Analytical synthesis of brachistochrones in an arbitrary velocity field

Abstract

Consider the problem of finding the brachistochrone between two specified points on the boundary of an area having a spatially varying velocity field. The arbitrary velocity field can be synthesized from a set of simple functions, for example, by a Fourier representation or a power‐series expansion. Now consider the analogous brachistochrone problem, using the corresponding end points, for each of the simple functions. Each of these problems is more easily solved than the prototype problem. The solution to the original brachistochrone problem can be expressed in terms of the solutions to the set of brachistochrone problems on the simpler surfaces. This synthetic approach has the advantage that each of the specified sets of functions becomes the solution for some extreme condition.