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Fundamental lemma of calculus of variations
From Wikipedia, the free encyclopedia
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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The fundamental lemma of the calculus of variations states that if f is a smooth function over the interval [a,b], and
for every function h ∈ 
[a,b] with h(a) = h(b) = 0, then f(x) is identically zero in the open interval (a,b).
In other words, the test functions h (smooth functions vanishing at the endpoints) separate smooth functions: ![C^\infty[a,b]](http://upload.wikimedia.org/math/4/7/b/47bd5a14c0ad0c9eef40b6778b11e28b.png)
is a Hausdorff space in the weak topology of pairing against smooth functions that vanish at the endpoints.
Let f be a function satisfying 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 satisfies the hypotheses, so
.
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].
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 
. If
for all 
then f(x) = 0 for almost all x in Ω. Here, 
is the space of all infinitely differentiable functions defined on Ω whose support is a compact set contained in Ω.
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](http://upload.wikimedia.org/math/d/6/5/d65452f09a8ff74f1e7e384f7ef74dda.png)
are weak solutions of the Euler-Lagrange equation
The Euler-Lagrange equation plays a prominent role in classical mechanics and differential geometry.
- 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. Retrieved on 2007-04-17.
This article incorporates material from Fundamental lemma of calculus of variations on PlanetMath, which is licensed under the GFDL.