Modern problems of mechanics, Collected papers
Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, 351 pp.
Abstract
The volume presents studies on various issues in mechanics and dynamical systems theory, including the self-similar piston problem in a Prandtl–Reuss elastoplastic medium with special properties, homogenization of acoustic equations for a heterogeneous layered medium consisting of creep materials, spectral stability of shock waves in singular limits of smooth heteroclinic solutions to an extended system of equations, and stability of periodic orbits of a planar Birkhoff billiard. The problem of Arnold diffusion, dynamics of nonholonomic systems, integrable systems in analytical mechanics, and problems of the KAM theory in infinite-dimensional Hamiltonian systems are also discussed.
The volume is of interest to researchers, postgraduates, and students specializing in analytical mechanics and continuum mechanics.
Citation:
Kozlov V. V., Sergeev A. G., Modern problems of mechanics, Collected papers, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, 351 pp.
Dynamical systems on honeycombs
Traffic and granular flow '13, Springer, 2015, pp. 441–452
Abstract
Stochastic and deterministic versions of a discrete dynamical system on a necklace network are investigated. This network contains several contours. There are three cells and a particle on each contour. The particle occupies one of the cells and, at each step, it makes an attempt to move to the next cell in the direction of movement. As well as on neighboring contours the particles move in accordance with rules of stochastic or deterministic type. We prove that the behavior of the model with a rule of the first type is stochastic only at the beginning, and after a time interval the behavior becomes purely deterministic. The system with a rule of the first type reaches a stationary mode which depends on the initial state. The average velocity of particles and other characteristics of the dynamical systems are studied.
Citation:
Kozlov V. V., Buslaev A. P., Tatashev A. G., Yashina M. V., Dynamical systems on honeycombs, Traffic and granular flow '13, Springer, 2015, pp. 441–452
Asymptotic solutions of strongly nonlinear systems of differential equations
Translated from the 2009 Russian second edition [first edition: MR1457796] by Lester J. Senechal. Springer Monographs in Mathematics. Springer, Heidelberg, 2013, xx+262 pp.
Abstract
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Citation:
Kozlov V. V., Furta S. D., Asymptotic solutions of strongly nonlinear systems of differential equations, Translated from the 2009 Russian second edition [first edition: MR1457796] by Lester J. Senechal. Springer Monographs in Mathematics. Springer, Heidelberg, 2013, xx+262 pp.
Osnovaniya statisticheskoi mekhaniki i raboty Puankare, Erenfestov i fon Neimana
in A. Puankare, T. and P. Erenfesty, Dzh. fon Neiman, Raboty po statisticheskoi mekhanike, Moscow–Izhevsk: Institute of Computer Science, 2011, 249–279 pp.
Abstract
Citation:
Kozlov V. V., Smolyanov O. G., Osnovaniya statisticheskoi mekhaniki i raboty Puankare, Erenfestov i fon Neimana, in A. Puankare, T. and P. Erenfesty, Dzh. fon Neiman, Raboty po statisticheskoi mekhanike, Moscow–Izhevsk: Institute of Computer Science, 2011, 249–279 pp.
Dynamics of liquid and gas ellipsoids
Izhevsk: Regular and Chaotic Dynamics, 2010, 364 pp.
Abstract
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This book is a collection of the most significant classical results on the dynamics of liquid and gaseous ellipsoids, starting with the fundamental investigations of Dirichlet and Riemann.
The papers of the collection deal primarily with the derivation of various forms of the equations of motion and the investigation of qualitative properties of the dynamics of ellipsoidal figures.
The book addresses specialists and graduate students interested in mechanics, mathematical physics and the history of science.
Citation:
Borisov A. V., Mamaev I. S., Dynamics of liquid and gas ellipsoids, Izhevsk: Regular and Chaotic Dynamics, 2010, 364 pp.
Asymptotic expansions of solutions of strongly nonlinear systems of differential equations
Moscow–Izhevsk: Regular and Chaotic Dynamics, Institute of Computer Science, 2009, 312 pp.
