Arnold: Swimming Against the Tide - download pdf or read online

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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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.

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