Sergey Ramodanov

We study a mechanical system that consists of a 2D rigid body interacting dynamically with two point vortices in an unbounded volume of an incompressible, otherwise vortex-free, perfect fluid. The system has four degrees of freedom. The governing equations can be written in Hamiltonian form, are invariant under the action of the group $E$(2) and thus, in addition to the Hamiltonian function, admit three integrals of motion. Under certain restrictions imposed on the system’s parameters these integrals are in involution, thus rendering the system integrable (its order can be reduced by three degrees of freedom) and allowing for an analytical analysis of the dynamics.

The dynamical behavior of a heavy circular cylinder and a point vortex in an unbounded volume of ideal liquid is considered. The liquid is assumed to be irrotational and at rest at infinity. The circulation about the cylinder is different from zero. The governing equations are Hamiltonian and admit an evident autonomous integral of motion — the horizontal component of the linear momentum. Using the integral we reduce the order and thereby obtain a system with two degrees of freedom. The stability of equilibrium solutions is investigated and some remarkable types of partial solutions of the system are presented.

We study the system of a 2D rigid body moving in an unbounded volume of incompressible, vortex-free perfect fluid which is at rest at infinity. The body is equipped with a gyrostat and a so-called Flettner rotor. Due to the latter the body is subject to a lifting force (Magnus effect). The rotational velocities of the gyrostat and the rotor are assumed to be known functions of time (control inputs). The equations of motion are presented in the form of the Kirchhoff equations. The integrals of motion are given in the case of piecewise continuous control. Using these integrals we obtain a (reduced) system of first-order differential equations on the configuration space. Then an optimal control problem for several types of the inputs is solved using genetic algorithms.

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.

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.

A planar analog of the Bjeknes problem of interaction of two spheres in a perfect fluid is considered. For the case of equal circulations around the cylinders, the equations of motion in the Poincare–Chetaev form are obtained, the integrals of motion are indicated. The problem is then reduced to a problem with two degrees of freedom. Most probably, this reduced problem is not integrable.

The motion of a rigid circular cylinder and $N$ point vortices in an unbounded volume of perfect fluid is treated here on the basis of a potential framework. The formulas for the hydrodynamic force and moment acting upon a cylinder of arbitrary cross section are obtained. The equations governing the motion of a circular cylinder interacting with vortices are derived. For the greater part this paper coincides with [4] in which, however, only the case of one vortex was treated. It so happened that (due to a makeup man`s fault maybe) [4] was printed with no pictures in it. In this paper we reproduce those gures and extend the previous results.

The motion of a circular cylinder and a point vortex in an unbounded ideal fluid is treated here on the basis of a potential framework. The formulas for the hydrodynamic force and moment acting upon a cylinder of arbitrary cross section are obtained. The equations governing the motion of a circular cylinder are derived and partially investigated.