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A. V. Borisov

Publications and source records attributed to A. V. Borisov.

At least 37 records · Page 2Linked to original sources
arXiv2005

Superintegrable systems on sphere

We consider various generalizations of the Kepler problem to three-dimensional sphere $S^3$, a compact space of constant curvature. These generalizations include, among other things, addition of a spherical analog of the magnetic monopole (the Poincaré--Appell system) and addition of a more complicated field, which itself is a generalization of the MICZ-system. The mentioned systems are integrable -- in fact, superintegrable. The latter is due to the vector integral, which is analogous to the Laplace--Runge--Lenz vector. We offer a classification of the motions and consider a trajectory isomorphism between planar and spatial motions. The presented results can be easily extended to Lobachevsky space $L^3$.

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arXiv2005

Kovalevskaya Top and Generalizations of Integrable Systems

Generalizations of the Kovalevskaya, Chaplygin, Goryachev-Chaplygin and Bogoyavlensky systems on a bundle are considered in this paper. Moreover, a method of introduction of separating variables and action-angle variables is described. Another integration method for the Kovalevskaya top on the bundle is found. This method uses a coordinate transformation that reduces the Kovalevskaya system to the Neumann system. The Kolosov analogy is considered. A generalization of a recent Gaffet system to the bundle of Poisson brackets is obtained at the end of the paper.

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arXiv2005

Stability of Thomson's Configurations of Vortices on a Sphere

In this work stability of polygonal configurations on a plane and sphere is investigated. The conditions of linear stability are obtained. A nonlinear analysis of the problem is made with the help of Birkhoff normalization. Some problems are also formulated.

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arXiv2005

Lie algebras in vortex dynamics and celestial mechanics - IV

The work of A.V. Borisov, A.E. Pavlov, Dynamics and Statics of Vortices on a Plane and a Sphere - I (Reg. & Ch. Dynamics, 1998, Vol. 3, No 1, p.28-39) introduces a naive description of dynamics of point vortices on a plane in terms of variables of distances and areas which generate Lie-Poisson structure. Using this approach a qualitative description of dynamics of point vortices on a plane and a sphere is obtained in the works Dynamics of Three Vortices on a Plane and a Sphere - II. General compact case by A.V. Borisov, V.G. Lebedev (Reg. & Ch. Dynamics, 1998, Vol. 3, No 2, p.99-114), Dynamics of three vortices on a plane and a sphere - III. Noncompact case. Problem of collaps and scattering by A.V. Borisov, V.G. Lebedev (Reg. & Ch. Dynamics, 1998, Vol. 3, No 4, p.76-90). In this paper we consider more formal constructions of the general problem of n vortices on a plane and a sphere. The developed methods of algebraization are also applied to the classical problem of the reduction in the three-body problem.

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arXiv2005

Kovalevskaya Exponents and Poisson Structures

We consider generalizations of pairing relations for Kovalevskaya exponents in quasihomogeneous systems with quasihomogeneous tensor invariants. The case of presence of a Poisson structure in the system is investigated in more detail. We give some examples which illustrate general theorems.

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arXiv2005

Dynamics of three vortices on a plane and a sphere -- III. Noncompact case. Problems of collaps and scattering

In this article we considered the integrable problems of three vortices on a plane and sphere for noncompact case. We investigated explicitly the problems of a collapse and a scattering of vortices and obtained the conditions of its realization. We completed the bifurcation analysis and investigated for collinear and Thomson's configurations the dependence of stability in linear approximation and frequency of rotation in relative coordinates from value of a full moment. We indicated the geometric interpretation for characteristic situations. We constructed a phase portrait and geometric projection for the integrable configuration of four vortices on a plane.

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arXiv2005

On the motion of a heavy rigid body in an ideal fluid with circulation

Chaplygin's equations describing the planar motion of a rigid body in an unbounded volume of an ideal fluid involved in a circular flow around the body are considered. Hamiltonian structures, new integrable cases, and partial solutions are revealed, and their stability is examined. The problems of non-integrability of the equations of motion because of a chaotic behavior of the system are discussed.

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arXiv2005

Dynamics and statics of vortices on a plane and a sphere - I

In the present paper a description of a problem of point vortices on a plane and a sphere in the "internal" variables is discussed. The hamiltonian equations of motion of vortices on a plane are built on the Lie-Poisson algebras, and in the case of vortices on a sphere on the quadratic Jacobi algebras. The last ones are obtained by deformation of the corresponding linear algebras. Some partial solutions of the systems of three and four vortices are considered. Stationary and static vortex configurations are found.

