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part II: A brief history of dynamics
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Analysis
The mathematical study of motion has been a a problem since earliest times. Zeno of Elea (490-430 bc) developed many mathematical arguments to show that motion is impossible. The invention of the calculus by Newton (and independently by Leibniz) made the logical treatment of motion, continuity and infinity live issues in mathematics. Analysis is the department of mathematics that arose to deal with the difficult relationship between discrete and continuous quantities.
Zeno's most graphic paradox concerns Achilles and the Tortoise. He proves that Achilles, for all his speed, cannot catch a tortoise, the slowest of animals. For suppose Achilles starts 100 metres behind the tortoise, and that while Achilles runs that 100 metres the tortoise walks 10 metres further down the track. While Achilles runs the next ten metres, the tortoise walks a metre, and while Achilles covers that metre, the tortoise walks one tenth of a metre, and so on, ad infinitum. Since this infinite series of motions never ends, Achilles can ever catch the tortoise.
Although such an argument is obviously faulty to anyone who has chased and caught a tortoise (or anything else), the logical conundrums involved waited thousands of years for solution. The literature of analysis is vast, and its subtleties are indeed subtle, but a few simple ideas capture the bulk of analytical wisdom.
The first is that the sum of an infinite series can be finite if its terms decrease fast enough. Thus if Achilles can run 100 metres in ten seconds, he covers the next ten metres in 1 second, the next metre in one tenth of a second, and so on. The sum of the infinite series of times 10 + 1 + 0.1 + ... is not infinite, but can be shown by relatively simple arithmetic to be 11.111 ... seconds. On the distance and time given, Achilles will catch the tortoise 11 and 1/9 seconds after they start.
The second, very similar, is that an infinite process can have a definite result. Here we concentrate not on the sums of series of times or distances, but on the process of summing the series. If the changes arising from each step of the process get smaller fast enough, then the process, though infinite, converges to a finite result. The processes of analysis were the harbingers of the theory of computation, which made it possible (in principle) for mathematics to analyze any process whatever.
The modern mathematical treatment of motion concentrates of two transformations, differentiation and integration. Differentiation reveals the rate at which a process is going. Integration tells us how far a process moving at a certain rate will take us. These simple ideas (with all their devilish details) have become very powerful tools in the hands of mathematicians and physicists.
In the three centuries since Newton wrote, we have been able to find very succinct and precise mathematical descriptions of almost all the motion in the universe. At present, the outcome of all this work is called the 'standard model'. Although many look forward to the day when the standard model will be replaced by something better, we can be pretty sure that the mathematics of the new system will evolve from current analytical ideas.
Books
| Berberian, Sterling K, A First Course in Real Analysis, Springer Verlag 1994 Jacket: 'This book offers an initiation into mathematical reasoning and into the mathematicians's mind-set and reflexes. Specifically, the fundamental operations of calculus - differentiation and integration of functions, and the summation of infinite series - are built, with logical continuity (i.e., rigor"), starting from the real number system. The first chapter sets down the axioms for the real number system, from which all else is derived using the logical tools summarized in the Appendix. The discussion of the "fundamental theorem of calculus", the focal point of the book, is especially thorough. The concluding chapter establishes a significant beachhead in the theory of the Lebesgue integral by elementary means.' http://www.amazon.com/exec/obidos/ASIN/0387942173/tnrp">Amazon back |
| Gaukroger, Stephen, Descartes: An Intellectual Biography, Clarendon Press 1995 Jacket: 'Rene Descartes (1596-1650) is the father of modern philsoophy and one of the greatest of all thinkers. This is the first intellectual biography of Descartes in English; it offers a fundamental reassessment of all aspects of his life and work. ... Descares' early work in mathematics and science produced ground-breaking theories, methods and tools still in use today. This book gives the first full acount of how this work infomred and influenced the later phisosophical studies for which, above all, Descartes is renowned.... [It] offers for the first time a full understanding of how Descartes developed his revolutionary ideas. It will be a landmark publication, welcomed by all readers interested in the origins of modern thought.' Amazon back |
| Kolmogorov, A N, and S V Fomin, Elements of the Theory of Functions and Functional Analysis Volumes 1 and 2, (Two volumes bound as one), Dover 1999 Jacket: Beginning with a brief introduction to set theory and mappings, the authors offer a clear presentation of the theory of metric and complete metric spaces, The principle of contraction mappings and its applications to the proof of existence theorems in the theory of differential and integral equations receives detailed analysis, as do continuous curves in metric spaces - a topic seldom discussed in textbooks. ... Part two focusses on an exposition of measure theory, the Lebesgue interval and Hilbert space. Both parts feature numerous exercises at the end of each section and include helpful lists of symbols, definitions and theorems.' Amazon back |
| Kreyszig , Erwin, Introductory Functional Analysis with Applications, John Wiley and Sons 1989 Amazon: 'Kreyszig's "Introductory Functional Analysis with Applications", provides a great introduction to topics in real and functional analysis. This book is part of the Wiley Classics Library and is extremely well written, with plenty of examples to illustrate important concepts. It can provide you with a solid base in these subjects, before one takes on the likes of Rudin and Royden. I had purchased a copy of this book, when I was taking a graduate course on real analysis and can only strongly recommend it to anyone else.' Krishnan S. Kartik Amazon back |
| Pour-El, Marian B, and Jonathan I Richards, Computability in Analysis and Physics, Springer-Verlag 1989 Author's Preface: 'This book is concerned with the computability or noncomputability of standard processes in analysis and physics. ... The book is written for a mixed audience. Although it is intended primarily for logicians and analysts, it should be of interest to physicists and computer scientists ... The work is self-contained. ... The reasoning used is classical - i.e. in the tradition of classical mathematics. Thus it is not intuitionist or constructivist in the sense of Brouwer or Bishop.' Amazon back |
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