Herman. H. Goldstine's A History of the Calculus of Variations from the 17th PDF

By Herman. H. Goldstine

ISBN-10: 1461381061

ISBN-13: 9781461381068

ISBN-10: 1461381088

ISBN-13: 9781461381082

The calculus of diversifications is a topic whose starting should be accurately dated. it would be stated to start in the interim that Euler coined the identify calculus of diversifications yet this is often, in fact, now not the genuine second of inception of the topic. it should 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. 2 hundred B. C. ) and preserved by means of Pappus (c. three hundred A. D. ). i have never performed this seeing that those difficulties have been solved via geometric capability. as an alternative i've got arbitrarily selected to start with Fermat's stylish precept of least time. He used this precept in 1662 to teach how a gentle ray used to be refracted on the interface among optical media of other densities. This research of Fermat turns out to me particularly applicable as a place to begin: He used the tools of the calculus to lessen the time of passage cif a gentle ray in the course of the media, and his approach used to be tailored by means of John Bernoulli to unravel the brachystochrone challenge. there were a number of different histories of the topic, yet they're now hopelessly archaic. One by means of Robert Woodhouse seemed in 1810 and one other via Isaac Todhunter in 1861.

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He then proposes the more general problem in which PZ is an arbitrary function of PF. 55 He goes on to say that "As regards the other problem in which one asks among all cycloids which start from the same point and the same horizontal base, that one down which a heavy body arrives in the least time at a given vertical line; it is true that this is properly the problem for whose solution the generous 'no name' promised me the prize of fifty silver ducats (ecus blancs) .... " He proceeds to say that he has sent his solutions to Leibniz and asked him to serve as their judge.

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. It is easy to see that Bernoulli's synchrones are transversal to his cycloids. In this same connection it is important to recall that John Bernoulli discovered geodesics on surfaces and discussed this subject in a 1698 letter to Leibniz.

16 I. 12) 4' since GR = Yo csc GRB. We need to make the assumption that our arcs are representable parametrically as x = x(t),y = yet) {II <; t <; tJ with x;;> O,j;;> 0 at each point. Newton realized this and made provision for it. Otherwise, as was first shown by Legendre, Newton's problem has for a solution a sawtoothed curve giving a total resistance of zero. We discuss this later (pp. 144). Under the assumptions above we can show that all extremals are representable as x = x(y) with x single-valued, continuous, and having a piecewise continuous derivative.

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A History of the Calculus of Variations from the 17th through the 19th Century by Herman. H. Goldstine


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