Valery Kozlov

Valery Kozlov
Gubkina st. 8, Moscow, 119991 Russia
Steklov Mathematical Institute, Russian Academy of Sciences
D.Sc., Professor

Born 1 January 1950 in Ryazanskaya district, Russia.


1967-1972: Undergraduate: M.V. Lomonosov Moscow State University, Department of Mechanics and Mathematics. Title of Graduation Thesis: «Nonintegrability of the equations of motion of a heavy rigid body about a fixed point»
1972-1973: Graduate: M.V. Lomonosov Moscow State University, Department of Mechanics and Mathematics.
1974: Kandidate in physics and mathematics. Thesis title: «Qualitative analysis of motion of a rigid body in integrable cases». M.V. Lomonosov Moscow State University
1978: Doctor in physics and mathematics. Title of thesis: «On the problems of qualitative analysis in the dynamics of a rigid body». M.V. Lomonosov Moscow State University

Positions held:

1974-1983: Assistant, Dozent, Senior scientist, M.V. Lomonosov Moscow State University, Department of Mechanics and Mathematics
Since 1983: Professor of the Chair of Theoretical Mechanics, M.V. Lomonosov Moscow State University
1980-1987: The Deputy Dean for science and research, Department of Mechanics and Mathematics, M.V. Lomonosov Moscow State University
1989-1998: Vice-rector, M.V. Lomonosov Moscow State University
1997-2001: The Deputy Minister of the Education of the Russian Federation
since 2001: Vice-President of Russian Academy of Science
since 2002: Head of Department of Mathematical Statistics and Random Processes, M.V. Lomonosov Moscow State University
since 2003: Head of Department of Mechanics, Steklov Mathematical Institute, Russian Academy of Sciences
since 2004: The Deputy Director Steklov Mathematical Institute Russian Academy of Sciences

Professional Societies:

Member of the Commission under the Russian Federation President on awarding State Prizes of the Russian Federation.

Founder and Editor-in-Chief of the international scientific journal «Regular and Chaotic Dynamics», Editor-in-Chief of «Izvestiya RAN, Seriya Matematicheskaya» (Izvestiya: Mathematics), Associate Editor of «Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika», member of editorial boards of «Matematicheskie Zametki» (Mathematical Notes) and «Russian Journal of Mathematical Physics».

Research supervision of 29 Kandidates of sciences and 5 Doctors of sciences (one of them, Dmitry V. Treschev, is now the Corresponding Member of the Russian Academy of Sciences).


1955: Member of the Russian National Committee on Theoretical and Applied Mechanics
1995: Full Member of the Russian Academy of Natural Sciences
1997: Corresponding Member of the Division of Machine Engineering, Mechanics and Control Processes Problems, Russian Academy of Sciences
2000: Full Member (Academician) of the Russian Academy of Sciences
2003: Foreign member of the Serbian Learned Society
2011: Foreign member of the Montenegrian Academy of Sciences and Arts
2012: Foreign member of the Serbian Academy of Sciences and Arts
2016: European Academy of Sciences and Arts

Awards & Honours:

1973: Lenin Komsomol Prize (the major prize for young scientists in USSR)
1986: M.V. Lomonosov 1st Degree Prize (the major prize awarded by M.V. Lomonosov Moscow State University)
1988: S. A. Chaplygin prize, Russian Academy of Sciences
1994: State Prize of the Russian Federation
1995: Peter The First Golden Medal of the International Academy of Environmental and Human Society Sciences
1996: The Breast Badge «For Distinguished Services» of the Russian Academy of Natural Sciences
2000: S.V. Kovalevskaya prize, Russian Academy of Sciences
2007: L. Euler Gold Medal, Russian Academy of Sciences
2009: Order of Service to the Fatherland III class
2009: The Gilli–Agostinelli International Prize of the Turin Academy of Sciences
2010: Russian Federation Government Prize in Education
2010: Prize "Triumph"
2014: Order of Service to the Fatherland II class
2015: S.A. Chaplygin Gold Medal, Russian Academy of Sciences