Abstract
pdf (2.14 Mb)
Citation:
Kozlov V. V., Furta S. D., Asymptotic expansions of solutions of strongly nonlinear systems of differential equations, Moscow–Izhevsk: Regular and Chaotic Dynamics, Institute of Computer Science, 2009, 312 pp.
The Clebsch System. Separation of Variables and Explicit Integration?
Izhevsk: Regular and Chaotic Dynamics, 2009, 288 pp.
Abstract
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This book is a collection of the main classical works concerned with the Clebsch problem and related integrable systems. These papers, written by outstanding mathematicians of XIX centry, had a substantial influence on the development of many fields of modern mathematics and physics. The study of the Clebsch case and equivalent systems is far from completion, therefore, this problem remains one of the central in the theory of integrable systems.
The book addresses researchers, undergraduate and graduate students interested in theoretical mechanics, mathematical physics and the history of science.
Citation:
Borisov A. V., Tsiganov A. V., The Clebsch System. Separation of Variables and Explicit Integration?, Izhevsk: Regular and Chaotic Dynamics, 2009, 288 pp.
Proceedings of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow August 25-30, 2006)
Dordrecht Springer, 2006, 512 pp.
Abstract
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This work brings together previously unpublished notes contributed by participants of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence held in Moscow, August 25-30, 2006. The study of vortex motion is of great interest to fluid and gas dynamics; since all real flows are vortical in nature, applications of the vortex theory are extremely diverse, many of them (e.g. aircraft dynamics, atmospheric and ocean phenomena) being especially important. The last few decades have shown that serious possibilities for progress in the research of real turbulent vortex motions are essentially related to the combined use of mathematical methods, computer simulation and laboratory experiments. These approaches have led to a series of interesting results which allow us to study these processes from new perspectives. Based on this principle, the papers collected in this proceedings volume present new results on theoretical and applied aspects and processes of formation and evolution of various flows, wave and coherent structures in gas and fluid. Much attention is given to the studies of nonlinear regular and chaotic regimes of vortex interactions, advective and convective motions. The contributors are leading scientists engaged in fundamental and applied aspects of the above mentioned fields.
Citation:
Borisov A. V., Kozlov V. V., Mamaev I. S., Proceedings of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow August 25-30, 2006), Dordrecht Springer, 2006, 512 pp.
Mathematical aspects of classical and celestial mechanics
Dynamical systems. III, Encyclopaedia Math. Sci., 3, Ed. 3, Springer-Verlag, Berlin, 2006, xiv+518 pp.
Abstract
pdf (4.59 Mb)
Citation:
Arnold V. I., Kozlov V. V., Neishtadt A. I., Mathematical aspects of classical and celestial mechanics, Dynamical systems. III, Encyclopaedia Math. Sci., 3, Ed. 3, Springer-Verlag, Berlin, 2006, xiv+518 pp.
Rigid body dynamics. Hamiltonian methods, integrability, chaos
Moscow–Izhevsk: Institute of Computer Science, 2005, 576 pp.
Abstract
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In this book we discuss the main forms of equations of motion of rigid-body systems, such as motion of rigid bodies in potential fields, in fluid (Kirchhoff equation) and rigid bodies with fluid-filled cavities. All the systems considered in the book can be described within the framework of the Hamiltonian formalism. Almost all known integrable cases and methods of explicit integration are included. Compared to the previous volume, new sections dealing with non-integrability analysis and chaos in various problems of rigid body dynamics are added. Computer-based visualization of motion is widely used. Some results are obtained by the authors.
Citation:
Borisov A. V., Mamaev I. S., Rigid body dynamics. Hamiltonian methods, integrability, chaos, Moscow–Izhevsk: Institute of Computer Science, 2005, 576 pp.
Mathematical methods in the dynamics of vortex structures
Moscow–Izhevsk: Institute of Computer Science, 2005, 368 pp.
