2 1. Introduction a b ya yb A B x y M Figure 1.1. Curve of descent energy and potential energy remains constant. If our particle starts from rest, we may thus write 1 2 mv2 + mgy = mgya . (1.2) The particle’s speed is then v = 2g(ya y) . (1.3) We now wish to find the brachistochrone (from βραχιστoς, short- est, and χρoνoς, time John Bernoulli originally, but erroneously, wrote brachystochrone). That is, we wish to find the curve y = y(x) ya (1.4) that minimizes the integral T = 1 2g b a 1 + y 2 ya y dx . (1.5) Several famous mathematicians responded to John Bernoulli’s challenge. Solutions were submitted by Gottfried Wilhelm Leibniz
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