Fundamental lemma of calculus of variations

In mathematics, specifically in the calculus of variations, the fundamental lemma in the calculus of variations is a lemma that is typically used to transform a problem from its weak formulation (variational form) into its strong formulation (differential equation).

Statement

A function is said to be of class C^k if it is k-times continuously differentiable. For example, class C⁰ consists of continuous functions, and class $C^\infty$ consists of infinitely smooth functions.

Let f be of class C^k on the interval [a,b]. Assume furthermore that

$\int_a^b f(x) \, h(x) \, dx = 0$

for every function h that is of class C^k on [a,b] with h(a) = h(b) = 0. Then the fundamental lemma of the calculus of variations states that f(x) is identically zero on [a,b].

In other words, the test functions h (C^k functions vanishing at the endpoints) separate C^k functions: C^k[a,b] is a Hausdorff space in the weak topology of pairing against C^k functions that vanish at the endpoints.

Proof

Let f satisfy the hypotheses. Let r be any smooth function that is 0 at a and b and positive on (a, b); for example, r = − (x − a)(x − b). Let h = rf. Then h is of class C^k on [a,b], so

$0 = \int_a^b f h \; dx = \int_a^b r f^2 \; dx.$

The integrand is nonnegative, so it must be 0 except perhaps on a subset of [a,b] of measure 0. However, by continuity if there are points where the integrand is non-zero, there is also some interval around that point where the integrand is non-zero, which has non-zero measure, so it must be identically 0 over the entire interval. Since r is positive on (a, b), f is 0 there and hence on all of [a, b].

The du Bois-Reymond lemma

The du Bois-Reymond lemma (named after Paul du Bois-Reymond) is a more general version of the above lemma. It defines a sufficient condition to guarantee that a function vanishes almost everywhere. Suppose that f is a locally integrable function defined on an open set $\Omega \subset \mathbb{R}^n$. If

$\int_\Omega f(x) h(x) dx = 0\,$

for all $h \in C^\infty_0(\Omega),$ then f(x) = 0 for almost all x in Ω. Here, $C^\infty_0(\Omega)$ is the space of all infinitely differentiable functions defined on Ω whose support is a compact set contained in Ω.

Applications

This lemma is used to prove that extrema of the functional

$J[L(t,y,\dot y)] = \int_{x_0}^{x_1} L(t,y,\dot y) \, dt$

are weak solutions of the Euler-Lagrange equation

${\partial L(t,y,\dot y) \over \partial y} = {d \over dt} {\partial L(t,y,\dot y) \over \partial \dot y} .$

The Euler-Lagrange equation plays a prominent role in classical mechanics and differential geometry.

References

• L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer; 2nd edition (September 1990) ISBN 0-387-52343-X.

• Lang, Serge (1969). Analysis II. Addison-Wesley. 

• Leitmann, George (1981). The Calculus of Variations and Optimal Control: An Introduction. Springer. ISBN 0306407078. http://books.google.com/books?id=ActDgnJsvvoC. Retrieved 2007-04-17. 

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