The brachistochrone problem consists in determining the curve of minimum time when a particle starting at a point A = ([x.sub.0], [y.sub.0]) of a vertical plane goes to a point B = ([x.sub.1], [y.sub.1]) in the same plane under the action of the gravity force and with no initial velocity.
[T.sub.b]: the brachistochrone for the problem with ([x.sub.0], [x.sub.1]) = (0, 10), ([y.sub.0], [y.sub.1]) = (10, 0) has parameters a [??] 5:72917 and [[theta].sub.1] = 2.41201; the time is [T.sub.b] [??] 1:84421;
[T.sub.o]: a piecewise linear function with 20 segments defined over the brachistochrone; the time is [T.sub.o] = 1:85075.
1a one can see that the piecewise linear solution is made of points that are not over the brachistochrone, because that is not the best solution for piecewise functions.
The brachistochrone problem with restrictions, 1999
Ramm  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.e.) and 0 [less than or equal to] y(x) [less than or equal to] [y.sub.0](x), where [y.sub.0] is a straight line between A = (0, 1) and B = (b, 0), b > 0.
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]2.1).
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 ; 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 .