The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on $S^2$ are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author?s previous research on interaction of rigid bodies and point vortices in a plane.

Ašnew concept ofšdynamic advection isšintroduced. The model ofšdynamic advection deals with the motion ofšmassive particles inšaš2Dšflow ofšanšideal incompressible liquid. Unlike the standard advection problem, which isšwidely treated inšthe modern literature, our equations ofšmotion account not only for particles? kinematics, governed byšthe Euler equations, but also for their dynamics (which isšobviously neglected ifšthe mass ofšparticles isštaken tošbešzero). Ašfew simple model problems are considered.

The paper isšconcerned with ašclass ofšproblems which involves the dynamical interaction ofšašrigid body with point vortices onšthe surface ofšaštwo-dimensional sphere. The general approach tošthe 2Dšhydrodynamics isšfurther developed. The problem ofšmotion ofšašdynamically symmetric circular body interacting with ašsingle vortex isšshown tošbešintegrable. Mass vortices onš$S^2$ are introduced and the related issues (such asšequations ofšmotion, integrability, partial solutions, etc.) are discussed. This paper isšašnatural progression ofšthe author?s previous research onšinteraction ofšrigid bodies and point vortices inšašplane.

The paper develops further the algebraic-reduction method for $SO(4)$-symmetric systems onšthe three-dimensional sphere. Canonical variables for the reduced system are constructed both onštwo-dimensional and three-dimensional spheres. The method isšillustrated byšapplying itštošthe two-body problem onšašsphere (the bodies are assumed tošinteract with ašpotential that depends only onšthe geodesic distance between them) and the three-vortex problem onšaštwo-dimensional sphere.

The paper deals with the derivation ofšthe equations ofšmotion for two spheres inšanšunbounded volume ofšideal and incompressible fluid inš3DšEuclidean space. Reduction ofšorder, based onšthe use ofšnew variables that form ašLie algebra, isšoffered. Aštrivial case ofšintegrability isšindicated.

In this paper we consider the system of an arbitrary two-dimensional cylinder interacting with point vortices in a perfect fluid. We present the equations of motion and discuss their integrability. Simulations show that the system of an elliptic cylinder (with nonzero eccentricity) and a single point vortex already exhibits chaotic features and the equations of motion are nonintegrable. We suggest a Hamiltonian form of the equations. The problem we study here, namely, the equations of motion, the Hamiltonian structure for the interacting system of a cylinder of arbitrary cross-section shape, with zero circulation around it, and $N$ vortices, has been addressed by Shashikanth [Regular Chaotic Dyn. 10, 1 (2005)]. We slightly generalize the work by Shashikanth by allowing for nonzero circulation around the cylinder and offer a different approach than that by Shashikanth by using classical complex variable theory.

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.

Inšthis paper wešconsider the system ofštwo 2Dšrigid circular cylinders immersed inšanšunbounded volume ofšinviscid perfect fluid. The circulations around the cylinders are assumed tošbešequal inšmagnitude and opposite inšsign. Special cases ofšthis system (the cylinders move along the line through their centers and the circulation around each cylinder isšzero) are considered. Ašsimilar system ofštwo interacting spheres was originally considered inšclassical works ofšCarl and Vilhelm Bjerknes, G.šLamb and N.E. Joukowski.

Byšmaking the radii ofšthe cylinders infinitesimally small, wešhave obtained ašnew mechanical system which consists ofštwo regular point vortices but with non-zero masses. The study ofšthis system can bešreduced tošthe study ofšthe motion ofšašparticle subject tošpotential and gyroscopic forces. Ašnew integrable case isšfound. The Hamiltonian equations ofšmotion for this system have been generalized tošthe case ofšanšarbitrary number ofšmass vortices with arbitrary intensities. Some first integrals have been obtained. These equations expand upon the classical Kirchhoff equations ofšmotion for nšpoint vortices.

The paper studies the system of a rigid body interacting dynamically with point vortices in a perfect fluid. For arbitrary value of vortex strengths and circulation around the cylinder the system is shown to be Hamiltonian (the corresponding Poisson bracket structure is rather complicated). We also reduced the number of degrees of freedom of the system by two using the reduction by symmetry technique and performed a thorough qualitative analysis of the integrable system of a cylinder interacting with one vortex.

The paper studies the system of a rigid body interacting dynamically with point vortices in a perfect fluid. For arbitrary value of vortex strengths and circulation around the cylinder the system is shown to be Hamiltonian (the corresponding Poisson bracket structure is rather complicated). We also reduced the number of degrees of freedom of the system by two using the reduction by symmetry technique and performed a thorough qualitative analysis of the integrable system of a cylinder interacting with one vortex.