С 25 по 30 августа 2006 г. в Математическом институте им. В.А. Стеклова РАН был проведен IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence — Международный симпозиум «Гамильтонова динамика. Вихревые структуры. Турбулентность» под эгидой Международного союза по теоретической и прикладной механике (IUTAM). Со-организаторами симпозиума выступили Российская академия наук, Математический институт им. В.А. Стеклова и Институт компьютерных исследований. За шесть дней работы симпозиума было сделано 19 пленарных (40 мин) и 47 секционных (20 мин) докладов. Кроме этого, была проведена мемориальная секция, посвященная памяти профессора В.И. Юдовича, и был выслушан доклад М.Питерса (M. Peters), представителя издательства Springer-Verlag, Heidelberg. В работе симпозиума приняли участие ученые из 11 стран — Бразилии, Франции, Израиля, Италии, Японии, Литвы, Нидерландов, России, Великобритании, Украины, США.

The year 2007 will mark the centenary of the death of William Thomson (Lord Kelvin), one of the great nineteenth-century pioneers of vortex dynamics. Kelvin was inspired by Hermann von Helmholtz’s (1858) famous paper «Uber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen», translated by P.G. Tait and published in English (1867) under the title «On Integrals of the Hydrodynamical Equations, which express Vortex-motion». Kelvin conceived his «Vortex theory of Atoms» (1867-1875) on the basis that, since vortex lines are frozen in the flow of an ideal fluid, their topology should be invariant. We now know that this invariance is encapsulated in the conservation of helicity in suitably defined Lagrangian fluid subdomains. Kelvin’s efforts were thwarted by the realisation that all but the very simplest three-dimensional vortex structures are dynamically unstable, and his vortex theory of atoms perished in consequence before the dawn of the twentieth century. The course of scientific history might have been very different if Kelvin had formulated his theory in terms of magnetic flux tubes in a perfectly conducting fluid, instead of vortex tubes in an ideal fluid; for in this case, stable knotted structures, of just the kind that Kelvin envisaged, do exist, and their spectrum of characteristic frequencies can be readily defined. This introductory lecture will review some aspects of these seminal contributions of Helmholtz and Kelvin, in the light of current knowledge.

Several problems related to the dynamics of vortex patterns as observed in wake flows are addressed. These include: The universal Strouhal-Reynolds number relation. The Hamiltonian dynamics of point vortices in a periodic strip, both the classical two-vortices-in-a-strip problem, which gives the structure and self-induced velocity of the traditional vortex street, and the three-vortices- in-a-strip problem, which is argued to be relevant to the wake behind an oscillating body. The bifurcation diagram for wake structure found experimentally by Williamson and Roshko is addressed theoretically.

The issues discussed in this paper relate to the description of developed two-dimensional turbulence, when the mean values of characteristics of steady flow stabilize. More exactly, the problem of a weak limit of vortex distribution in two-dimensional flow of an ideal fluid at time tending to infinity is considered. Relations between the vorticity equation and the well-known Vlasov equation are discussed.

The system of two interacting dynamically 2D rigid circular cylinders in an infinite volume of perfect fluid was considered in [4,5], while the pioneering contribution is due to Hicks [1,2]. An allied problem, the motion of two spheres in perfect fluid, was studied by Stokes, Hicks, Carl and Vilhelm Bjerknes, Kirhhoff, and Joukowski (the references can be found in [3] and [7]). Assuming the circulations around the cylinders to be constant and making the radii of the cylinders infinitely small result in new 2D hydrodynamic objects called mass vortices [5]. The equations of motion for mass vortices expand upon the classical Kirhhoff equations governing the motion of ordinary point vortices. In this paper the motion of two mass vortices is examined in greater detail (some results have been obtained already in [5]). A reduction of order is performed; using the Poincare surfaсe-of-section technique the system is shown to be generally non-integrable. Some integrable cases are indicated. In conclusion the motion of a single mass vortex and the motion of cylinder in a half plane are briefly investigated.

Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, Koiller and Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.

We consider the analogue of the 4th Appelrot class of the Kowalevskaya top for the case of double force field. The trajectories of this family fill the 4-dimensional surface in the 6-dimensional phase space. We point out two almost everywhere independent partial integrals that give the regular parametrization of the corresponding sheet of the bifurcation diagram in the complete problem. Projections of the Liouville tori onto the plane of auxiliary variables are investigated. The bifurcation diagram of the partial integrals is found. The region of existence of motions in terms of the integral constants is established. We introduce the change of variables that separate the system of differential equations for this case.