Kozlov V. V.
An example of an analytic system of differential equations in $\mathbb{R}^6$ with an equilibrium formally stable and stable for most initial conditions is presented. By means of a divergent formal transformation this system is reduced to a Hamiltonian system with three degrees of freedom. Almost all its phase space is foliated by three-dimensional invariant tori carrying quasi-periodic trajectories. These tori do not fill all phase space. Though the ``gap'' between these tori has zero measure, this set is everywhere dense in $\mathbb{R}^6$ and unbounded phase trajectories are dense in this gap. In particular, the formally stable equilibrium is Lyapunov unstable. This behavior of phase trajectories is quite consistent with the diffusion in nearly integrable systems. The proofs are based on the Poincaré–Dulac theorem, the theory of almost periodic functions, and on some facts from the theory of inhomogeneous Diophantine approximations. Some open problems related to the example are presented.
Keywords: analytic systems, formal stability, stability for most initial conditions, Lyapunov instability, diffusion, normal forms, almost periodic functions, inhomogeneous Diophantine approximations, Hamiltonian systems, Poisson stability
Citation: Kozlov V. V.,  Formal Stability, Stability for Most Initial Conditions and Diffusion in Analytic Systems of Differential Equations, Regular and Chaotic Dynamics, 2023, vol. 28, no. 3, pp. 251-264
Kozlov V. V.
On the Integrability of Circulatory Systems
2022, vol. 27, no. 1, pp.  11-17
This paper discusses conditions for the existence of polynomial (in velocities)
first integrals of the equations of motion of mechanical systems in a nonpotential force field
(circulatory systems). These integrals are assumed to be single-valued smooth functions on
the phase space of the system (on the space of the tangent bundle of a smooth configuration
manifold). It is shown that, if the genus of the closed configuration manifold of such a system
with two degrees of freedom is greater than unity, then the equations of motion admit no
nonconstant single-valued polynomial integrals. Examples are given of circulatory systems with
configuration space in the form of a sphere and a torus which have nontrivial polynomial laws
of conservation. Some unsolved problems involved in these phenomena are discussed.
Keywords: circulatory system, polynomial integral, genus of surface
Citation: Kozlov V. V.,  On the Integrability of Circulatory Systems, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, pp. 11-17
Kozlov V. V.
This paper addresses the problem of conditions for the existence of conservation laws (first integrals) of circulatory systems which are quadratic in velocities (momenta), when the external forces are nonpotential. Under some conditions the equations of motion are reduced to Hamiltonian form with some symplectic structure and the role of the Hamiltonian is played by a quadratic integral. In some cases the equations are reduced to a conformally Hamiltonian rather than Hamiltonian form. The existence of a quadratic integral and its properties allow conclusions to be drawn on the stability of equilibrium positions of circulatory systems.
Keywords: circulatory system, polynomial integrals, Hamiltonian system, property of being conformally Hamiltonian, indices of inertia, asymptotic trajectories, Ziegler’s pendulum
Citation: Kozlov V. V.,  Integrals of Circulatory Systems Which are Quadratic in Momenta, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 647-657
Kozlov V. V.
The properties of the Gibbs ensembles of Hamiltonian systems describing the motion along geodesics on a compact configuration manifold are discussed.We introduce weakly ergodic systems for which the time average of functions on the configuration space is constant almost everywhere. Usual ergodic systems are, of course, weakly ergodic, but the converse is not true. A range of questions concerning the equalization of the density and the temperature of a Gibbs ensemble as time increases indefinitely are considered. In addition, the weak ergodicity of a billiard in a rectangular parallelepiped with a partition wall is established.
Keywords: Hamiltonian system, Liouville and Gibbs measures, Gibbs ensemble, weak ergodicity, mixing, billiard in a polytope
Citation: Kozlov V. V.,  Nonequilibrium Statistical Mechanics of Weakly Ergodic Systems, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 674-688
Kozlov V. V.
A chain of quadratic first integrals of general linear Hamiltonian systems that have not been represented in canonical form is found. Their involutiveness is established and the problem of their functional independence is studied. The key role in the study of a Hamiltonian system is played by an integral cone which is obtained by setting known quadratic first integrals equal to zero. A singular invariant isotropic subspace is shown to pass through each point of the integral cone, and its dimension is found. The maximal dimension of such subspaces estimates from above the degree of instability of the Hamiltonian system. The stability of typical Hamiltonian systems is shown to be equivalent to the degeneracy of the cone to an equilibrium point. General results are applied to the investigation of linear mechanical systems with gyroscopic forces and finite-dimensional quantum systems.
