v Ĭalculus of variations - is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. This article incorporates material from Fundamental lemma of calculus of variations on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. The Calculus of Variations and Optimal Control: An Introduction. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer 2nd edition (September 1990) ISBN 3-X. The Euler-Lagrange equation plays a prominent role in classical mechanics and differential geometry. This lemma is used to prove that extrema of the functionalĪre weak solutions of the Euler-Lagrange equation Here, is the space of all infinitely differentiable functions defined on Ω whose support is a compact set contained in Ω. Ifįor all then f( x) = 0 for almost all x in Ω. Suppose that f is a locally integrable function defined on an open set. It defines a sufficient condition to guarantee that a function vanishes almost everywhere. The du Bois-Reymond lemma (named after Paul du Bois-Reymond) is a more general version of the above lemma. Since r is positive on ( a, b), f is 0 there and hence on all of. 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. The integrand is nonnegative, so it must be 0 except perhaps on a subset of of measure 0. 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). In other words, the test functions h ( C k functions vanishing at the endpoints) separate C k functions: C k is a Hausdorff space in the weak topology of pairing against C k functions that vanish at the endpoints. Then the fundamental lemma of the calculus of variations states that f( x) is identically zero on. Assume furthermore thatįor every function h that is of class C k on with h( a) = h( b) = 0. For example, class C 0 consists of continuous functions, and class consists of infinitely smooth functions. A function is said to be of class C k if it is k-times continuously differentiable.
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