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.
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2014 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.
