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2013
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# Valery Kozlov

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

## Publications:

 Kozlov V. V. On the Extendability of Noether’s Integrals for Orbifolds of Constant Negative Curvature 2016, vol. 21, no. 7-8, pp.  821-831 Abstract 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, no. 7-8, pp. 821-831 DOI:10.1134/S1560354716070054
 Kozlov V. V. The Dynamics of Systems with Servoconstraints. II 2015, vol. 20, no. 4, pp.  401-427 Abstract 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 DOI:10.1134/S1560354715040012
 Kozlov V. V. The Dynamics of Systems with Servoconstraints. I 2015, vol. 20, no. 3, pp.  205-224 Abstract 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 DOI:10.1134/S1560354715030016
 Kozlov V. V. On Rational Integrals of Geodesic Flows 2014, vol. 19, no. 6, pp.  601-606 Abstract 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 DOI:10.1134/S156035471406001X
 Kozlov V. V. Remarks on Integrable Systems 2014, vol. 19, no. 2, pp.  145-161 Abstract 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 DOI:10.1134/S1560354714020014
 Kozlov V. V. The Euler–Jacobi–Lie Integrability Theorem 2013, vol. 18, no. 4, pp.  329-343 Abstract 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 DOI:10.1134/S1560354713040011
 Kozlov V. V. An Extended Hamilton–Jacobi Method 2012, vol. 17, no. 6, pp.  580-596 Abstract 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 DOI:10.1134/S1560354712060093
 Kozlov V. V. On Invariant Manifolds of Nonholonomic Systems 2012, vol. 17, no. 2, pp.  131-141 Abstract 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 DOI:10.1134/S1560354712020037
 Kozlov V. V. The Vlasov Kinetic Equation, Dynamics of Continuum and Turbulence 2011, vol. 16, no. 6, pp.  602-622 Abstract 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 DOI:10.1134/S1560354711060049
 Kozlov V. V. Statistical Irreversibility of the Kac Reversible Circular Model 2011, vol. 16, no. 5, pp.  536-549 Abstract 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 DOI:10.1134/S1560354711050091
 Kozlov V. V. Kinetics of collisionless gas: Equalization of temperature, growth of the coarse-grained entropy and the Gibbs paradox 2009, vol. 14, no. 4-5, pp.  535-540 Abstract 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, no. 4-5, pp. 535-540 DOI:10.1134/S1560354709040091
 Kozlov V. V. Gauss Principle and Realization of Constraints 2008, vol. 13, no. 5, pp.  431-434 Abstract 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 DOI:10.1134/S1560354708050055
 Kozlov V. V. Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat 2008, vol. 13, no. 3, pp.  141-154 Abstract 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 DOI:10.1134/S1560354708030015
 Kozlov V. V. Lagrange’s Identity and Its Generalizations 2008, vol. 13, no. 2, pp.  71-80 Abstract 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 DOI:10.1134/S1560354708020019
 Borisov A. V., Kozlov V. V., Mamaev I. S. Asymptotic stability and associated problems of dynamics of falling rigid body 2007, vol. 12, no. 5, pp.  531-565 Abstract 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 DOI:10.1134/S1560354707050061
 Kozlov V. V. Billiards, invariant measures, and equilibrium thermodynamics. II 2004, vol. 9, no. 2, pp.  91-100 Abstract 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 DOI:10.1070/RD2004v009n02ABEH000268
 Kozlov V. V. Notes on diffusion in collisionless medium 2004, vol. 9, no. 1, pp.  29-34 Abstract 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 DOI:10.1070/RD2004v009n01ABEH000262
 Kozlov V. V., Mitrofanova M. Y. Galton board 2003, vol. 8, no. 4, pp.  431-439 Abstract 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 DOI:10.1070/RD2003v008n04ABEH000255
 Kozlov V. V. On the Integration Theory of Equations of Nonholonomic Mechanics 2002, vol. 7, no. 2, pp.  161-176 Abstract 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 DOI:10.1070/RD2002v007n02ABEH000203
 Kozlov V. V. On Justification of Gibbs Distribution 2002, vol. 7, no. 1, pp.  1-10 Abstract 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 DOI:10.1070/RD2002v007n01ABEH000190
 Kozlov V. V. Kinetics of Collisionless Continuous Medium 2001, vol. 6, no. 3, pp.  235-251 Abstract 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 DOI:10.1070/RD2001v006n03ABEH000175
 Kozlov V. V. Billiards, Invariant Measures, and Equilibrium Thermodynamics 2000, vol. 5, no. 2, pp.  129-138 Abstract 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 DOI:10.1070/RD2000v005n02ABEH000136
 Kozlov V. V. Canonical Gibbs distribution and thermodynamics of mechanical systems with a finite number of degrees of freedom 1999, vol. 4, no. 2, pp.  44-54 Abstract 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 DOI:10.1070/RD1999v004n02ABEH000106
 Kozlov V. V. Averaging in a neighborhood of stable invariant tori 1997, vol. 2, nos. 3-4, pp.  41-46 Abstract 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 DOI:10.1070/RD1997v002n03ABEH000046
 Kozlov V. V. Closed Orbits and Chaotic Dynamics of a Charged Particle in a Periodic Electromagnetic Field 1997, vol. 2, no. 1, pp.  3-12 Abstract 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 DOI:10.1070/RD1997v002n01ABEH000021
 Kozlov V. V. Symmetries and Regular Behavior of Hamilton's Systems 1996, vol. 1, no. 1, pp.  3-14 Abstract 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 DOI:10.1070/RD1996v001n01ABEH000001