The following variation of a classical brachistochrone is considered: the particle must start and end at the same elevation with horizontal velocities, and must never "fly-off" the curve of its own inertia. First, it is shown that the problem can be reduced to that of finding brachistochrones that start horizontally and go to a specified vertical line, the corresponding particles constrained to never "fly-off" of their own inertia. This problem is then formulated as an optimal control problem with a state-dependent inequality constraint on the control. It is then shown that, strictly speaking, the problem has no solution but that interesting local extrema can be found. Graphs of these extrema are given and the resulting times are compared with those for the corresponding classical brachistochrone. 18 pp. Ref. (Author)
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