On the general problem of tautochrones

The cycloid is the solution to two classic problems: the brachistochrone and the tautochrone in a uniform gravitational field. The deep difference between the two problems is reflected in the techniques commonly used for their solution: calculus of variations for the brachistochrone, which usually leads to differential equations with boundary values, and integral equations for the tautochrone. Generalizations of both problems have been proposed through the years in the literature, including treatments of various types of force fields other than the uniform gravitational one. This work focuses on a general definition of tautochrone, basically equivalent to the standard one, and shows that under very broad hypotheses the existence, uniqueness, and computation of the curve can be easily reduced to the study of an initial value problem for a set of two differential equations of the first order, simply solvable by standard analytical or numerical techniques. Examples are discussed to illustrate the method, which is physically interesting because it can be applied to any positional force field, including non-conservative force fields, and also extended to the case of an additional frictional force proportional to velocity or a relativistic point mass.