Abstract
A new brachistochrone problem of the motion of a material point on a transcendental surface between two given points in a vertical uniform field of gravity is considered. The surface is a cut horizontal cylinder whose directrix is a cycloid, and generatrix is straight lines parallel to the horizontal axis. Energy dissipation is neglected. A time functional is obtained and a differential equation is derived for a single parameter that describes the form of brachistochrons in the problem. Spatial parametric equations of the brachistochronic motion of a material point on a surface are obtained. The fast response time is determined.
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