Keywords: Hamiltonian system, quadratic integrals, integral cones, degree of instability, quantum systems, Abelian integrals
Citation: Kozlov V. V.,  Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 26-46
Kozlov V. V.
This paper is concerned with the problem of the integrable behavior of geodesics on homogeneous factors of the Lobachevsky plane with respect to Fuchsian groups (orbifolds). Locally the geodesic equations admit three independent Noether integrals linear in velocities (energy is a quadratic form of these integrals). However, when passing along closed cycles the Noether integrals undergo a linear substitution. Thus, the problem of integrability reduces to the search for functions that are invariant under these substitutions. If a Fuchsian group is Abelian, then there is a first integral linear in the velocity (and independent of the energy integral). Conversely, if a Fuchsian group contains noncommuting hyperbolic or parabolic elements, then the geodesic flow does not admit additional integrals in the form of a rational function of Noether integrals. We stress that this result holds also for noncompact orbifolds, when there is no ergodicity of the geodesic flow (since nonrecurrent geodesics can form a set of positive measure).
Keywords: Lobachevsky plane, Fuchsian group, orbifold, Noether integrals
Citation: Kozlov V. V.,  On the Extendability of Noether’s Integrals for Orbifolds of Constant Negative Curvature, Regular and Chaotic Dynamics, 2016, vol. 21, nos. 7-8, pp. 821-831
Kozlov V. V.
The Dynamics of Systems with Servoconstraints. II
2015, vol. 20, no. 4, pp.  401-427
This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servoconstraint, which implies that the projection of the body’s angular velocity on some body-fixed direction is zero.
Keywords: servoconstraints, symmetries, Lie groups, left-invariant constraints, systems with quadratic right-hand sides, vakonomic systems
Citation: Kozlov V. V.,  The Dynamics of Systems with Servoconstraints. II, Regular and Chaotic Dynamics, 2015, vol. 20, no. 4, pp. 401-427
Kozlov V. V.
The Dynamics of Systems with Servoconstraints. I
2015, vol. 20, no. 3, pp.  205-224
The paper discusses the dynamics of systems with Béghin’s servoconstraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servoconstraint — the projection of the angular velocity onto some direction fixed in the body is equal to zero (a generalization of the nonholonomic Suslov problem) — and the motion of the Chaplygin sleigh with servoconstraints of a certain type. The dynamics of systems with Béghin’s servoconstraints is richer and more varied than the more usual dynamics of nonholonomic systems.
Keywords: servoconstraints, symmetries, Lie groups, left-invariant constraints, systems with quadratic right-hand sides
Citation: Kozlov V. V.,  The Dynamics of Systems with Servoconstraints. I, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 205-224
Kozlov V. V.
On Rational Integrals of Geodesic Flows
2014, vol. 19, no. 6, pp.  601-606
This paper is concerned with the problem of first integrals of the equations of geodesics on two-dimensional surfaces that are rational in the velocities (or momenta). The existence of nontrivial rational integrals with given values of the degrees of the numerator and the denominator is proved using the Cauchy–Kovalevskaya theorem.
Keywords: conformal coordinates, rational integral, irreducible integrals, Cauchy–Kovalevskaya theorem
Citation: Kozlov V. V.,  On Rational Integrals of Geodesic Flows, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 601-606
Kozlov V. V.
Remarks on Integrable Systems
2014, vol. 19, no. 2, pp.  145-161
The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension $\geqslant n$. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.
Keywords: integrability by quadratures, adjoint system, Hamiltonian equations, Euler–Jacobi theorem, Lie theorem, symmetries
Citation: Kozlov V. V.,  Remarks on Integrable Systems, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 145-161
Kozlov V. V.
The Euler–Jacobi–Lie Integrability Theorem
2013, vol. 18, no. 4, pp.  329-343
This paper addresses a class of problems associated with the conditions for exact integrability of systems of ordinary differential equations expressed in terms of the properties of tensor invariants. The general theorem of integrability of the system of $n$ differential equations is proved, which admits $n−2$ independent symmetry fields and an invariant volume $n$-form (integral invariant). General results are applied to the study of steady motions of a continuum with infinite conductivity.
Keywords: symmetry field, integral invariant, nilpotent group, magnetic hydrodynamics
Citation: Kozlov V. V.,  The Euler–Jacobi–Lie Integrability Theorem, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 329-343
Kozlov V. V.
An Extended Hamilton–Jacobi Method
2012, vol. 17, no. 6, pp.  580-596
