32 2. The First Variation c a constant. If we integrate once again and set our new constant of integration to d, we conclude that y = cx + d . (2.16) This is the equation of a straight line. The constants c and d can be determined from the boundary conditions. 2.3. Lagrange’s approach Let us return to the problem of minimizing or maximizing the func- tional J[y] = b a f(x, y(x),y (x)) dx (2.17) subject to the boundary conditions y(a) = ya , y(b) = yb . (2.18) Euler derived the Euler–Lagrange equation by varying a single ordi- nate. Lagrange realized that he could derive this same equation while simultaneously varying all of the (free) ordinates. x y a b ˆ(x ) ˆ(x ) + h(x ) Figure 2.2. A small variation

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