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Mikhail P. Kharlamov

Publications and source records attributed to Mikhail P. Kharlamov.

At least 19 recordsLinked to original sources
arXiv2015

Topological analysis of integrable problems in the dynamics of a rigid body

The book contains the results obtained by the author in 1975-1982 and presents new constructive methods of the topological analysis of integrable systems having non-linear integrals in involution. The phase topology of the classical integrable cases of the rigid body dynamics is investigated including the cases of Euler-Zhukovsky, Goryachev-Chaplygin-Sretenski and Kovalevskaya. All types of bifurcations of two-dimensional tori in these problems are revealed.

nlin.SI↗
arXiv2014

Regions of existence of critical motions for the generalized Kowalevski top and bifurcation diagrams

The paper concludes the cycle of investigations on the bifurcation diagrams of the system with three degrees of freedom which describes the motion of an axially symmetric top with the Kowalevski conditions in a double force field. The explicit inequalities are obtained defining the conditions for the existence of the critical motions on the surfaces bearing the bifurcation diagram (see M.P.Kharlamov, Mekh. Tverd. Tela, 2004, No. 34). We fulfill the construction of all diagrams on iso-energy levels having a stable type with respect to the physical parameters and the energy value.

nlin.SI↗
arXiv2014

Topological atlas of the Kovalevskaya top in a double field

We fulfill the rough topological analysis of the problem of the motion of the Kovalevskaya top in a double field. This problem is described by a completely integrable system with three degrees of freedom not reducible to a family of systems with two degrees of freedom. The notion of a topological atlas of an irreducible system is introduced. The complete topological analysis of the critical subsystems with two degrees of freedom is given. We calculate the types of all critical points. We present the parametric classification of the equipped iso-energy diagrams of the complete momentum map pointing out all chambers, families of 3-tori, and 4-atoms of their bifurcations. Basing on the ideas of A.T. Fomenko, we introduce the notion of the simplified net iso-energy invariant. All such invariants are constructed. Using them, we establish, for all parametrically stable cases, the number of critical periodic solutions of all types and the loop molecules of all rank 1 singularities. The work was supported by the grants of the RFBR No. 13-01-97025 and 14-01-00119.

nlin.SI↗
arXiv2014

Topological atlas of the Kowalevski--Yehia gyrostat: analytical results and topological analysis

We present a review of the results obtained during the last fifty years in the problem of the motion of a heavy gyrostat under the conditions of the Kowalevski type. Hamad M. Yehia in 1986 has proved that the problem is complete integrable. Since then, a lot of works were devoted to different aspects of integrating this problem and of its topological investigation. The main idea of this work is to prove strictly all the facts of qualitative analysis and to fix different mistakes and inaccuracies. The work was supported by the grants of the RFBR No. 10-01-00043, 10-01-97001, 13-01-97025, and 14-01-00119.

nlin.SI↗
arXiv2014

Integral manifolds of the reduced system in the problem of inertial motion of a rigid body about a fixed point

The reduced system in the problem of the inertial motion of a rigid body with a fixed point (the Euler case) is equivalent, by the Maupertuis principle, to some geodesic flow on the 2-sphere. We describe the phase topology of this case including the types of the bifurcations of the integral tori. We establish the topology of the singular integral surfaces. Using the geometrical interpretation of $SO(3)$ as a 3-ball with opposite points of the boundary sphere identified we show how the singular integral surfaces and the families of 2-tori are settled in an iso-energy level.Using the contemporary language, one can say that this is the first description of the bifurcation of the type $C_2$ ($2T^2 \to 2T^2$).

nlin.SI↗
arXiv2014

Phase topology of one integrable case of the rigid body motion

The reduced system in the Clebsch problem of the motion of a rigid body in fluid treated as the motion of a rigid body about its fixed mass center in a central Newtonian field with zero value of the area integral is a completely integrable natural mechanical system with two degrees of freedom. We find out the phase topology of this system including constructing the bifurcation set of two integrals quadratic with respect to velocities and describing all types of bifurcations of the integral tori. We establish the topology of all singular integral surfaces. Using the contemporary language, one can say that this is the first description of the bifurcations of the types $B$ ($T^2\to 2T^2$) and $C_2$ ($2T^2 \to 2T^2$), for the latter see also arXiv:1408.4548. This investigation includes the description of the foliation into integral surfaces of an invariant neighborhood of a saddle type singularity having two equilibriums on the connected critical integral surface.

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arXiv2014

Phase topology of one system with separated variables and singularities of the symplectic structure

We consider an example of a system with two degrees of freedom admitting separation of variables but having a subset of codimension 1 on which the 2-form defining the symplectic structure degenerates. We show how to use separation of variables to calculate the exact topological invariant of non-degenerate singularities and singularities appearing due to the symplectic structure degeneration. New types of non-orientable 3-atoms are found.

