CHAPTER III - THE BRACHISTOCHRONE PROBLEM

Summary

Its significance as an illustration. The brachisto-chrone problem is historically the most interesting of all the special problems mentioned in Chapter I since as we have there seen it gave the first impetus to systematic research in the calculus of variations. Since the time of the Bernoulli brothers it has been used with great regularity as an illustration by writers on the subject, and it is in many respects a most excellent one. Unfortunately in the forms originally proposed by the Bernoullis it does not require the application of an important necessary condition for a minimum which was first described by Jacobi in 1837, more than a century after the calculus of variations began to be systematically studied. A special case of this condition is the restriction on the position of the center of curvature in the problem of finding the shortest arc from a point to a curve, as described in the theorem on page 33 of the last chapter. It is perhaps at first surprising that the significance of such a simple instance of the condition escaped the early students of the calculus of variations, but a study of the older memoirs soon impresses one with a realization of the serious difficulties encountered with the methods originally used. Throughout the eighteenth century, investigators in the calculus of variations for the most part desisted when they had found the forms, or in many cases the differential equations only, of the minimizing curves which they were seeking.