Abstract
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The book describes the main mathematical methods of investigation of vortex structures in an ideal incompressible fluid. All the methods of analysis of integrability and non-integrable systems are based on a systematic use of the Hamiltonian formalism and qualitative analysis. Some topics discussed in the book are: motion of point vortices on a plane and a sphere, interaction of vortex patches and some fresh issues concerned with dynamical interaction between rigid bodies and vortex structures in an ideal fluid. The appendices contain some new results, obtained by the authors in cooperation with their students and colleagues.
Citation:
Borisov A. V., Mamaev I. S., Mathematical methods in the dynamics of vortex structures, Moscow–Izhevsk: Institute of Computer Science, 2005, 368 pp.
Classical dynamics in non-Eucledian spaces
Moscow–Izhevsk: Institute of Computer Science, 2004, 348 pp.
Abstract
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The book is a collecton of recent and classical works of dynamics in spaces of constant curvature. The Kepler problem and its extentions, the two- and three-body problem and the rigid body dinamics in curved spaces are considered. Many classical works by W. Killing, H.Liebmann, etc. have been practically not available for a wide audience and almost forgotten. The recent papers collected here discuss stochasticity and integrability issues, various generalizations of some results from classical and celestial mechanics, and the Newton potential theory.
The book is for undergraduate and postgraduate students and specialists in dynamical systems. We hope that it will be interesting for scientific historians.
Citation:
Borisov A. V., Mamaev I. S., Classical dynamics in non-Eucledian spaces, Moscow–Izhevsk: Institute of Computer Science, 2004, 348 pp.
Fundamental and Applied Problems in the Theory of Vortices
Moscow–Izhevsk: Institute of Computer Science, 2003, 704 pp.
Abstract
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The book includes works of national and abroad authors examining the dynamics of vortex structures in fluid. The collected articles show this domain of study as an intensively evolving branch of fluid dynamics. With this aim in view, there are given both the results that have become well known, and the last achievements of the authors. The first part of the book is dedicated to the statement and solution of problems formulated in the frames of the classic hydrodynamic theory. Vortex problems of the geophysical hydrodynamics are stated in the second part of the book.
This collection of articles will be useful for the specialists in the field of dynamic systems and hydrodynamics, for lecturers, post-graduates and students in this scientific branch.
Citation:
Borisov A. V., Mamaev I. S., Sokolovskiy M. A., Fundamental and Applied Problems in the Theory of Vortices, Moscow–Izhevsk: Institute of Computer Science, 2003, 704 pp.
Modern Methods of the Theory of Integrable Systems
Moscow–Izhevsk: Institute of Computer Science, 2003, 296 pp.
Abstract
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The book studies integrable systems of the Hamiltonian mechanics within the context of the Lax representation and the explicit integration procedures. The authors introduce new methods of separation of variables and formulate the universal algorithm for constructing L-A pairs based on bi-hamiltonianity. In the book are also discussed multidimensional analogues of the integrable problems in the rigid body dynamics, generalized Toda lattices, geodesic flows and other problems in mechanics and geometry.
Citation:
Borisov A. V., Mamaev I. S., Modern Methods of the Theory of Integrable Systems, Moscow–Izhevsk: Institute of Computer Science, 2003, 296 pp.
Nonholonomic Dynamical Systems
Moscow–Izhevsk: Institute of Computer Science, 2002, 324 pp.
Abstract
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The book is a collection of papers concerned with dynamical effects in the motion of nonholonomic systems. Most of the contributions have been exclusively written for this book by leading Russian experts and involve novel results. These include, among others, new geometrical images of dynamics and various hierarchies of systems behavior. Also three-dimensional mappings in the problems of rolling motion of bodies are investigated numerically.
Citation:
Borisov A. V., Mamaev I. S., Nonholonomic Dynamical Systems, Moscow–Izhevsk: Institute of Computer Science, 2002, 324 pp.
Dynamics of the Rigid Body
Izhevsk: Regular and Chaotic Dynamics, 2001, 384 pp.
