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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. Isoperimetric Inequalities for Moments of Inertia and Stability of Stationary Motions of a Flexible Thread 2019, Vol. 15, no. 4, pp.  513-523 Abstract It is well known that the maximal value of the central moment of inertia of a closed homogeneous thread of fixed length is achieved on a curve in the form of a circle. This isoperimetric property plays a key role in investigating the stability of stationary motions of a flexible thread. A discrete variant of the isoperimetric inequality, when the mass of the thread is concentrated in a finite number of material particles, is established. An analog of the isoperimetric inequality for an inhomogeneous thread is proved. Keywords: moment of inertia, Sundman and Wirtinger inequalities, articulated polygon Citation: Kozlov V. V.,  Isoperimetric Inequalities for Moments of Inertia and Stability of Stationary Motions of a Flexible Thread, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp.  513-523 DOI:10.20537/nd190410
 Kozlov V. V. The dynamics of systems with servoconstraints. II 2015, Vol. 11, No. 3, pp.  579-611 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 servo-constraint, 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, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 3, pp.  579-611 DOI:10.20537/nd1503007
 Kozlov V. V. The dynamics of systems with servoconstraints. I 2015, Vol. 11, No. 2, pp.  353-376 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, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 2, pp.  353-376 DOI:10.20537/nd1502010
 Kozlov V. V. Principles of dynamics and servo-constraints 2015, Vol. 11, No. 1, pp.  169-178 Abstract It is well known that in the Béghin– Appel theory servo-constraints are realized using controlled external forces. In this paper an expansion of the Béghin–Appel theory is given in the case where servo-constraints are realized using controlled change of the inertial properties of a dynamical system. The analytical mechanics of dynamical systems with servo-constraints of general form is discussed. The key principle of the approach developed is to appropriately determine virtual displacements of systems with constraints. Keywords: servo-constraints, d’Alembert–Lagrange principle, virtual displacements, Gauss principle, Noether theorem Citation: Kozlov V. V.,  Principles of dynamics and servo-constraints, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 1, pp.  169-178 DOI:10.20537/nd1501009
 Kozlov V. V. On Rational Integrals of Geodesic Flows 2014, Vol. 10, No. 4, pp.  439-445 Abstract This paper is concerned with the problem of first integrals of the equations of geodesics on twodimensional 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, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp.  439-445 DOI:10.20537/nd1404004
 Kozlov V. V. Notes on integrable systems 2013, Vol. 9, No. 3, pp.  459-478 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 permit 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 momentums 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, Hamilton equations, Euler–Jacobi theorem, Lie theorem, symmetries Citation: Kozlov V. V.,  Notes on integrable systems, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 3, pp.  459-478 DOI:10.20537/nd1303005
 Kozlov V. V. The Euler–Jacobi–Lie integrability theorem 2013, Vol. 9, No. 2, pp.  229-245 Abstract This paper addresses a class of problems associated with the conditions for exact integrability of a system 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 continuous medium with infinite conductivity. Keywords: symmetry field, integral invariant, nilpotent group, magnetic hydrodynamics Citation: Kozlov V. V.,  The Euler–Jacobi–Lie integrability theorem, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 2, pp.  229-245 DOI:10.20537/nd1302003
 Kozlov V. V. An extended Hamilton–Jacobi method 2012, Vol. 8, No. 3, pp.  549-568 Abstract We develop a new method for solving Hamilton’s canonical differential equations. The method is based on the search of 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, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp.  549-568 DOI:10.20537/nd1203010
 Kozlov V. V. On invariant manifolds of nonholonomic systems 2012, Vol. 8, No. 1, pp.  57-69 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, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp.  57-69 DOI:10.20537/nd1201004
