There's some useful information here mainly about the brachistochrone property, but also about the tautochrone property. If you can get through the writing style: < http://www.maa.org/pubs/Calc_articles/ma060.pdf >. Maybe there's a way to use this to show the two properties are equivalent. --Dan On 2012-11-22, at 9:07 AM, Andy Latto wrote:
This reminds me of a question I've wondered about. The cycloid is both the tautochrone and the brachistochrone. I've seen proofs of both of these facts, but it seems a remarkable coincidence that the same curve satisfies both these properties. Is there a proof that the tautochrone and the brachistochrone are the same curve that is simpler than just finding the equations of both, and proving that they are in fact the same?
(Brachistochrone = Given points x and y, find the curve so that a ball rolling from x to y gets there fastest. Tautochrone = Find a curve so that the time to roll from an arbitrary starting point on the curve to a fixed ending point is the same, regardless of the choice of starting point).