Var The problem of Newton

Section 1.4 The problem of Newton

It was remarked above that the earliest problems of the calculus of variations were not by any means the simplest. In his Principia (1686) Newton states without proof certain conditions which must be satisfied by a surface of revolution which is so formed that it will encounter a minimum resistance when moved in the direction of its axis through a resisting medium. A particular case of the problem of finding such a surface is the well-known one of determining the form of a projectile which for a specified initial velocity will give the longest range. In practical ballistics it turns out that one of the most difficult parts of the investigation of this question lies in the experimental determination of the retardation law for bodies moving in the air at high rates of speed. Newton assumed a relatively simple law of resistance for bodies moving in a resisting medium which does not agree well with our experience with bodies moving in the air, but on the basis of which he was able to find a condition characterizing the meridian curves of the surfaces of revolution which encounter minimum resistance. From a letter written by Newton to Professor David Gregory, probably in 1694, Bolza has reconstructed in most interesting fashion the arguments which Newton used in attaining his results.

It is sufficient for the purposes of this introductory chapter to say that when the surface is generated by rotating about the \(x\)-axis an arc with an equation of the form \(y = y(x)\) the resistance experienced by the surface when moved in the direction of the \(x\)-axis will, except for a constant factor, be

\begin{equation*} I=\int_{x_{1}}^{x_{2}} \frac{y (y')^3}{1+(y')^2}\,dx. \end{equation*}

Newton's problem in analytical form is then that of determining among all the arcs \(y = y(x)\) joining two given points 1 and 2 one which makes this integral a minimum. We could equally well of course ask to determine the curve so that the resistance should be a maximum. If the law of resistance of Newton is replaced by another the methods which we now know of attacking the problem will still be applicable, though the results may be different, as a number of writers have shown.