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The calculus of variations is a subject whose beginning can be precisely dated. It might be said to begin at the moment that Euler coined the name calculus of variations but this is, of course, not the true moment of inception of the subject. It would not have been unreasonable if I had gone back to the set of isoperimetric problems considered by Greek mathemati­ cians such as Zenodorus (c. 200 B. C. ) and preserved by Pappus (c. 300 A. D. ). I have not done this since these problems were solved by geometric means. Instead I have arbitrarily chosen to begin with Fermat's elegant principle of least time. He used this principle in 1662 to show how a light ray was refracted at the interface between two optical media of different densities. This analysis of Fermat seems to me especially appropriate as a starting point: He used the methods of the calculus to minimize the time of passage cif a light ray through the two media, and his method was adapted by John Bernoulli to solve the brachystochrone problem. There have been several other histories of the subject, but they are now hopelessly archaic. One by Robert Woodhouse appeared in 1810 and another by Isaac Todhunter in 1861.

Inhaltsverzeichnis

Frontmatter

1. Fermat, Newton, Leibniz, and the Bernoullis

Abstract
Fermat wrote nine interesting and important papers on the method of maxima and minima which are grouped together in his collected works. The last two in this set were sent by him in 1662 as attachments to a letter to a colleague, Marin Cureau de la Chambre.1 As their titles, “The analysis of refractions” and “The synthesis of refractions,” imply, they are companion derivations of the law of refraction, now usually known as Snell’s law. These papers are fundamental for us because Fermat enunciates in them his principle that “nature operates by means and ways that are ‘easiest and fastest.’” He goes on to state that it is not generally true that “nature always acts along shortest paths” (this was the assumption of de la Chambre). Indeed, he cites the example of Galileo that when particles move under the action of gravity they proceed along paths that take the least time to traverse, and not along ones that are of the least length.2 This enunciation by Fermat is, as far as I am aware, the first one to appear in correct form and to be used properly.
Herman H. Goldstine

2. Euler

Abstract
It is not quite certain when Euler first became seriously interested in the calculus of variations. Carathéodory, who edited Euler’s magnificent 1744 opus, The Method of Finding Plane Curves that Show Some Property of Maximum or Minimum … , believed it unlikely that it occurred during his period in Basel with John Bernoulli.1 However, we should note that Euler considered in 1732 and 1736 problems more or less arising out of James Bernoulli’s isoperimetric problems; and even as early as the end of 1728 or early in 1729, as Eneström showed, he wrote “On finding the equation of geodesic curves.” In effect, Euler in 1744, following John Bernoulli, examined the question of end-curves that cut a family of geodesics so that they have equal length. He showed that the end-curves must be orthogonal to the geodesics. This is the precursor of the so-called envelope theorem of Chapter 7 below and plays a key part in Carathéodory’s work.
Herman H. Goldstine

3. Lagrange and Legendre

Abstract
On 12 August 1755 a 19-year-old, one Ludovico de la Grange Tournier of Turin, wrote Euler a brief letter to which was attached an appendix containing mathematical details of a very beautiful and revolutionary idea (see Lagrange [1755] and Euler [1755]). He saw how to eliminate from Euler’s methods of 1744 the tedium and need for geometrical insight and to reduce the entire process to a quite analytic machine or apparatus, which could turn out the necessary condition of Euler and more, almost automatically. This basic idea of Lagrange ushered in a new epoch in the calculus of variations. Indeed after seeing Lagrange’s work, Euler dropped his own method, espoused that of Lagrange, and renamed the subject the calculus of variations.1
Herman H. Goldstine

4. Jacobi and His School

Abstract
Before proceeding to a detailed examination of Jacobi’s 1836 paper, it is probably desirable that we review the situation surrounding the second variation from the point of view of hindsight. This is particularly true in the case of Jacobi and his commentators since Jacobi stated his results with little or no indication as to why they are so. I do not know why he chose this means of announcing his results, unless he was anxious to publish before someone preceded him. In any case let us look at some results in connection with the second variation that should help to illuminate Jacobi’s paper.
Herman H. Goldstine

5. Weierstrass

Abstract
These lectures by Weierstrass on the calculus of variations were written up by a number of his students and were made available in the Mathematische Verein in Berlin and the Mathematische Lesezimmer in Göttingen. They were given by Weierstrass during the summer semesters of 1875, 1879, and 1882 and represent his contributions to the field.1 The editor, Rothe, compiled his text principally from lecture notes prepared by Burckhardt based on the 1882 summer-semester lectures and from those by Schwarz. Since these lectures were not formally published, presumably they did not reach the entire mathematical community; as a result, much went on in our subject without reference to this monumental achievement of Weierstrass. In fact, Weierstrass’s result became known in considerable measure through the dissertations of his students.
Herman H. Goldstine

6. Clebsch, Mayer, and Others

Abstract
Much work went on in the calculus of variations without reference to that of Weierstrass’s for quite a while. One reason for this was that Weierstrass’s great accomplishments were generally made known to the mathematical community through the dissertations of his students. The other reason was the development of a school interested in generalizing the scope of the calculus of variations to more general problems than Weierstrass was considering.
Herman H. Goldstine

7. Hilbert, Kneser, and Others

Abstract
In his world-famous presentation of mathematical problems at the 1900 International Mathematical Congress, Hilbert ([1900], p. 473) said the following by way of introduction to his discussion of the calculus of variations: “Nevertheless, I should like to close with a general problem, namely the indication of a branch of mathematics repeatedly mentioned in this lecture—which, in spite of the considerable advancement lately given it by Weierstrass, does not receive the general appreciation which, in my opinion, is its due—I mean the calculus of variations.”
Herman H. Goldstine

Backmatter

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