2.2. Euler’s approach 31 leaving us with the Euler–Lagrange equation ∂f ∂y − d dx ∂f ∂y = 0 . (2.10) This condition must be modified if the minimizing curve lies on the boundary rather than in the interior of the region of interest. More- over, the Euler–Lagrange equation is only a necessary condition, in the same sense that f (x) = 0 is a necessary, but not a suﬃcient, condition in calculus. I should, perhaps, add that the above discussion is misleading to the extent that the formal notion of a variational or functional deriv- ative was not introduced until much later, by Vito Volterra (1887), in the early stages of the development of functional analysis. See the recommended reading at the end of this chapter for more information about variational derivatives. Example 2.1 (Shortest curve in the plane). Let’s see what the Euler–Lagrange equation has to say about the shape of the shortest curve between two points, (a, ya) and (b, yb), in the plane. We clearly wish to minimize the arc-length functional J[y] = b a ds = b a 1 + y 2 dx . (2.11) The integrand, f(x, y, y ) = 1 + y 2 , (2.12) does not depend on y and so the Euler–Lagrange equation reduces to d dx y 1 + y 2 = 0 . (2.13) Integrating once produces y 1 + y 2 = constant (2.14) and we quickly conclude that y = c , (2.15)

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