Abstract
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Here the fundamental forms of the equations of motion for rigid body are discussed. These involve motion in potential fields and in fluid (the Kirchhoff equations), motion of a body with cavities filled with fluid. Conditions under which the reduction of the order of these equations is possible and cyclic variables exist are given. In addition, the book collects almost all known to date integrable cases along with methods of their explicit integration. Throughout the book results of computer simulations are copiously presented to help visualize motions' peculiarities. Most results discussed in the book are obtained by the authors.
Citation:
Borisov A. V., Mamaev I. S., Dynamics of the Rigid Body, Izhevsk: Regular and Chaotic Dynamics, 2001, 384 pp.
Lyapunov's first method for strongly nonlinear systems of differential equations
Nonlinear mechanics, eds. V. M. Matrosov, V. V. Rumyantsev, A. V. Karapetyan. Moscow: Fizmatlit, 2001, 89–113 pp.
Abstract
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Citation:
Kozlov V. V., Furta S. D., Lyapunov's first method for strongly nonlinear systems of differential equations, Nonlinear mechanics, eds. V. M. Matrosov, V. V. Rumyantsev, A. V. Karapetyan. Moscow: Fizmatlit, 2001, 89–113 pp.
Methods of qualitative analysis in the dynamics of a rigid body, Second edition
Izhevsk: Regular and Chaotic Dynamics, 2000, 248 pp.
Abstract
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Citation:
Kozlov V. V., Methods of qualitative analysis in the dynamics of a rigid body, Second edition, Izhevsk: Regular and Chaotic Dynamics, 2000, 248 pp.
Poisson Structures and Lie Algebras in Hamiltonian Mechanics
Izhevsk: Izd. UdSU, 1999, 464 pp.
Abstract
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The book is about Poisson structures and their application to various problems of Hamiltonian mechanics arising in a number of areas, such as dynamics of rigid body, celestial mechanics, the theory of vortices, cosmological models. The equations governing the motion of such systems can be written in a convenient polynomial (algebraic) form. This form is in close connection with the possibility of representing the equations of motion as a set of Hamiltonian equations with linear Poisson structure associated with some Lie algebra. The authors also discuss nonlinear Poisson structures defined by infinite-dimensional Lie algebras and consider most typical situations in which such structures occur. The equations obtained are studied using the Painleve–Kovalevskaya method. Also the book presents some new integrable cases and establishes isomorphisms between integrable problems.
Citation:
Borisov A. V., Mamaev I. S., Poisson Structures and Lie Algebras in Hamiltonian Mechanics, Izhevsk: Izd. UdSU, 1999, 464 pp.
Principle of relativity and forces of inertia
Theoretical Mechanics, Collection of scientific and methodological papers, no. 21, MPI, Moscow, 1999, pp. 16–23
Abstract
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Citation:
Kozlov V. V., Harin A. O., Principle of relativity and forces of inertia, Theoretical Mechanics, Collection of scientific and methodological papers, no. 21, MPI, Moscow, 1999, pp. 16–23
Hamiltonian systems with three degrees of freedom and hydrodynamics
Hamiltonian systems with three or more degrees of freedom (S'Agaró, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 1999, vol. 533, pp. 127–133
Abstract
According to Stephan Banach, the possibilities of a mathematician are determined by his abilities to operate with analogies of different levels. The first level is the ability to see and to use the analogies between different problems. The next level is constituted by the analogies between theories. And finally only can be considered as high class mathematicians those who are able to use analogies between analogies.
Citation:
Kozlov V. V., Hamiltonian systems with three degrees of freedom and hydrodynamics, Hamiltonian systems with three or more degrees of freedom (S'Agaró, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 1999, vol. 533, pp. 127–133
Asymptotic expansions of solutions of strongly nonlinear systems of differential equations
Izdatel'stvo Moskovskogo Universiteta, Moscow, 1996, 244 pp.
