2.3. Lagrange’s approach 41 Euler–Lagrange condition: Every ˆ(x) C2[a, b] that produces a relative extremum of the integral J[y] = b a f(x, y, y ) dx (2.56) satisfies the Euler–Lagrange differential equation ∂f ∂y d dx ∂f ∂y = 0 . (2.57) Lagrange’s simplification forces us to assume that our solutions have continuous second derivatives. Can we loosen this assumption? Let us start with the necessary condition that the first variation must vanish, δJ[η] = b a [fy(x, ˆ ˆ ) η + fy (x, ˆ ˆ ) η ] dx = 0 , (2.58) and try a different approach. 2.3.2. Du Bois-Reymond’s simplification. Let us now assume that the functions ˆ(x) and η(x) are merely continuously differen- tiable, ˆ(x), η(x) C1[a, b]. Since fy (x, ˆ ˆ ) depends on ˆ (x), this function need not be differentiable. As a result, we cannot integrate the second term in integrand (2.41) by parts. Let us instead integrate the first term in integrand (2.41) by parts. Doing so, we obtain b a fy(x, ˆ ˆ ) η dx = [η(x)φ(x)]x = b x = a b a φ(x) η (x) dx , (2.59) where φ(x) = x a fy(u, ˆ(u), ˆ (u)) du . (2.60)
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