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arXiv2005

Dynamics of rolling disk

In the paper we present the qualitative analysis of rolling motion without slipping of a homogeneous round disk on a horisontal plane. The problem was studied by S.A. Chaplygin, P. Appel and D. Korteweg who showed its integrability. The behavior of the point of contact on a plane is investigated and conditions under which its trajectory is finit are obtained. The bifurcation diagrams are constructed.

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arXiv2005

On the History of the Development of the Nonholonomic Dynamics

The main directions in the development of the nonholonomic dynamics are briefly considered in this paper. The first direction is connected with the general formalizm of the equations of dynamics that differs from the Lagrangian and Hamiltonian methods of the equations of motion's construction. The second direction, substantially more important for dynamics, includes investigations concerning the analysis of the specific nonholonomic problems. We also point out rather promising direction in development of nonholonomic systems that is connected with intensive use of the modern computer-aided methods.

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arXiv2005

Euler-Poisson Equations and Integrable Cases

In this paper we propose a new approach to the study of integrable cases based on intensive computer methods' application. We make a new investigation of Kovalevskaya and Goryachev-Chaplygin cases of Euler-Poisson equations and obtain many new results in rigid body dynamics in absolute space. Also we present the visualization of some special particular solutions.

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arXiv2005

Absolute and relative choreographies in the problem of point vortices moving on a plane

We obtained new periodic solutions in the problems of three and four point vortices moving on a plane. In the case of three vortices, the system is reduced to a Hamiltonian system with one degree of freedom, and it is integrable. In the case of four vortices, the order is reduced to two degrees of freedom, and the system is not integrable. We present relative and absolute choreographies of three and four vortices of the same intensity which are periodic motions of vortices in some rotating and fixed frame of reference, where all the vortices move along the same closed curve. Similar choreographies have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the n-body problem in celestial mechanics [6, 7, 17]. Nevertheless, the choreographies that appear in vortex dynamics have a number of distinct features.

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arXiv2005

Motion of a circular cylinder and n point vortices in a perfect fluid

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.

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arXiv2005

Two-body problem on a sphere. Reduction, stochasticity, periodic orbits

We consider the problem of two interacting particles on a sphere. The potential of the interaction depends on the distance between the particles. The case of Newtonian-type potentials is studied in most detail. We reduce this system to a system with two degrees of freedom and give a number of remarkable periodic orbits. We also discuss integrability and stochastization of the motion.

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arXiv2003

The Rolling Body Motion Of a Rigid Body on a Plane and a Sphere. Hierarchy of Dynamics

In this paper we consider cases of existence of invariant measure, additional first integrals, and Poisson structure in a problem of rigid body's rolling without sliding on plane and sphere. The problem of rigid body's motion on plane was studied by S.A. Chaplygin, P. Appel, D. Korteweg. They showed that the equations of motion are reduced to a second-order linear differential equation in the case when the surface of dynamically symmetric body is a surface of revolution. These results were partially generalized by P. Woronetz, who studied the motion of body of revolution and the motion of round disk with sharp edge on the surface of sphere. In both cases the systems are Euler-Jacobi integrable and have additional integrals and invariant measure. It turns out that after some change of time defined by reducing multiplier, the reduced system is a Hamiltonian system. Here we consider different cases when the integrals and invariant measure can be presented as finite algebraic expressions. We also consider the generalized problem of rolling of dynamically nonsymmetric Chaplygin ball. The results of studies are presented as tables that describe the hierarchy of existence of various tensor invariants: invariant measure, integrals, and Poisson structure in the considered problems.

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arXiv2003

The Rolling Motion of a Ball on a Surface. New Integrals and Hierarchy of Dynamics

The paper is concerned with the problem on rolling of a homogeneous ball on an arbitrary surface. New cases when the problem is solved by quadratures are presented. The paper also indicates a special case when an additional integral and invariant measure exist. Using this case, we obtain a nonholonomic generalization of the Jacobi problem for the inertial motion of a point on an ellipsoid. For a ball rolling, it is also shown that on an arbitrary cylinder in the gravity field the ball's motion is bounded and, on the average, it does not move downwards. All the results of the paper considerably expand the results obtained by E. Routh in XIX century.

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