Abstract
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Citation:
Kozlov V. V., Furta S. D., Asymptotic expansions of solutions of strongly nonlinear systems of differential equations, Izdatel'stvo Moskovskogo Universiteta, Moscow, 1996, 244 pp.
Symmetries, topology and resonances in Hamiltonian mechanics
Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 31. Springer-Verlag, Berlin, 1996, xii+378 pp.
Abstract
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Citation:
Kozlov V. V., Symmetries, topology and resonances in Hamiltonian mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 31. Springer-Verlag, Berlin, 1996, xii+378 pp.
Symmetries, topology and resonances in Hamiltonian mechanics
Izhevsk: Publishing House of Udmurt State University, 1995, 429 pp.
Abstract
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Citation:
Kozlov V. V., Symmetries, topology and resonances in Hamiltonian mechanics, Izhevsk: Publishing House of Udmurt State University, 1995, 429 pp.
Nonintegrability and Stochasticity in Rigid Body Dynamics
Izhevsk: Publishing House of Udmurt State University, 1995, 57 pp.
Abstract
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In this paper we present analytical and numerical data on nonintegrability of the equation of motion and the stochastic behaviour of trajectories in a range of classical and modern problems of dynamics of rigid body. There are considered analytical methods of proof of non-integrability in the systems close to integrable and their relation with data of numerical simulation. The observed spliting of separatrixes leads to absence of the additional first integrals and formation of stochastic layers within chaos in dynamics of rigid body.
Citation:
Borisov A. V., Emel'yanov K. V., Nonintegrability and Stochasticity in Rigid Body Dynamics, Izhevsk: Publishing House of Udmurt State University, 1995, 57 pp.
Mathematical aspects of classical and celestial mechanics
Dynamical systems. III, Encyclopaedia Math. Sci., 3, 2nd ed., Springer-Verlag, Berlin, 1993, xiv+291 pp.
Abstract
pdf (4.59 Mb)
Citation:
Arnold V. I., Kozlov V. V., Neishtadt A. I., Mathematical aspects of classical and celestial mechanics, Dynamical systems. III, Encyclopaedia Math. Sci., 3, 2nd ed., Springer-Verlag, Berlin, 1993, xiv+291 pp.
Billiards. A genetic introduction to the dynamics of systems with impacts
Izdatel'stvo Moskovskogo Universiteta, Moscow, 1991, 168 pp.
Abstract
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Citation:
Kozlov V. V., Treschev D. V., Billiards. A genetic introduction to the dynamics of systems with impacts, Izdatel'stvo Moskovskogo Universiteta, Moscow, 1991, 168 pp.
Billiards. A genetic introduction to the dynamics of systems with impacts
Translations of Mathematical Monographs, 89. Providence, RI: American Mathematical Society (AMS), 1991, viii+171 pp.
Abstract
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Billiard systems arise in many problems of mechanics and physics, some of them being described in the introduction. The authors have chosen a genetic approach, which consists in replacing a single-sided constraint imposed on the system by a field of elastic and dissipative forces. When the elastic and dissipative coefficients tend to infinity, the motion with fixed initial values tends on finite time intervals to a motion with impacts. The celebrated theorem of Birkhoff on the estimating of the number of distinct periodic trajectories of a convex elastic billiard is proved in a variational way, and some conditions for stability of the periodic trajectories are given. A chapter is dedicated to the Hill equation, obtained in the problem of the motion of the perigee of the moon and encountered also in problems of the stability of periodic motions with impacts. An important problem is the integrability of systems of billiard type. Old and new integrable problems, as the elliptic billiard, the billiard in affine Weyl cells, a harmonic oscillator inside an ellipse, billiards on surfaces of constant curvatures are described. The book is written in a clear and careful style and completely describes the problems from the physical phenomenon to the mathematical solution.
Citation:
Kozlov V. V., Treschev D. V., Billiards. A genetic introduction to the dynamics of systems with impacts, Translations of Mathematical Monographs, 89. Providence, RI: American Mathematical Society (AMS), 1991, viii+171 pp.