This note discusses reformulations the brachistochrone problem suitable for solution via NLP. The availability of solvers and modeling languages such as AMPL makes it tempting to formulate discretized optimization problems and get solutions to the discretized versions of trajectory optimization problems. We use the famous brachistochrone problem to warn that the resulting solutions may be far different from the true optimal trajectory. Actually, we use our knowledge of the brachistochrone to argue that without this knowledge, we could not distinguish the true solution (a cycloid) from spurious solutions obtained by a natural discretization.