30 2. The First Variation Here, yi = y(xi). We can now approximate the functional J[y] by the sum J(y1,...,yn) ≡ n i = 0 f xi,yi, yi+1 − yi Δx Δx , (2.6) a function of n variables. (Remember that y0 = ya and yn+1 = yb are fixed.) What is the effect of raising or lowering one of the free yi? To answer this question, let us choose one of the free yi, yk, and take the partial derivative with respect to yk. Since yk appears in only two terms in our sum, the partial derivative is just ∂J ∂yk = fy xk,yk, yk+1 − yk Δx Δx (2.7) + fy xk−1,yk−1, yk − yk−1 Δx − fy xk,yk, yk+1 − yk Δx . To find an extremum, we would ordinarily set this partial deriva- tive equal to zero for each k. We also, however, want to take the limit as n → ∞. In this limit, Δx → 0 and the right-hand side of equation (2.7) goes to zero. The equation 0 = 0, while true, is, sadly, not very helpful. To obtain a nontrivial result, we must first divide by Δx, 1 Δx ∂J ∂yk = fy xk,yk, yk+1 − yk Δx (2.8) − 1 Δx fy xk,yk, yk+1 − yk Δx − fy xk−1, yk−1, yk − yk−1 Δx . As we now let n → ∞ and Δx → 0, equation (2.8) yields the variational derivative δJ δy = fy(x, y, y ) − d dx fy (x, y, y ) . (2.9) This variational derivative plays the same role for functionals that the partial derivative plays for multivariate functions. For a relative (or local) minimum, we expect this derivative to vanish at each point,

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