# Fundemental Lemma of the Calculus of Variations

• Niles
In summary, the conversation is discussing the Fundamental Lemma of the Calculus of Variations, which states that if f is a continuous function on [a,b] and the integral of f(t)h(t) is 0 for all h in a specific space of parametrized curves, then f(x) = 0 for all x in [a,b]. It is also mentioned that this lemma does not work in higher dimensions without additional conditions.
Niles
Hi all.

In my notes I wrote down from the blackboard, I wrote

[Fundemental Lemma of the Calculus of Variations] Let f : [a,b] -> R be continuous and suppose that

$$\int_a^b f(t)h(t)dt = 0$$

for all $h\in C_{0,0}^1([a,b], R)$, where $C_{0,0}^1([a,b], R)$ is the space of C1 parametrized curves O : [a,b] -> R that start and end in 0.

I suspect that I missed some k's when writing this down from the blackboard. Am I correct when I say that this also works if we are in Rk?

What you are missing is the conclusion of the lemma. You have only stated the hypotheses. IF f is a function having those properties then- f(x)= 0 of all x in [a,b].

No, you are not correct that this also works in Rk. In higher dimensions just integrating on paths is not enough. You would also have to know that integral was 0 on all surfaces, all three dimensional regions, etc. up to the dimension of the space.

Last edited by a moderator:
Thank you. Yes, for some reason I didn't type in the conclusion, even though I did write it down in class. Thanks.

## 1. What is the Fundamental Lemma of the Calculus of Variations?

The Fundamental Lemma of the Calculus of Variations is a fundamental theorem that states that the necessary and sufficient conditions for a function to be an extremum of a functional can be expressed as an Euler-Lagrange equation.

## 2. What is a functional in the context of the Fundamental Lemma of the Calculus of Variations?

A functional is a mathematical expression that takes in a function as its input and returns a number as its output. In the context of the Fundamental Lemma, the functional represents a quantity that we want to minimize or maximize.

## 3. How is the Euler-Lagrange equation derived from the Fundamental Lemma of the Calculus of Variations?

The Euler-Lagrange equation is derived by considering the variation of the functional with respect to the function. This variation is then set to zero, and the resulting equation is the Euler-Lagrange equation, which represents the necessary condition for the function to be an extremum of the functional.

## 4. What is the importance of the Fundamental Lemma of the Calculus of Variations in mathematics?

The Fundamental Lemma of the Calculus of Variations plays a crucial role in mathematical analysis, optimization, and physics. It provides a powerful tool for finding the extrema of functionals and is used in many fields, including mechanics, economics, and engineering.

## 5. Are there any applications of the Fundamental Lemma of the Calculus of Variations in real-life problems?

Yes, the Fundamental Lemma of the Calculus of Variations has many practical applications, such as in the optimization of complex systems, the determination of optimal paths or trajectories, and the modeling of physical phenomena. It is also used in the field of machine learning for finding optimal solutions to problems.

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