Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics
Sbornik: Mathematics, 2016, vol. 207, no. 10, pp. 1435–1449
Abstract
For integrable systems with two degrees of freedom there are well-known inequalities connecting the Euler characteristic of the configuration space (as a closed two-dimensional surface) with the number of singular points of Newtonian type of the potential energy. On the other hand, there are results on conditions for ergodicity of systems on a two-dimensional torus with short-range potential depending only on the distance from an attracting or repelling centre. In the present paper we consider the problem of conditions for the existence of nontrivial first integrals that are polynomial in the momenta of the problem of motion of a particle on a multi-dimensional Euclidean torus in a force field whose potential has singularity points. These conditions depend only on the order of the singularity, and in the two-dimensional case they are satisfied by potentials with singularities of Newtonian type.
Keywords:
polynomial integrals, potentials with singularities, order of singularity, Poincaré condition
Citation:
Kozlov V. V., Treschev D. V., Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics, Sbornik: Mathematics, 2016, vol. 207, no. 10, pp. 1435–1449
On the equations of the hydrodynamic type
Journal of Applied Mathematics and Mechanics, 2016, vol. 80, no. 3, pp. 209–214
Abstract
The conditions of stability of ‘pure’ equilibrium states of finite-dimensional dynamical systems of the hydrodynamic type are indicated. Integrable chains allowing full sets of quadratic first integrals are found. The conditions of existence of invariant Gaussian measures in the infinite-dimensional case are discussed. The existence of such measures makes it possible to establish the properties of recurrence of solutions of infinite-dimensional dynamical systems of the hydrodynamic type.
Citation:
Kozlov V. V., On the equations of the hydrodynamic type, Journal of Applied Mathematics and Mechanics, 2016, vol. 80, no. 3, pp. 209–214
Invariant measures of smooth dynamical systems, generalized functions and summation methods
Izvestiya: Mathematics, 2016, vol. 80, no. 2, pp. 342–358
Abstract
pdf (289.71 Kb)
We discuss conditions for the existence of invariant measures of smooth dynamical systems on compact manifolds. If there is an invariant measure with continuously differentiable density, then the divergence of the vector field along every solution tends to zero in the Cesaro sense as time increases unboundedly. Here the Cesaro convergence may be replaced, for example, by any Riesz summation method, which can be arbitrarily close to ordinary convergence (but does not coincide with it). We give an example of a system whose divergence tends to zero in the ordinary sense but none of its invariant measures is absolutely continuous with respect to the `standard' Lebesgue measure (generated by some Riemannian metric) on the phase space. We give examples of analytic systems of differential equations on analytic phase spaces admitting invariant measures of any prescribed smoothness (including a measure with integrable density), but having no invariant measures with positive continuous densities. We give a new proof of the classical Bogolyubov-Krylov theorem using generalized functions and the Hahn-Banach theorem. The properties of signed invariant measures are also discussed.
Keywords:
invariant measures, generalized functions, summation methods, small denominators, Hahn–Banach theorem
Citation:
Kozlov V. V., Invariant measures of smooth dynamical systems, generalized functions and summation methods, Izvestiya: Mathematics, 2016, vol. 80, no. 2, pp. 342–358
Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas
Russian Mathematical Surveys, 2016, vol. 71, no. 2, pp. 253–290
Abstract
pdf (723.26 Kb)
The problem of conditions ensuring the existence of first integrals that are polynomials in the momenta (velocities) is considered for certain multidimensional billiard systems which play an important role in non-equilibrium statistical mechanics. These are the Lorentz gas, a particle in a Euclidean space with (not necessarily convex) scattering domains, and the Boltzmann–Gibbs gas, a system of small identical balls in a rectangular box which collide elastically with one another and the walls of the box. The ergodic properties of such systems are only partially understood: some problems are still waiting for solution, and in certain cases (for instance, when the scatterers are non-convex) the system is known not to be ergodic. An approach to showing the absence of a non-trivial polynomial first integral with continuously differentiable coefficients is developed. The known first integrals for integrable problems in dynamics are mostly polynomials in the momenta (or functions of polynomials). The investigation of multidimensional billiards with non-compact configuration space, when there is no hope for ergodic behaviour, is of particular interest. Applications of the general results on the absence of non-trivial polynomial integrals to problems in statistical mechanics are discussed.
Keywords:
Birkhoff billiards, Lorentz gas, Boltzmann–Gibbs gas, polynomial integral, topological obstructions to integrability, elastic reflection, KAM theory
Citation:
Kozlov V. V., Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas, Russian Mathematical Surveys, 2016, vol. 71, no. 2, pp. 253–290
The phenomenon of reversal in the Euler–Poincaré–Suslov nonholonomic systems
Journal of Dynamical and Control Systems, 2016, vol. 22, no. 4, pp. 713–724
Abstract
The development of robotics makes it necessary to study the problem of controlling nonholonomic systems (Svinin et al., Regul Chaotic Dyn. 2013; 18(1–2): 126–143, Borisov et al., Regul. Chaotic Dyn. 2013; 18(1–2): 144–158, Ivanova et al., Regul Chaotic Dyn. 2014; 19(1): 140–143). In this paper, the dynamics of nonholonomic systems on Lie groups with a left-invariant kinetic energy and left-invariant constraints are considered. Equations of motion form a closed system of differential equations on the corresponding Lie algebra. In addition, the effect of change in the stability of steady motions of these systems with the direction of motion reversed (the reversal found in rattleback dynamics) is discussed. As an illustration, the rotation of a rigid body with a fixed point and the Suslov nonholonomic constraint as well as the motion of the Chaplygin sleigh is considered.
