Brachistochrone


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brachistochrone

[brə′kis·tə‚krōn]
(mechanics)
The curve along which a smooth-sliding particle, under the influence of gravity alone, will fall from one point to another in the minimum time.

Brachistochrone

 

the curve of most rapid descent—that is, the one of all possible curves connecting two given points A and B of a potential force field that a mass point moving along the curve with an initial velocity equal to zero and acted upon only by the forces of the field will traverse from position A to position B in the shortest time. When the movement occurs in a homogeneous gravitational field, the brachistochrone is a cycloid with a horizontal base and a point of return that coincides with point A. The solution of the brachistochrone problem (Johann Bernoulli, 1696) served as the starting point for the development of the calculus of variations. The error of Galileo, who tried to prove that the brachistochrone is a circumferential arc, is instructive. (See G. Galilei, Izbrannye trudy, vol. 2, Moscow, 1964, pp. 298-301, note 465.)

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The brachistochrone problem consists in determining the curve of minimum time when a particle starting at a point A = ([x.
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Ramm [13] presents a conjecture about a brachistochrone problem over the set S of convex functions y (with y''(x) [greater than or equal to] 0 a.
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