We consider the problem of finding curves of minimum time of descent, joining two given points over a given frictionless surface and under the influence of gravity. We discuss the existence and minimality of extremals for the corresponding time functional, and find explicit solutions in certain special cases, including a closed form solution for the problem on an inclined plane. A discussion of numerical methods for computing these minimizers is given with several numerical examples for which explicit solutions are not known. This paper is mostly expository. Our main contribution is to put together on a single reference different results concerning the existence and minimality of extremals, explicit solutions in special cases, and a discussion of numerical methods for computing the desired minimizers.
