38 2. The First Variation Since our variations from the idealized solution vanish at the end- points of the interval, η(a) = 0 , η(b) = 0 , (2.43) our first necessary condition reduces to b a η(x) ∂f ∂y − d dx ∂f ∂y ˆ ˆ dx = 0 (2.44) for all admissible η(x). The subscript in this last equation signifies that the expression in square brackets is evaluated at y = ˆ(x) and y = ˆ (x). Let us note, right away, that our use of integration by parts, in this way, pretty much forces us to assume that ˆ(x) is twice differen- tiable. The partial derivative fy is generally a function of y (as well as of y and x) and if y does not exist, the existence of d dx ∂f ∂y (2.45) becomes doubtful. We shall see, momentarily, that Lagrange’s sim- plification actually forces us to assume that ˆ (x) ∈ C[a, b] or ˆ(x) ∈ C2 [a, b]. Lagrange claimed, without proof, that the coeﬃcient of η(x) in equation (2.44) must vanish, yielding the Euler–Lagrange equation, ∂f ∂y − d dx ∂f ∂y = 0 . (2.46) Euler pointed out, in a communication to Lagrange, that Lagrange’s statement was not self-evident and that he really ought to prove that the coeﬃcient of η(x) must vanish. This proof was eventually sup- plied by du Bois-Reymond (1879a). Du Bois-Reymond’s result is now known as the fundamental lemma of the calculus of variations.

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