TAUTOCHRONE CURVES IN SPACE

We consider the tautochrone problem in three-dimensional space. First, we derive the sliding time along a frictionless space curve from rest under the influence of gravity in terms of fractional calculus. We then give the condition for an unrestricted space curve to satisfy tautochronism. As examples, we study the tautochrone curve restricted to various vertical cylindrical surfaces, in a tilted plane and on a general surface. A tautochrone curve with glide time T on a vertical cylindrical surface can stretch upwards and reach its maximum height H=2gT2∕π2 like an unrestricted tautochrone curve, where g is the acceleration of gravity, while a tautochrone curve restricted on a general space surface cannot reach such a height in general. The tautochrone problems on the conical surface and the elliptic paraboloid lead to nonlinear ordinary differential equations worthy of further study.