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Vol. 1, No. 1

Vol. 1, No. 1, 2005

Borisov A. V.,  Mamaev I. S.,  Ramodanov S. M.
Abstract
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. Special cases of this system (the cylinders move along the line through their centers and the circulation around each cylinder is zero) are considered. A similar system of two interacting spheres was originally considered in classical works of Carl and Vilhelm Bjerknes, G. 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.
Keywords: perfect fluid, circulation, rigid body, qualitative analysis
Citation: Borisov A. V.,  Mamaev I. S.,  Ramodanov S. M., Interaction of two circular cylinders in a perfect fluid, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 1, pp. 3-21
DOI:10.20537/nd0501001
Zotev D. B.,  Kharlamov M. P.
Abstract
The motion of a rigid body about a fixed point in a double constant force field is governed by a Hamiltonian system with three degrees of freedom. We consider the general case when there are no one-dimensional symmetry groups. We point out the critical points of the Hamilton function and corresponding critical values of energy. Using the Morse theory, we have found the smooth type of non-degenerate five-dimensional iso-energetic levels and find their projections onto the configuration space, diffeomorphic to a three-dimensional projective space. The analogs of classical motion possibility regions, the projections of iso-energetic manifolds onto one of the Poisson spheres, are studied.
Keywords: rigid body, double constant force fields, iso-energetic manifolds, Poisson spheres
Citation: Zotev D. B.,  Kharlamov M. P., Iso-energetic manifolds and motion possibility regions of rigid body in double force field, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 1, pp. 23-31
DOI:10.20537/nd0501002
Gashenenko I. N.
Abstract
We study the topological structure of a common level surfaces of the first integrals in the problem on motion of a heavy gyrostat about a fixed point. We consider the special case when the gyrostatic momentum is collinear with the center-of-mass vector. With this supposition the axes of steady rotations can be directed only along generatrices of the Staude cone. We investigate the critical points of the effective potential, classify the bifurcation diagrams on the plane of constants of the first integrals and give a complete description of the topology of nonsingular integral manifolds of this problem.
Keywords: gyrostat, integral manifolds, bifurcation set, steady rotations
Citation: Gashenenko I. N., Bifurcations of the integral manifolds in the problem on motion of a heavy gyrostat, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 1, pp. 33-52
DOI:10.20537/nd0501003
Sokolov V. V.,  Marikhin V. G.
Abstract
A 8-parametric pair of commuting Hamiltonians of two degrees of freedom, quadratic in moments and coefficients depending only on coordinates is constructed. The Schottky-Manakov and the Clebsch spinning tops are particular cases of this model. The action function as an integral on a non-hyperelliptic curve of genus 4 is found.
Keywords: Separation of variables on non-hiperelliptic curve
Citation: Sokolov V. V.,  Marikhin V. G., Separation of variables on non-hiperelliptic curve, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 1, pp. 53-67
DOI:10.20537/nd0501004
Meleshko V. V.,  Krasnopolskaya T. S.
Abstract
The paper presents a new methodology for investigating and evaluating the basic properties of distributive laminar mixing in creeping flows. Our analysis is based upon conservation of some topological properties (e.g. connectedness and orientation) of the Lagrangian interface line under continuous transformations induced by an Eulerian velocity field. The principal advantage of our approach for line tracking is that despite complicated stretching and folding, the area of the blob, enclosed by the contour, is preserved. This gives possibility to develop three criteria for estimating the quality of mixing. The objective of this article is to expound the methodology of investigation of distributive laminar mixing of highly viscous materials by considering a typical example of two-dimensional Stokes flow in an annular wedge cavity.

Our methodology is based on the following steps: (1) determination of an analytical solution for the velocity field in the cavity; (2) observations on the deformation of the interface contour line of the stirring blob; (3) finding and classification of periodic points; (4) construction of statistical quantity measures for estimation of the quality of mixing at any given moment of time.
Keywords: mixing, stirring, coarse grained density
Citation: Meleshko V. V.,  Krasnopolskaya T. S., Mixing of viscous fluids, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 1, pp. 69-109
DOI:10.20537/nd0501005
Tronin K. G.
Abstract
A system of Liouville’s equations with slowly varying time-periodic parameters is considered. The system is shown to exhibit chaotic behavior with unique features. We start with the presentation of some general theoretical methods based on the analysis of abrupt changes of adiabatic invariants and separatrix splitting and then compare theoretical results with results obtained from numeric experiments.
Keywords: adiabatic invariant, Liouville's equations, numeric experiment
Citation: Tronin K. G., Adiabatic chaos in Liouville's equations, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 1, pp. 111-122
DOI:10.20537/nd0501006
Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
Abstract
For the classical problem of motion of a rigid body about a fixed point with zero integral of areas, the paper presents a family of solutions which are periodic in the absolute space. Such solutions are known as choreographies. The family includes the famous Delaunay solution in the case of Kovalevskaya, some particular solutions in the Goryachev-Chaplygin case and Steklov’s solution. The «genealogy» of the solutions of the family, arising naturally from the energy continuation, and their connection with the Staude rotations are considered.

It is shown that if the integral of areas is zero, the solutions are periodic but with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).
Keywords: rigid body dynamics, periodic solutions, continuation by a parameter, bifurcation
Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., Absolute and relative choreographies in rigid body dynamics, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 1, pp. 123-141
DOI:10.20537/nd0501007
Chenciner A.
Abstract
On November 30th 1896, Poincare published a note entitled «On the periodic solutions and the least action principle» in the «Comptes rendus de l’Academie des Sciences». He proposed to find periodic solutions of the planar Three-Body Problem by minimizing the Lagrangian action among loops in the configuration space which satisfy given constraints (the constraints amount to fixing their homology class). For the Newtonian potential, proportional to the inverse of the distance, the «collision problem» prevented him from realizing his program; hence he replaced it by a «strong force potential» proportional to the inverse of the squared distance.

In the lecture, the nature of the difficulties met by Poincare is explained and it is shown how, one century later, these have been partially resolved for the Newtonian potential, leading to the discovery of new remarkable families of periodic solutions of the planar or spatial n-body problem.
Keywords: Poincare, three-body problem, action minimizing periodic solutions
Citation: Chenciner A., A note by Poincare, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 1, pp. 143-154
DOI:10.20537/nd0501008

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