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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.



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.)

References in periodicals archive ?
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.
The classical brachistochrone solution holds for cases 1 and 2 only.
2 permit us to make conclusions similar to the ones obtained for the pure brachistochrone problem ([sections]2.
Newton's problem turns out to be more complex than previously studied brachistochrone problems.
More precisely, we considered the 1696 brachistochrone problem (B); the 1687 Newton's aerodynamical problem of minimal resistance (N); a recent brachistochrone problem with restrictions (R) studied by Ramm in 1999, and where some open questions still remain [13]; and finally a generalized aerodynamical minimum resistance problem with non-parallel flux of particles (P), Recently studied by Plakhov and Torres [11;14] and which gives rise to other interesting questions [15].
First, for the brachistochrone and Ramm problems the functional value for the ES approximation was better than the linear interpolation over the exact solution, showing that the ES algorithm is capable of good precision.
We note that the per iteration "step" was [sigma] = 0:001 in the brachistochrone (-type) problems and [sigma] = 0:01 for the Newton(-type) problems.