functionals?

Yes, the lemma is applicable to all types of functionals, including those that involve multiple variables and higher-order derivatives. However, it is important to note that it only applies to functionals that are differentiable at the point of interest.

Are there any prerequisites for understanding the Fundamental Lemma of Variational Calculus?

To fully understand the lemma, one should have a strong understanding of basic calculus, especially the concept of derivatives and integrals. It is also helpful to have some knowledge of variational calculus and the Euler-Lagrange equation.

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