Fundamental lemma of calculus of variations

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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).

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[edit] Statement

A function is said to be of class Ck 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 Ck on the interval [a,b]. Assume furthermore that

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

for every function h that is Ck 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 in the open interval (a,b).

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

[edit] 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 Ck on [a,b], so

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

But the integrand is nonnegative, so it must be identically 0. Since r is positive on (a, b), f is 0 there and hence on all of [a, b].

[edit] The DuBois-Reymond lemma

The DuBois-Reymond lemma 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.

[edit] 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.

[edit] 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. 

This article incorporates material from Fundamental lemma of calculus of variations on PlanetMath, which is licensed under the GFDL.

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