Download A History of the Calculus of Variations from the 17th by Herman. H. Goldstine PDF

By Herman. H. Goldstine

The calculus of diversifications is a topic whose starting may be accurately dated. it'd be acknowledged to start for the time being that Euler coined the identify calculus of adaptations yet this is often, after all, now not the real second of inception of the topic. it will no longer were unreasonable if I had long past again to the set of isoperimetric difficulties thought of via Greek mathemati­ cians corresponding to Zenodorus (c. two hundred B. C. ) and preserved through Pappus (c. three hundred A. D. ). i have never performed this considering those difficulties have been solved by means of geometric ability. in its place i've got arbitrarily selected first of all Fermat's based precept of least time. He used this precept in 1662 to teach how a mild ray used to be refracted on the interface among optical media of alternative densities. This research of Fermat turns out to me specifically acceptable as a kick off point: He used the equipment of the calculus to lessen the time of passage cif a gentle ray in the course of the media, and his process was once tailored via John Bernoulli to resolve the brachystochrone challenge. there were a number of different histories of the topic, yet they're now hopelessly archaic. One via Robert Woodhouse seemed in 1810 and one other by way of Isaac Todhunter in 1861.

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26 1. 17). 18') qi = X~2 = 1. 12'), the same as for the earlier one. Since Y = a(l + q2 I q, YI = 0 would imply that a = 0 and the minimizing curve would be the line CB, but this does not pass through point D; x~ = 0 would again imply a = O. , the curve x = x(y) intersects the line x = XI at an angle of 45° or 135°, as Newton asserted. In his next paragraph Newton takes up the question of finding a parametric representation for the extremals in terms of the parameter q = BEl BG. 12 in terms of this quantity q.

D~ = (a1J/aa)/(a~/aa). It is easy to see that Cc:p - sinc:p) + aCI - cosC:P)C:Pa d~= (l-cosc:p)+asinc:p·c:pa fir! This must be the negative reciprocal of dy / dx, the slope of the cycloid. This implies directly that S2Curiously, Stroik says that Bernoulli noted that his synchrone is a cycloid. I cannot find this assertion in the original text, nor is it true. 44 I. 24) along the family of cycloids, we find li T = lEi fP, and hence we see that the orthogonality of the curve C to the family of cycloids is equivalent to the fact that the points of intersection correspond to a constant time of descent.

James Bernoulli's Solution The solution as given by James Bernoulli to his brother's problem is quite different and is a good deal more like Newton's solution to the least resistance problem; moreover, it probably influenced Euler in developing his earlier techniques-until, in fact, Euler saw Lagrange's superior method of variations at which point Euler shifted over and renamed the subject the calculus of variations. 53 In the preface to his paper James Bernoulli says that he was persuaded to solve his brother's problem by a letter on 13 September 1696 from Leibniz urging him to work on it.

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