We develop a new method for solving Hamilton’s canonical differential equations. The method is based on the search for invariant vortex manifolds of special type. In the case of Lagrangian (potential) manifolds, we arrive at the classical Hamilton–Jacobi method.
Keywords: generalized Lamb’s equations, vortex manifolds, Clebsch potentials, Lagrange brackets
Citation: Kozlov V. V.,  An Extended Hamilton–Jacobi Method, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 580-596
Kozlov V. V.
On Invariant Manifolds of Nonholonomic Systems
2012, vol. 17, no. 2, pp.  131-141
Invariant manifolds of equations governing the dynamics of conservative nonholonomic systems are investigated. These manifolds are assumed to be uniquely projected onto configuration space. The invariance conditions are represented in the form of generalized Lamb’s equations. Conditions are found under which the solutions to these equations admit a hydrodynamical description typical of Hamiltonian systems. As an illustration, nonholonomic systems on Lie groups with a left-invariant metric and left-invariant (right-invariant) constraints are considered.
Keywords: invariant manifold, Lamb’s equation, vortex manifold, Bernoulli’s theorem, Helmholtz’ theorem
Citation: Kozlov V. V.,  On Invariant Manifolds of Nonholonomic Systems, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 131-141
Kozlov V. V.
We consider a continuum of interacting particles whose evolution is governed by the Vlasov kinetic equation. An infinite sequence of equations of motion for this medium (in the Eulerian description) is derived and its general properties are explored. An important example is a collisionless gas, which exhibits irreversible behavior. Though individual particles interact via a potential, the dynamics of the continuum bears dissipative features. Applicability of the Vlasov equations to the modeling of small-scale turbulence is discussed.
Keywords: kinetic Vlasov’s equation, Euler’s equation, continuum, turbulence
Citation: Kozlov V. V.,  The Vlasov Kinetic Equation, Dynamics of Continuum and Turbulence, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 602-622
Kozlov V. V.
The Kac circular model is a discrete dynamical system which has the property of recurrence and reversibility. Within the framework of this model M.Kac formulated necessary conditions for irreversibility over "short" time intervals to take place and demonstrated Boltzmann’s most important exploration methods and ideas, outlining their advantages and limitations. We study the circular model within the realm of the theory of Gibbs ensembles and offer a new approach to a rigorous proof of the "zeroth" law of thermodynamics based on the analysis of weak convergence of probability distributions.
Keywords: reversibility, stochastic equilibrium, weak convergence
Citation: Kozlov V. V.,  Statistical Irreversibility of the Kac Reversible Circular Model, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 536-549
Kozlov V. V.
The Poincaré model for dynamics of a collisionless gas in a rectangular parallelepiped with mirror walls is considered. The question on smoothing of the density and the temperature of this gas and conditions for the monotone growth of the coarse-grained entropy are discussed. All these effects provide a new insight of the classical paradox of mixing of gases.
Keywords: collisionless gas, coarse-grained entropy, Gibbs paradox
Citation: Kozlov V. V.,  Kinetics of collisionless gas: Equalization of temperature, growth of the coarse-grained entropy and the Gibbs paradox, Regular and Chaotic Dynamics, 2009, vol. 14, nos. 4-5, pp. 535-540
Kozlov V. V.
Gauss Principle and Realization of Constraints
2008, vol. 13, no. 5, pp.  431-434
The paper generalizes the classical Gauss principle for non-constrained dynamical systems. For large anisotropic external forces of viscous friction our statement transforms into the common Gauss principle for systems with constraints.
Keywords: Gauss principle, constraints, anisotropic friction
Citation: Kozlov V. V.,  Gauss Principle and Realization of Constraints, Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 431-434
Kozlov V. V.
The paper develops an approach to the proof of the "zeroth" law of thermodynamics. The approach is based on the analysis of weak limits of solutions to the Liouville equation as time grows infinitely. A class of linear oscillating systems is indicated for which the average energy becomes eventually uniformly distributed among the degrees of freedom for any initial probability density functions. An example of such systems are sympathetic pendulums. Conditions are found for nonlinear Hamiltonian systems with finite number of degrees of freedom to converge in a weak sense to the state where the mean energies of the interacting subsystems are the same. Some issues related to statistical models of the thermostat are discussed.
Keywords: Hamiltonian system, sympathetic oscillators, weak convergence, thermostat
Citation: Kozlov V. V.,  Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 141-154
Kozlov V. V.
Lagrange’s Identity and Its Generalizations
2008, vol. 13, no. 2, pp.  71-80