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arXiv2014

Regions of possible motion in mechanical systems

A method to study the topology of the integral manifolds basing on their projections to some other manifold of lower dimension is proposed. These projections are called the regions of possible motion and in mechanical systems arise in a natural way as the regions on a space of configuration variables. To classify such regions we introduce the notion of a generalized boundary of a region of possible motion and give the equation to find the generalized boundaries. The inertial motion of a gyrostat (the Euler--Zhukovsky case) is considered as an example. Explicit parametric equations of generalized boundaries are obtained. The investigation gives the main types of connected components of the regions of possible motion (including the sets of the admissible velocities over each point of the region). From this information, the phase topology of the case is established.

math-ph↗
arXiv2014

Reduction in mechanical systems with symmetry

The first part of the article is, in fact, the classical Routh method delivered in the language of contemporary theory of Lagrangian systems. But the Routh method deals only with concrete equations and, therefore, can be applied only in the case when the configuration spaces of the initial and the reduced systems are open submanifolds in Euclidean spaces. The global approach gives a possibility to find the structure of these manifolds in the general case and also to reveal some properties of the reduced system, first of all, the existence for this system of a global Lagrange function. We use the notion of a mechanical system introduced by S. Smale. The described method is applied to the global reduction in the problem of the motion of a rigid body having a fixed point in the potential force field with an axial symmetry. We present the complete proof of the theorem formulated by G.V. Kolosov on the equivalence of the reduced system in this case to the problem of the motion of a material point over an ellipsoid and also some corollaries of this theorem based on the results of L.A. Lyusternik and L.G. Shnirelman.

math.DS↗
arXiv2014

Characteristic class of a bundle and the existence of a global Routh function

The possibility of the global Lagrangian reduction of a mechanical system with symmetry is shown to be connected with the characteristic class of a principal fiber bundle of the configuration space over the factor manifold. It is proved that the reduced system is globally Lagrangian if and only if the product of the momentum constant with this characteristic class is zero. In the case of a rigid body rotating about a fixed point in an axially symmetric force field the bundle over a 2-sphere is non-trivial, therefore the reduced system admits a global Routh function if and only if the momentum constant is zero.

nlin.SI↗
arXiv2013

Bifurcation of common levels of first integrals of the Kovalevskaya problem

The structure of integral manifolds in the Kovalevskaya problem of the motion of a heavy rigid body about a fixed point is considered. An analytic description of a bifurcation set is obtained, and bifurcation diagrams are constructed. The number of two-dimensional tori is indicated for each connected component of the supplement to the bifurcation set in the space of the first integrals constants. The main topological bifurcations of the regular tori are described.

nlin.SI↗
arXiv2013

Extensions of the Appelrot classes for the generalized gyrostat in a double force field

For the integrable system on $e(3,2)$ found by Sokolov and Tsiganov we obtain explicit equations of some invariant 4-dimensional manifolds on which the induced systems are almost everywhere Hamiltonian with two degrees of freedom. These subsystems generalize the famous Appelrot classes of critical motions of the Kowalevski top. For each subsystem we point out a commutative pair of independent integrals, describe the sets of degeneration of the induced symplectic structure. With the help of the obtained invariant relations, for each subsystem we calculate the outer type of its points considered as critical points of the initial system with three degrees of freedom.

nlin.SI↗
arXiv2013

The atlas of the diagrams for the generalization of the 4th Appelrot class of especially remarkable motions to a gyrostat in a double force field

For the system with two degrees of freedom, which is an analogue of the 4th Appelrot class for a gyrostat of the Kowalevski type in a double force field the problem of the classification of bifurcation diagrams is solved. The separating set is built and its completeness is proved. All transformations taking place in the diagrams are shown. The results serve as a necessary part of solving the problem of obtaining the topological invariants for the Reyman - Semenov-Tian-Shansky system.

nlin.SI↗
arXiv2013

Analytical classification of the permanent rotations of the Kowalevski-Yehia gyrostat

The complete investigation of the permanent rotations of a gyrostat in the integrable case of Kowalevski-Yehia is presented. The notion of equivalence classes is given with respect to the defining parameters, the separating set is constructed. For each class the type of a singularity is calculated as the type of a fixed point in the reduced system. The detailed character of stability is obtained, and the structure of local Liouville foliation is shown.

nlin.SI↗
arXiv2013

Generalized 4th Appelrot class: phase topology

The article continues the author's publication in [Mech. Tverd. Tela, No. 35, 2005 and No. 38, 2008], in which we investigate the integrable dynamical system induced on one four-dimensional submanifold of the phase space of the problem of a rigid body motion in a double force field. When the intensity of one of the fields tends to zero this systems turns into the family of the espesially remarkable motions of the Kowalevski top belonging to the 4th Appelrot class. We introduce a method to describe the phase topology in the case when algebraic dependency is known of the phase variables in terms of separation variables. This method is based on some construction with Boolean vector functions. For the considered system with two degrees of freedom we fulfil the rough topological analysis.

nlin.SI↗
arXiv2013

Topological analysis and Boolean functions. I. Methods and application to classical systems

We aim to completely formalize the rough topological analysis of integrable Hamiltonian systems admitting analytical solutions such that the initial phase variables along with the time derivatives of the auxiliary variables are expressed as rational functions (in fact, as polynomials) in some set of radicals depending on one variable each. We suggest a method to define the admissible regions in the integral constants space, the segments of oscillation of the separated variables and the number of connected components of integral manifolds and critical integral surfaces. This method is based on some algorithms of processing the tables of some Boolean vector-functions and of reducing the matrices of linear Boolean vector-functions to some canonical form. From this point of view we consider here the topologically richest classical problems of the rigid body dynamics. The article will be continued with the investigation of some new integrable problems.

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