2.2. Euler’s approach 29 x y a xk b Δx Figure 2.1. Polygonal curves handling these problems. Euler dropped his method for Lagrange’s more elegant “method of variations” after receiving Lagrange’s (Au- gust 12, 1755) letter. Euler also named this subject the calculus of variations in Lagrange’s honor. Euler’s essential idea was to first go from a variational problem to an n-dimensional problem and to then pass to the limit as n ∞. We will borrow from the modernized treatment of Euler’s method found in Elsgolc (1961) and Gelfand and Fomin (1963). See Goldstine (1980) and Fraser (2003) for more on the original approach. Let us divide the closed interval [a, b] into n+1 equal subintervals (see Figure 2.1). We will assume that the subintervals are bounded by the points x0 = a, x1,..., xn, xn+1 = b . (2.3) Each subinterval is of width Δx = xi+1 xi = (b a) n + 1 . (2.4) We will also replace the smooth function y(x) by the polygonal curve with vertices (x0,y0), (x1,y1),..., (xn,yn), (xn+1,yn+1) . (2.5)
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