The famous Lagrange identity expresses the second derivative of the moment of inertia of a system of material points through the kinetic energy and homogeneous potential energy. The paper presents various extensions of this brilliant result to the case 1) of constrained mechanical systems, 2) when the potential energy is quasi-homogeneous in coordinates and 3) of continuum of interacting particles governed by the well-known Vlasov kinetic equation.
Keywords: Lagrange's identity, quasi-homogeneous function, dilations, Vlasov’s equation
Citation: Kozlov V. V.,  Lagrange’s Identity and Its Generalizations, Regular and Chaotic Dynamics, 2008, vol. 13, no. 2, pp. 71-80
Borisov A. V., Kozlov V. V., Mamaev I. S.
We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.
Keywords: rigid body, ideal fluid, non-holonomic mechanics
Citation: Borisov A. V., Kozlov V. V., Mamaev I. S.,  Asymptotic stability and associated problems of dynamics of falling rigid body, Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, pp. 531-565
Kozlov V. V.
The kinetics of collisionless continuous medium is studied in a bounded region on a curved manifold. We have assumed that in statistical equilibrium, the probability distribution density depends only on the total energy. It is shown that in this case, all the fundamental relations for a multi-dimensional ideal gas in thermal equilibrium hold true.
Citation: Kozlov V. V.,  Billiards, invariant measures, and equilibrium thermodynamics. II, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 91-100
Kozlov V. V.
Notes on diffusion in collisionless medium
2004, vol. 9, no. 1, pp.  29-34
A collisionless continuous medium in Euclidean space is discussed, i.e. a continuum of free particles moving inertially, without interacting with each other. It is shown that the distribution density of such medium is weakly converging to zero as time increases indefinitely. In the case of Maxwell's velocity distribution of particles, this density satisfies the well-known diffusion equation, the diffusion coefficient increasing linearly with time.
Citation: Kozlov V. V.,  Notes on diffusion in collisionless medium, Regular and Chaotic Dynamics, 2004, vol. 9, no. 1, pp. 29-34
Kozlov V. V., Mitrofanova M. Y.
Galton board
2003, vol. 8, no. 4, pp.  431-439
In this paper, we present results of simulations of a model of the Galton board for various degrees of elasticity of the ball-to-nail collision.
Citation: Kozlov V. V., Mitrofanova M. Y.,  Galton board, Regular and Chaotic Dynamics, 2003, vol. 8, no. 4, pp. 431-439
Kozlov V. V.
The paper deals with the problem of integration of equations of motion in nonholonomic systems. By means of well-known theory of the differential equations with an invariant measure the new integrable systems are discovered. Among them there are the generalization of Chaplygin's problem of rolling nonsymmetric ball in the plane and the Suslov problem of rotation of rigid body with a fixed point. The structure of dynamics of systems on the invariant manifold in the integrable problems is shown. Some new ideas in the theory of integration of the equations in nonholonomic mechanics are suggested. The first of them consists in using known integrals as the constraints. The second is the use of resolvable groups of symmetries in nonholonomic systems. The existence conditions of invariant measure with analytical density for the differential equations of nonholonomic mechanics is given.
Citation: Kozlov V. V.,  On the Integration Theory of Equations of Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 161-176
Kozlov V. V.
On Justification of Gibbs Distribution
2002, vol. 7, no. 1, pp.  1-10
The paper develop a new approach to the justification of Gibbs canonical distribution for Hamiltonian systems with finite number of degrees of freedom. It uses the condition of nonintegrability of the ensemble of weak interacting Hamiltonian systems.
Citation: Kozlov V. V.,  On Justification of Gibbs Distribution, Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, pp. 1-10
Kozlov V. V.
Kinetics of Collisionless Continuous Medium
2001, vol. 6, no. 3, pp.  235-251
In this article we develop Poincaré ideas about a heat balance of ideal gas considered as a collisionless continuous medium. We obtain the theorems on diffusion in nondegenerate completely integrable systems. As a corollary we show that for any initial distribution the gas will be eventually irreversibly and uniformly distributed over all volume, although every particle during this process approaches arbitrarily close to the initial position indefinitely many times. However, such individual returnability is not uniform, which results in diffusion in a reversible and conservative system. Balancing of pressure and internal energy of ideal gas is proved, the formulas for limit values of these quantities are given and the classical law for ideal gas in a heat balance is deduced. It is shown that the increase of entropy of gas under the adiabatic extension follows from the law of motion of a collisionless continuous medium.
Citation: Kozlov V. V.,  Kinetics of Collisionless Continuous Medium, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 235-251
Kozlov V. V.
