By ᾠA. A. Bolibruch, Yu. S. Osipov, Ya. G. Sinai, V. I. Arnold, A. A. Bolibruch, A. M. Vershik, Yu. I. Manin, Yu. S. Osipov, Ya.G. Sinai, V.M. Tikhomirov, L.D. Faddeev, V. B. Philippov (eds.)

Russian arithmetic (later Soviet arithmetic, and Russian arithmetic once more) occupies a unique position in twentieth-century arithmetic. as well as its famous achievements, Russian arithmetic demonstrated a distinct variety of examine in response to the lifestyles of famous mathematical colleges. those faculties have been headed via well-known leaders, who turned well-known as a result of their abilities and amazing contributions to technology. the current assortment is meant essentially to collect in a single e-book the t- timonies of the members within the improvement of arithmetic during the last century. of their articles the authors have expressed their very own issues of view at the occasions that happened. The editors haven't felt they had a correct to make any alterations, except stylistic ones, or so as to add any in their personal statement to the textual content. clearly, the issues of view of the authors shouldn't be construed as these of the editors. The record of mathematicians invited to take part within the current variation was once fairly lengthy. regrettably, many of the authors for numerous purposes didn't settle for our invitation, and regretfully a few components of study aren't absolutely represented the following. however, the cloth that has been assembled is of serious worth not just within the medical experience, but in addition in its historic context. we want to convey our gratitude to the entire authors who contributed.

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A few weeks later, Leontovich (who was, as far as I remember, the vice-chairman of the editorial board) invited me to his home, near the Atomic Energy (now Kurchatov) Institute to discuss the paper. Leontovich, who headed the theoretical physics division of the thermonuclear controlled reaction project, was a friend of Kolmogorov and also of my father (he helped our family to survive when my father died and I was 11 years old). " He proposed that I should rewrite the article. Now I understand how right he was trying to defend a physical journal from the Bourbakist mathematical style.

I. Arnold In trying to understand this phenomenon, Kolmogorov investigated its abstract variant: a dynamical system on a torus given by a divergence-free field with respect to a volume element. In 1953, in a short paper published in Doklady, Kolmogorov proved that such a generic vector field (which satisfies some Diophantine conditions on mean frequencies that are valid almost always) is equivalent to a standard (translationally invariant) field on the torus. 10 Such a field determines quasi-periodic motion ("conditionally periodic," as Kolmogorov said using old-fashioned terminology).

Consider, for instance, the motion of a charged particle along a surface under the influence of a strong magnetic field which is orthogonal to the surface. Mathematically, this involves the problem of the description of the curves of prescribed large geodesic curvature on the surface. In the first approximation, such a curve is a circle of small radius, the so-called Larmor circle. But in the next approximation (provided by the theory of adiabatic invariants), the center of the Larmor circle starts to move along the surface.

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