By H.S. Dumas
This can be a semi-popular arithmetic e-book aimed toward a wide readership of mathematically literate scientists, specifically mathematicians and physicists who're no longer specialists in classical mechanics or KAM conception, and scientific-minded readers. components of the ebook also needs to attract much less mathematically knowledgeable readers with an curiosity within the historical past or philosophy of technology.
The scope of the booklet is huge: it not just describes KAM thought in a few element, but additionally offers its ancient context (thus exhibiting why it used to be a "breakthrough"). additionally mentioned are purposes of KAM conception (especially to celestial mechanics and statistical mechanics) and the elements of arithmetic and physics during which KAM thought is living (dynamical platforms, classical mechanics, and Hamiltonian perturbation theory).
even though a few resources on KAM thought at the moment are on hand for specialists, this e-book makes an attempt to fill a long-standing hole at a extra descriptive point. It sticks out very in actual fact from present courses on KAM idea since it leads the reader via an obtainable account of the idea and areas it in its right context in arithmetic, physics, and the historical past of science.
Readership: Undergraduates, graduates, and researchers greatly drawn to Hamiltonian perturbation idea, statistical mechanics, ergodic concept, Nekhoroshev concept, Arnold diffusion, nonlinear dynamics, dynamical structures, chaos concept, classical mechanics, and the background of those matters.
Read Online or Download The KAM Story: A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov-Arnold-Moser Theory PDF
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Additional resources for The KAM Story: A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov-Arnold-Moser Theory
Although Poincaré’s work was amply recognized in his lifetime, many parts remained enigmatic long afterward. As more nuggets were dug from his Nachlass and it was realized how far ahead of his time he was, there emerged a tendency to read his works line-by-line, like sacred texts. These ‘exegetical Poincaré studies’ did (and do) occasionally lead to new insights into the master’s meaning, but on the whole it’s amusing to imagine what Poincaré might make of the close scrutiny given to writings that he himself took less seriously38 It’s also interesting to ponder the severe and mostly no-fun reaction to Poincaré’s intuitionist style in the decades after his death.
5 A proxy r ω2 /ω1 for the frequency map (I1, I2) (ω1,ω2) showing (a) a nondegenerate case, (b) a degenerate case. In case (b), the interval [r1,r2] is collapsed to the point p under the proxy map. Finally here’s a key fact we’ll return to below: For a nondegenerate system with n = 2, as we move out radially in r (Fig. 2), the resonant tori are distributed among the nonresonant tori in the same way that the rational numbers are distributed among the irrationals. 31 This will be important, because one of the main messages of KAM theory is that, when a nondegenerate integrable system is perturbed, its resonant tori (along with some others ‘close to resonant’) break up chaotically, while a large subset of its nonresonant tori survive and are only slightly distorted.
This has been told many times and at various levels of detail (it is described in Chapter 7 of [Pete93] by I. Peterson, in Chapter 1 of [DiH96] by F. Di-acu and P. Holmes, and it is the essential core of the book [Barro97] by J. Barrow-Green), so I give here only the barest outline, in a present-tense, stage-direction style. The stage opens in Berlin in the early 1880s where we find Karl Weierstrass (1815– 1897) keenly interested in the n body problem. Weierstrass is probably the world’s leading figure in mathematics, rising from humble beginnings as a Gymnasium teacher in Münster (1841) to full professor at the University of Berlin (1856).