The questions of justification of the Gibbs canonical distribution for systems with elastic impacts are discussed. A special attention is paid to the description of probability measures with densities depending on the system energy.
Citation: Kozlov V. V.,  Billiards, Invariant Measures, and Equilibrium Thermodynamics, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 129-138
Kozlov V. V.
Traditional derivation of Gibbs canonical distribution and the justification of thermodynamics are based on the assumption concerning an isoenergetic ergodicity of a system of n weakly interacting identical subsystems and passage to the limit $n \to\infty$. In the presented work we develop another approach to these problems assuming that n is fixed and $n \geqslant 2$. The ergodic hypothesis (which frequently is not valid due to known results of the KAM-theory) is substituted by a weaker assumption that the perturbed system does not have additional first integrals independent of the energy integral. The proof of nonintegrability of perturbed Hamiltonian systems is based on the Poincare method. Moreover, we use the natural Gibbs assumption concerning a thermodynamic equilibrium of bsystems at vanishing interaction. The general results are applied to the system of the weakly connected pendula. The averaging with respect to the Gibbs measure allows to pass from usual dynamics of mechanical systems to the classical thermodynamic model.
Citation: Kozlov V. V.,  Canonical Gibbs distribution and thermodynamics of mechanical systems with a finite number of degrees of freedom, Regular and Chaotic Dynamics, 1999, vol. 4, no. 2, pp. 44-54
Kozlov V. V.
Averaging in a neighborhood of stable invariant tori
1997, vol. 2, nos. 3-4, pp.  41-46
We analyse the operation of averaging of smooth functions along exact trajectories of dynamic systems in a neighborhood of stable nonresonance invariant tori. It is shown that there exists the first integral after the averaging; however in the typical situation the mean value is discontinuous or even not everywhere defind. If the temporal mean were a smooth function it would take its stationary values in the points of nondegenerate invariant tori. We demonstrate that this result can be properly derived if we change the operations of averaging and differentiating with respect to the initial data by their places. However, in general case for nonstable tori this property is no longer preserved. We also discuss the role of the reducibility condition of the invariant tori and the possibility of the generalization for the case of arbitrary compact invariant manifolds on which the initial dynamic system is ergodic.
Citation: Kozlov V. V.,  Averaging in a neighborhood of stable invariant tori, Regular and Chaotic Dynamics, 1997, vol. 2, nos. 3-4, pp. 41-46
Kozlov V. V.
We study motion of a charged particle on the two dimensional torus in a constant direction magnetic field. This analysis can be applied to the description of electron dynamics in metals, which admit a $2$-dimensional translation group (Bravais crystal lattice). We found the threshold magnetic value, starting from which there exist three closed Larmor orbits of a given energy. We demonstrate that if there are n lattice atoms in a primitive Bravais cell then there are $4+n$ different Larmor orbits in the nondegenerate case. If the magnetic field is absent the electron dynamics turns out to be chaotic, dynamical systems on the corresponding energy shells possess positive entropy in the case that the total energy is positive.
Citation: Kozlov V. V.,  Closed Orbits and Chaotic Dynamics of a Charged Particle in a Periodic Electromagnetic Field, Regular and Chaotic Dynamics, 1997, vol. 2, no. 1, pp. 3-12
Kozlov V. V.
Symmetries and Regular Behavior of Hamilton's Systems
1996, vol. 1, no. 1, pp.  3-14
The paper discusses relationship between regular behavior of Hamilton's systems and the existence a sufficient number of fields of symmetry. Some properties of quite regular schemes and their relationship with various characteristics of stochastic behavior are studied.
Citation: Kozlov V. V.,  Symmetries and Regular Behavior of Hamilton's Systems, Regular and Chaotic Dynamics, 1996, vol. 1, no. 1, pp. 3-14
Kozlov V. V.
This paper discusses a range of questions concerning the application of solvable Lie algebras of vector fields to exact integration of systems of ordinary differential equations. The set of $n$ independent vector fields generating a solvable Lie algebra in $n$-dimensional space is locally reduced to some “canonical” form. This reduction is performed constructively (using quadratures), which, in particular, allows a simultaneous integration of $n$ systems of differential equations that are generated by these fields. Generalized completely integrable systems are introduced and their properties are investigated. General ideas are applied to integration of the Hamiltonian systems of differential equations.
Keywords: quadratures, solvable alebra, Frobenius theorem, completely integrable systems, Lie theorem, Hamiltonian systems
Kozlov V. V.
Integrals of circulatory systems which are quadratic in momenta
, , pp. 

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