A History of the Calculus of Variations from the 17th - download pdf or read online

By Herman. H. Goldstine

ISBN-10: 1461381061

ISBN-13: 9781461381068

ISBN-10: 1461381088

ISBN-13: 9781461381082

The calculus of adaptations is a topic whose starting may be accurately dated. it'd be stated to start in the intervening time that Euler coined the identify calculus of adaptations yet this is often, in fact, no longer the real second of inception of the topic. it is going to no longer were unreasonable if I had long gone again to the set of isoperimetric difficulties thought of by way of Greek mathemati­ cians corresponding to Zenodorus (c. 2 hundred B. C. ) and preserved via Pappus (c. three hundred A. D. ). i haven't performed this for the reason that those difficulties have been solved through geometric ability. in its place 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 mild ray used to be refracted on the interface among optical media of other densities. This research of Fermat turns out to me particularly acceptable as a place to begin: He used the equipment of the calculus to reduce the time of passage cif a mild ray during the media, and his procedure used to be tailored via John Bernoulli to resolve the brachystochrone challenge. there were numerous different histories of the topic, yet they're now hopelessly archaic. One by means of Robert Woodhouse seemed in 1810 and one other through Isaac Todhunter in 1861.

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Extra info for A History of the Calculus of Variations from the 17th through the 19th Century

Example text

Fl _ X 2 / 3 From this we find by integrating that y - c = -(2 + x 2/ 3)(1 _ x2/3)1/2 and a little manipulation shows this is expressible as an algebraic equation in x and y of degree 6. 49 John Bernoulli, 00, Vol. I, p. 191. To see that this last property defines a cycloid, let b = a12, arc LK .. bcp, and arc GL .. bw - bcp ... bt. Then sin t = sin qi = LO I b and CO = A G == b'IT - Y + M L + LO = Y + arc LK + b sin t; hence y = b'IT - arc LK - b sin t - bw - bcp - b sin t = b(t - sin t). To find x, note that cos t = -cos qi = -(b - OK) and that x = AC == GK - OK - 2b - b - b cos t == b(1 - cos t).

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.

191. This is reference CR. 7. John Bernoulli's First Published Solution and Some Related Work 41 to the cycloid. fax - x 2 He then goes on to say49: However (a dx - 2x dx) : 2(ax - XX)I/2 is the differential quantity whose integral is (ax - XX)I/2 or LO; and a dx : 2(ax - XX)I/2 is the differential of the arc GL. For this reason by integrating the equation dy = dx(x/(a - X»I/2 one has y = eM = [arc]GL - LO, and therefore MO = CO - [arc]GL + LO. Since however CO - [arc]GL = [arc]LK (because CO = semicircle GLK), it follows that MO = [arc]LK + LO and by deducting the common value LO, that ML = [arc]LK, which implies that the curve KMA is a cycloid.

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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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