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PROC

We offer a new method of reduction for a system of point vortices
on a plane and a sphere. This method is similar to the classical node elimination
procedure. However, as applied to the vortex dynamics, it requires substantial
modification. Reduction of four vortices on a sphere is given in more detail.
We also use the Poincaré surface-of-section technique to perform the reduction
a four-vortex system on a sphere.

PROC

In this paper we describe new classes of periodic solutions for point
vortices on a plane and a sphere. They correspond to similar solutions (so-called
choreographies) in celestial mechanics.

DCDS

In this paper we consider the system of two 2D rigid circular
cylinders immersed in an unbounded volume of inviscid perfect fluid.
The circulations around the cylinders are assumed to be equal in
magnitude and opposite in sign. We also explore some special cases
of this system assuming that the cylinders move along the line
through their centers and the circulation around each cylinder is
zero. A similar system of two interacting spheres was originally
considered in the classical works of Carl and Vilhelm Bjerknes, H.
Lamb and N. E. Joukowski.

By making the radii of the cylinders infinitesimally small, we have obtained a new mechanical system which consists of two regular point vortices but with non-zero masses. The study of this system can be reduced to the study of the motion of a particle subject to potential and gyroscopic forces. A new integrable case is found. The Hamiltonian equations of motion for this system have been generalized to the case of an arbitrary number of mass vortices with arbitrary intensities. Some first integrals have been obtained. These equations expand upon the classical Kirchhoff equations of motion for $n$ point vortices.

By making the radii of the cylinders infinitesimally small, we have obtained a new mechanical system which consists of two regular point vortices but with non-zero masses. The study of this system can be reduced to the study of the motion of a particle subject to potential and gyroscopic forces. A new integrable case is found. The Hamiltonian equations of motion for this system have been generalized to the case of an arbitrary number of mass vortices with arbitrary intensities. Some first integrals have been obtained. These equations expand upon the classical Kirchhoff equations of motion for $n$ point vortices.

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