Whereas statics, we recall, is that part of mechanics which is concerned with the equilibrium of bodies, dynamics is that part which is concerned with the motion of bodies. The former, as we have had occasion to note, goes back to the Greeks; to Archimedes' discovery of the Law of the Lever and his application of it to the integral calculus. The latter is relatively new; it starts with Galileo.

SECTION 1. GALILEO

Galileo is known by his first name; his family name is Galilei. He was born in 1564 and died in 1642. To believers in the transmigration of souls the date of his death is important. Not only did he die in the year in which Newton was born, conveniently for their speculations, he died shortly before Newton was born. A much more important date is 1636, the year in which he completed the book on which his fame so securely rests, the Dialogue Concerning Two New Sciences. Although many of his brilliant predecessors, beginning with Aristotle, and including that most versatile of versatile geniuses, Leonardo da Vinci, had been interested in the free fall of heavy bodies, Galileo was incomparably the greatest dynamicist of them all. He inherited a dogma and bequeathed a science.

His tomb is to be found in Florence, in the Church of Santa Croce, among those of Leonardo and Michelangelo the artists, Dante the poet, and Machiavelli the politician.

To date, what have we done? First we discussed measurement, especially in astronomy; then simple but pervasive topics culled from the history of statics, and finally, great discoveries from the history of dynamics—so many of which hark back to the stars. We have seen something of the role played by mathematics in the development of science; that the aim of physics is to condense its knowledge into mathematical formulae; that, as Galileo so delightfully expressed it, the book of Nature is written in mathematical characters.

Yet this view, although undeniable, is one-sided—or should I say unidirectional? Of course mathematics helps physics. But you must not suppose that help always flows downstream from mathematics to physics; the river of thought is tidal. My object in this chapter is to navigate an incoming tide, to show how help flows also from physics to mathematics.

My lecture-room navigation will not be reproduced here as my upstream voyage is already carefully charted in my Mathematics and Plausible Reasoning, Vol. 1, pp. 142–167, to which the interested mariner is directed.

The following treatment of integral transforms in applied mathematics is directed primarily toward senior and graduate students in engineering and applied science. It assumes a basic knowledge of complex variables and contour integration, gamma and Bessel functions, partial differential equations, and continuum mechanics. Examples and exercises are drawn from the fields of electric circuits, mechanical vibration and wave motion, heat conduction, and fluid mechanics. It is not essential that the student have a detailed familiarity with all of these fields, but knowledge of at least some of them is important for motivation (terms that may be unfamiliar to the student are listed in the Glossary, p. 89). The unstarred exercises, including those posed parenthetically in the text, form an integral part of the treatment; the starred exercises and sections are rather more difficult than those that are unstarred.

I have found that all of the material, plus supplementary material on asymptotic methods, can be covered in a single quarter by first-year graduate students (the minimum preparation of these students includes the equivalent of one-quarter courses on each of complex variables and partial differential equations); a semester allows either a separate treatment of contour integration or a more thorough treatment of asymptotic methods. The material in Chapter 4 and Sections 5.5 through 5.7 could be omitted in an undergraduate course for students with an inadequate knowledge of Bessel functions.

The exercises and, with a few exceptions, the examples require only those transform pairs listed in the Tables in Appendix 2.