Abstract
In the preceding chapter, it was shown that in order to find an optimal solution to a particular design problem, three things were required: (a) to determine the class of all possible solutions to the problem, (b) to assign to each such solution a certain number, which represents the cost it involves, and (c) to search among the set of costs to find that which is least. The remainder of the book is concerned with stages (a) and (b), which belong mainly to the realm of natural science. The present chapter, however, is devoted to a discussion of (c), which is pure mathematics.
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Notes to Chapter 2
The real numbers form an algebraic system called a field. The ordering relation on the real numbers possesses the Archimedean property ; namely, given any real number a>0> there is a positive integer N so large that l/N<a. The result in question states that, up to isomorphism, the real numbers constitute the only order-complete Archimedean field; that is, any Archimedean ordered field is order-isomorphic to a subfield of the real numbers. See, for example, B. F. van der Waerden, Modern Algebra, Vol. I, Ungar, New York, 1950, p. 209 et seq.
The complement S of a set S is defined by the property that x∊S if and only if x∉S.
The terminology here arises from the fact that, if a function f(x) has a relative maximum (say) at the point x=x 0, and if h represents a first-order infinitesimal, then f(x 0 + h) =f(x 0 ) up to infinitesimals of higher order. In other words, up to higher order infinitesimals, the value of f(x) at x = x 0 is ‘stationary’ under sufficiently small displacements of the independent variable. This fact follows immediately, for example, from the Theorem of the Mean, which states that terms in higher powers of h, and noticing that by hypothesis f ‘(0) =0. This result should be compared with the corollary to Theorem 2.7, which makes the analogous assertion in the context of the calculus of variations, and justifies the terminology ‘stationary curve’.
In 1696, Johann Bernoulli posed the following problem: To find that curve, lying in a vertical plane, along which a particle moving under the influence of gravity will fall from a point A to a point B in minimal time. For fuller details on the brachistochrone problem and its solution (an arc of a cycloid) the reader is referred to any standard text on the calculus of variations (vide Note 7 below). It might be of interest to note that Johann Bernoulli solved the Brachistochrone problem by means of Fermat’s principle, identifying the curve of quickest descent with the path of a light ray moving in a medium of appropriately varying refractive index.
See, for example, R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol I, Interscience, New York, 1953, p. 65.
The property in question is usually called the Heine-Borel property, and may be stated as follows. Let there be given a (topological) space, and a family Ω of open subsets of S. This family is called a covering of S if, for every element x∈S, there exists an open set 0∈Ω such that x∈0. A covering Ω’ is called a subcovering of Ω if every open set in Ω’ is also in Ω. S is said to have the Heine-Borel property if, given any open covering of S, there exists a finite subcovering; i.e. if finitely many open sets of any covering of S already suffice to cover S. It can readily be shown that a set of real numbers is compact (i.e. closed and bounded) if, and only if, it possesses the Heine-Borel property. The same statement is true for compact subsets of Euclidean n-dimensional space for any finite dimension n.
It seems appropriate to add a bibliographical note at this point. An exhaustive discussion of existence problems connected with the calculus of variations may be found in A. R. Forsyth, Calculus of Variations, reprinted by Dover Publications, New York, 1950; a less detailed but instructive discussion of the calculus, in general, may be found in Chap. 4 of Courant and Hilbert, op. cit. There are numerous good older texts on the Calculus of Variations, notably those of Bliss, Bolza, Hadamard and Hancock, all more or less written in the same vein.
For finite dimensional problems, the reader may consult H. Hancock, Theory of Maxima and Minima, reprinted Dover, New York, 1960.
A modern treatment of the Calculus of Variations may be found in L. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, New York, 1963. A book with a similar title, stressing applications to physics and engineering, is that of R. Weinstock, McGraw-Hill, New York, 1952.
This form for the functional J(y) emerged very early from a study of the classical problems of the calculus of variations. It seems at first sight to be of a very special nature, but in reality, such an integral representation is possible for arbitrary functional under very general conditions.
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© 1967 Springer Science+Business Media New York
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Rosen, R. (1967). Basic Mathematical Techniques. In: Optimality Principles in Biology. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-6419-9_2
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