calculus of variations pdf

Calculus of Variations 1 Functional Derivatives The fundamental equation of the calculus of variations is the Euler-Lagrange equation d dt ∂f ∂x˙ − ∂f ∂x = 0. What is the Calculus of Variations “Calculus of variations seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum).” (MathWorld Website) Variational calculus … Fundamental lemma of variational calculus Suppose that H(x) is continuously differentiable with Z b a H(x)ϕ(x)dx= 0 for every test function ϕ. There are several ways to derive this result, and we … The field has drawn the attention of a remarkable range of mathematical luminaries, beginning with … 16|Calculus of Variations 3 In all of these cases the output of the integral depends on the path taken. )dt computes a cost, J, for a fuven function x(t). To prove this, consider an … It is a functional of the path, a scalar-valued function of a function variable. For example J(x)=∫ a b F(t,x,x. Then H(x) must be identically zero. Calculus of Variations is a branch of m ethematics dealing with optimizing functionals. The history of the calculus of variations is tightly interwoven with the history of mathematics, [9]. A functional is a function of functions. Denote the argument by square … …

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