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The Brachistochrone Problem. Find the curve y(x) connecting y(a) = A to y(b) = B (assume a < b and B < A), along which a

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The Brachistochrone Problem. Find the curve y(x) connecting y(a) = A to y(b) = B (assume a < b and B < A), along which a particle under the influence of gravity g will slide without friction in minimum time. This is the famous Brachistochrone problem discussed in Problem 31 of Section 2.2. Bernoulli showed that the appropriate functional for the sliding time is T = ? T 0 dt = ? L 0 dt ds ds = ? L 0 ds ? = 1 ?2g ? L 0 ds ?y = 1 ?2g ? b a ? 1 + y ?2 y dx. (i) (a) Justify each of the equalities in Eq. (i). (b) From the functional on the right in Eq. (i), find the Euler

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