2.3. Lagrange’s approach 35 The strong norm, in contrast, does place a restriction on the size of the derivative. Stating that ||h||s (2.25) implies not only max [a,b] |h(x)| (2.26) but also sup [a,b] |h (x)| . (2.27) An -neighborhood in a strongly-normed space contains only weak variations (see Figure 2.4) that are close to the optimal solution in both ordinate and slope. Since strong variations are a superset of weak variations, a func- tion that minimizes a functional relative to nearby strong variations also minimizes that functional relative to nearby weak variations. Conversely, a necessary condition for a weak relative minimum is also a necessary condition for a strong relative minimum. Lagrange’s approach uses weak variations. This is alright if we want necessary conditions but is a problem if we want suﬃcient conditions. In due course, we will encounter examples of functionals that have minima relative to weak variations, but not relative to strong variations. To make Lagrange’s assumption as explicit as possible, we will consider small weak variations h(x) = η(x) (2.28) where η(a) = 0 , η(b) = 0 (2.29) and h(x) and h (x) are of the same order of smallness. The function η(x) is thus assumed to be independent of the parameter . As tends to zero, the variation h(x) tends to zero in both ordinate and slope. For notational convenience, we will also think of the functional J[y] as a function of , J( ) ≡ J[ˆ + η] = b a f(x, ˆ + η, ˆ + η ) dx . (2.30)

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