A simplified approach to the brachistochrone problem

Ever since Johann Bernoulli put forward the challenge "Problema novum ad cujus solutionem Mathematice invitantur" in Acta Eruditorum Lipsiae of June, 1696, of finding the minimum time trajectory (the brachistochrone) described by an object moving from one point to another (not directly behind the first one) in a constant uniform gravitational field, many works have been published on this subject, and some books mention it as part of the applications of the Euler-Lagrange formalism. However, we have found only one reference of the problem related to the general inhomogeneous inverse square gravitational field (Supplee and Schmidt 1991 Am. J. Phys. 59 467). Even in this reference, the problem is treated for particular initial conditions. In this work, we develop a simplified method to arrive to the equation of the brachistochrone curve for an arbitrary potential and also to the inverse formulation of the problem: what type of potential energy function is associated with a specified brachistochrone curve