2.3. Lagrange’s approach 37 for all suﬃciently small . Since, however, the first variation is odd in , we can change its sign by changing the sign of . To prevent this change in sign, we require that δJ = 0 . (2.37) For a minimum, we also require that δ2J ≥ 0 . (2.38) If we want a relative maximum, we will, in turn, require δJ = 0 , δ2J ≤ 0 . (2.39) It is convenient, at this early stage of the course, to focus on the first variation. In light of the above arguments, we may safely say: First variation condition: A necessary condition for the functional J[y] to have a relative (or local) minimum or maximum at y = ˆ(x) is that the first variation of J[y] vanish, δJ = 0 , (2.40) for y = ˆ(x) and for all admissible variations η(x). The first variation, δJ = b a [fy(x, ˆ ˆ ) η + fy (x, ˆ ˆ ) η ] dx , (2.41) is rather unwieldy as written. We will rewrite the first variation so as to factor out the dependence on the admissible variations η(x). There are two different ways to do this. Both methods involve integration by parts. We start with Lagrange’s approach. 2.3.1. Lagrange’s simplification. Let us subject the second term in integrand (2.41) to integration by parts, b a fy (x, ˆ ˆ )η dx = η(x) ∂f ∂y x = b x = a − b a η d dx ∂f ∂y dx . (2.42)

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