An alternative solution to the general tautochrone problem

Abstract

In 1658, Blaise Pascal put forward a challenge for solving the area under a segment of a cycloid and also its center of gravity. In 1659, motivated by Pascal challenge, Huygens showed experimentally that the cycloid is the solution to the tautochrone problem, namely that of finding a curve such that the time taken by a particle sliding down to its lowest point, under uniform gravity, is independent of its starting point. Ever since, this problem has appeared in many books and papers that show different solutions. In particular, the fractional derivative formalism has been used to solve the problem for an arbitrary potential and also to put forward the inverse problem: what potential is needed in order for a particular trajectory to be a tautochrone? Unfortunately, the fractional derivative formalism is not a regular subject in the mathematics curricula for physics at most of the Universities we know. In this work we develop an approach that uses the well-known Laplace transform formalism together with the convolution theorem to arrive at similar results.