As a problem that dates back to the end of the seventieth century, the
brachistochrone problem is one of the oldest problems in the calculus of variations and as such, has generated a myriad of publications. However, in most
classical texts and in most papers, the favored way to solve the problem is to
make two a priori assumptions, viz., that the brachistochrone lies in a vertical
plane, and that it can be represented as the graph of a function in this plane;
besides, with few exceptions, the existence of a solution is not rigorously established: instead it is sometimes even taken for granted that the solution is that
of the associated Euler-Lagrange equations, even though these are well known
to be only necessary conditions for the existence of a minimizer. The objective of this article is to show how all these shortcomings can be very simply,
and rigorously, overcome, by means of arguments that do not need any a priori assumptions and that otherwise require only a modicum of basic notions
from calculus, so as to rigorously establish the existence and uniqueness of the
brachistochrone in full generality. One originality of our approach is that from
the outset we seek the brachistochrone as a parameterized curve in the three-dimensional space, i.e., that can be represented by means of three parametric
equations, instead of by means of a single graph in a vertical plane. Contrary
to expectations, this increase of generality renders the ongoing analysis much
simpler. Our objective is thus to show how the direct method of the calculus
of variations based on the Euler-Lagrange equations can be used to solve the
brachistochrone problem. Otherwise, there are other ways to solve this problem, for instance by means of convex optimization or optimal control; such
methods are briefly described at the end of the paper.