 Kozlov V. V. The Lorentz force and its generalizations 2011, Vol. 7, No. 3, pp.  627-634 Abstract The structure of the Lorentz force and the related analogy between electromagnetism and inertia are discussed. The problem of invariant manifolds of the equations of motion for a charge in an electromagnetic field and the conditions for these manifolds to be Lagrangian are considered. Keywords: Lorentz force, Maxwell equations, Coriolis force, symplectic structure, Lagrangian manifold Citation: Kozlov V. V.,  The Lorentz force and its generalizations, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp.  627-634 DOI:10.20537/nd1103015
 Kozlov V. V. Statistical irreversibility of the Kac reversible circular model 2011, Vol. 7, No. 1, pp.  101-117 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 basing 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, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp.  101-117 DOI:10.20537/nd1101005
 Kozlov V. V. Lagrangian mechanics and dry friction 2010, Vol. 6, No. 4, pp.  855-868 Abstract A generalization of Amantons’ law of dry friction for constrained Lagrangian systems is formulated. Under a change of generalized coordinates the components of the dry-friction force transform according to the covariant rule and the force itself satisfies the Painlevé condition. In particular, the pressure of the system on a constraint is independent of the anisotropic-friction tensor. Such an approach provides an insight into the Painlevé dry-friction paradoxes. As an example, the general formulas for the sliding friction force and torque and the rotation friction torque on a body contacting with a surface are obtained. Keywords: Lagrangian system, anisotropic friction, Painlevé condition Citation: Kozlov V. V.,  Lagrangian mechanics and dry friction, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 4, pp.  855-868 DOI:10.20537/nd1004009
 Kozlov V. V. Note on dry friction and non-holonomic constraints 2010, Vol. 6, No. 4, pp.  903-906 Abstract Citation: Kozlov V. V.,  Note on dry friction and non-holonomic constraints, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 4, pp.  903-906 DOI:10.20537/nd1004014
 Kozlov V. V. The Vlasov kinetic equation, dynamics of continuum and turbulence 2010, Vol. 6, No. 3, pp.  489-512 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: The Vlasov kinetic equation, dynamics of continuum and turbulence Citation: Kozlov V. V.,  The Vlasov kinetic equation, dynamics of continuum and turbulence, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 3, pp.  489-512 DOI:10.20537/nd1003002
 Kozlov V. V. Kinetics of collisionless gas: equalization of temperature, growth of the coarse-grained entropy and the Gibbs paradox 2009, Vol. 5, No. 3, pp.  377-383 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, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 3, pp.  377-383 DOI:10.20537/nd0903005
 Kozlov V. V. Gauss Principle and Realization of Constraints 2008, Vol. 4, No. 3, pp.  281-285 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, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 3, pp.  281-285 DOI:10.20537/nd0803002
 Kozlov V. V. Lagrange’s identity and its generalizations 2008, Vol. 4, No. 2, pp.  157-168 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 continuumof 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, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 2, pp.  157-168 DOI:10.20537/nd0802004
 Borisov A. V., Kozlov V. V., Mamaev I. S. Asymptotic stability and associated problems of dynamics of falling rigid body 2007, Vol. 3, No. 3, pp.  255-296 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: nonholonomic mechanics, rigid body, ideal fluid, resisting medium Citation: Borisov A. V., Kozlov V. V., Mamaev I. S.,  Asymptotic stability and associated problems of dynamics of falling rigid body, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 3, pp.  255-296 DOI:10.20537/nd0703002
 Kozlov V. V. Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat 2007, Vol. 3, No. 2, pp.  123-140 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 average 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, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 2, pp.  123-140 DOI:10.20537/nd0702001
 Kozlov V. V. Vorticity equation of 2D-hydrodynamics, Vlasov steady-state kinetic equation and developed turbulence 2006, Vol. 2, No. 4, pp.  425-434 Abstract The issues discussed in this paper relate to the description of developed two-dimensional turbulence, when the mean values of characteristics of steady flow stabilize. More exactly, the problem of a weak limit of vortex distribution in two-dimensional flow of an ideal fluid at time tending to infinity is considered. Relations between the vorticity equation and the well-known Vlasov equation are discussed. Keywords: vortex motion equation, vorticity, Vlasov equation Citation: Kozlov V. V.,  Vorticity equation of 2D-hydrodynamics, Vlasov steady-state kinetic equation and developed turbulence, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 4, pp.  425-434 DOI:10.20537/nd0604004