Fundamental lemma of calculus of variations
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In calculus of variations, the fundamental theorem in calculus of variation is a theorem that is typically used to transform a problem from its weak formulation (variational form) into its strong formulation (differential equation).
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).
More generally, the conclusion of this lemma still holds when f is locally integrable on an open subset U of Rn and the functions h are infinitely smooth and with compact support in U (except that the conclusion is replaced by f=0 almost everywhere).
[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](../../../math/d/6/5/d65452f09a8ff74f1e7e384f7ef74dda.png)
are weak solutions of the Euler-Lagrange equation
[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.
- S. Lang, Analysis II, Addison-Wesley Publishing Company Inc., 1969. ASIN B0006BZ3E6.
This article incorporates material from Fundamental lemma of calculus of variations on PlanetMath, which is licensed under the GFDL.
