2.3. Lagrange’s approach 43 Now, consider the variation η(x) defined by the equation η(x) = x a [M(u) μ] du . (2.68) It is easy to see that η(x) C1[a, b]. The function η(x) also vanishes at x = a and x = b. It is clearly an admissible variation. Moreover, η (x) = M(x) μ . (2.69) By hypothesis, b a M(x) η (x) dx = b a M(x) [M(x) μ] dx = 0 . (2.70) Also, b a M(x) [M(x) μ] dx μ b a [M(x) μ] dx = 0 . (2.71) But, this last equation may be rewritten b a [M(x) μ]2 dx = 0 . (2.72) Let x0 [a, b] be a point where M(x) is continuous. If M(x0) = μ, then there would have to exist a subinterval about x = x0 on which M(x) = μ. But this is clearly impossible in light of our last displayed equation. Thus M(x) = μ at all points of continuity. It follows that M(x) is constant for all x [a, b]. We now wish to apply this lemma to necessary condition (2.62), b a ∂f ∂y x a ∂f ∂y du⎠ ˆ ˆ η (x) dx = 0 . (2.73)
Previous Page Next Page