By Boris A. Khesin, Serge L. Tabachnikov
Vladimir Arnold, an eminent mathematician of our time, is understood either for his mathematical effects, that are many and sought after, and for his powerful evaluations, frequently expressed in an uncompromising and galvanizing demeanour. His dictum that "Mathematics is part of physics the place experiments are reasonable" is celebrated. This ebook includes components: chosen articles by means of and an interview with Vladimir Arnold, and a suite of articles approximately him written through his buddies, colleagues, and scholars. The e-book is generously illustrated by means of a wide choice of pictures, a few by no means prior to released. The booklet offers many an aspect of this amazing mathematician and guy, from his mathematical discoveries to his daredevil open air adventures.
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This selection of expository articles grew out of the workshop "Number concept and Physics" held in March 2009 on the Erwin Schr? ¶dinger overseas Institute for Mathematical Physics, Vienna. the typical subject matter of the articles is the impact of principles from noncommutative geometry (NCG) on matters starting from quantity conception to Lie algebras, index thought, and mathematical physics.
ZUR 2. AUFLAGE DER DEUTSCHEN AUSGABE Warum eröffnen wir eine Buchreihe über Wissenschafts theorie verschiedener Hauptdisziplinen mit einer Neu auflage von Hans Reichenbachs Buch "Aufstieg der wissen schaftlichen Philosophie", dessen 1. Auflage 1951 in den united states erschien? Weil es sich mit seiner präzisen und ver ständlichen Sprache nach siebzehn Jahren immer noch so frisch wie damals liest, weil seine Problematik zwischen Rationalismus und Empirismus, seine competition gegen Metaphysik und Ontologie als philosophische Grund legung der Naturwissenschaft immer noch in Mittel europa aktuell ist.
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L. Lidov whose students A. I. Neishtadt and M. L. Zieglin later made profound contributions to perturbation theory, averaging, adiabatic invariants, Hamiltonian chaos and materialization of resonances. The resulting theories are well known, and I shall only mention one small relevant detail. Before I turn to this small detail, let me remark that now you have almost the whole picture of all my mathematical subjects. They all are starting from this problem of superpositions and you now see how they are connected.
Unfortunately I was unable to ﬁnd the positive Lyapunov exponent numerically. At that time, computers produced very-very long tapes with numbers, kilometers of numbers. We were trying to imagine the orbit in 6-dimensional phase space looking at those numbers. I think that probably the Reynolds number was not suﬃciently high, so what I have observed was a 3dimensional torus in 6-dimensional space — a scenario predicted by Landau. But I was certain that with more work you might ﬁnd the positive Lyapunov exponents, perhaps even the geodesic ﬂow on a surface of negative curvature.
M. Alekseev during our weekly common “windows” (breaks between two classes) at Moscow University, I realized that that the problem of celestial mechanics has several diﬃculties which one might tackle separately. The ﬁrst diﬃculty (“the limit degeneration”) is already present in the simplest problem on the plane area-preserving diﬀeomorphisms near a ﬁxed point, the so called Birkhoﬀ problem. Suppose that the mapping linearized at a ﬁxed point is a plane rotation. A rotation is resonant if the rotation angle is commensurable with 2π.