Kozlov V. V., The phenomenon of reversal in the Euler–Poincaré–Suslov nonholonomic systems, Journal of Dynamical and Control Systems, 2016, vol. 22, no. 4, pp. 713–724
Invariant and quasi-invariant measures on infinite-dimensional spaces
Doklady Mathematics, 2015, vol. 92, no. 3, pp. 743–746
Abstract
Extensions of locally convex topological spaces are considered such that finite cylindrical measures which are not countably additive on their initial domains turn out to be countably additive on the extensions. Extensions of certain transformations of the initial spaces with respect to which the initial measures are invariant or quasi-invariant to the extensions of these spaces are described. Similar questions are considered for differentiable measures. The constructions may find applications in statistical mechanics and quantum field theory.
Citation:
Kozlov V. V., Smolyanov O. G., Invariant and quasi-invariant measures on infinite-dimensional spaces, Doklady Mathematics, 2015, vol. 92, no. 3, pp. 743–746
Homogeneous systems with quadratic integrals, Lie–Poisson quasi-brackets, and the Kovalevskaya method
Sbornik: Mathematics, 2015, vol. 206, no. 12, pp. 29–54
Abstract
pdf (481.65 Kb)
We consider differential equations with quadratic right-hand sides which admit two quadratic first integrals, one of which is a positive definite quadratic form. We present general conditions under which a linear change of variables reduces this system to some "canonical" form. Under these conditions the system turns out to be nondivergent and is reduced to Hamiltonian form, however, the corresponding linear Lie–Poisson bracket does not always satisfy the Jacobi identity. In the three-dimensional case the equations are reduced to the classical equations of the Euler top, and in the four-dimensional space the system turns out to be superintegrable and coincides with the Euler–Poincaré equations on some Lie algebra. In the five-dimensional case we find a reducing multiplier after multiplication with which the Poisson bracket satisfies the Jacobi identity. In the general case, we prove that there is no reducing multiplier for $n>5$. As an example, we consider a system of Lotka–Volterra type with quadratic right-hand sides, which was studied already by Kovalevskaya from the viewpoint of the conditions for uniqueness of its solutions as functions of complex time.
Keywords:
first integrals, conformally Hamiltonian system, Poisson bracket, Kovalevskaya system, dynamical systems with quadratic right-hand sides
Citation:
Bizyaev I. A., Kozlov V. V., Homogeneous systems with quadratic integrals, Lie–Poisson quasi-brackets, and the Kovalevskaya method, Sbornik: Mathematics, 2015, vol. 206, no. 12, pp. 29–54
Rational integrals of quasi-homogeneous dynamical systems
Journal of Applied Mathematics and Mechanics, 2015, vol. 79, no. 3, pp. 209–216
Abstract
Dynamical systems, described by quasi-homogeneous systems of differential equations with polynomial right-hand sides, are considered. The Euler–Poisson equations from solid-state dynamics, as well as the Euler–Poincaré equations in Lie algebras, which describe the dynamics of systems in Lie groups with a left-invariant kinetic energy, can be pointed out as examples. The conditions for the existence of rational first integrals of quasi-homogeneous systems are found. They include the conditions for the existence of invariant algebraic manifolds. Examples of systems with rational integrals which do not admit of first integrals that are polynomial with respect to the momenta are presented. Results of a general nature are also demonstrated in the example of a Hess–Appel’rot invariant manifold from the dynamics of an asymmetric heavy top.
Citation:
Kozlov V. V., Rational integrals of quasi-homogeneous dynamical systems, Journal of Applied Mathematics and Mechanics, 2015, vol. 79, no. 3, pp. 209–216
Generalized transport-logistic problem as class of dynamical systems
Mathematical Models and Computer Simulations, 2015, vol. 27, no. 12, pp. 65–87
Abstract
pdf (925.4 Kb)
Dynamical systems on network with discrete set of states and discrete time are considered. Sites, channels and particles are forming an abstract model of mass transport, information and so on, on the one hand, and another, they are forming dynamical system of deterministic or stochastic type. State of the system in the following discrete instant of time $S(T+1)$ is defined by transformation of the state at the moment $S(T)$ with given rules $L$, $S(T+1)=L(S(T))$. In this case, $S(T+1)$ does not necessarily belong to the admissible states set $A$. Then "judicial system" is activated, i.e. operator $P$ such that projects $S(T+1)$ to $A$. Thus, $S(T+1)=\{L(S(T))$, if $L(S(T))$ belongs $A$; $PL(S(T))$, if $L(S(T))$ does not belong $A\}$. Properties of these systems are researched, and applications for transport problems are discussed.
Bugaev A. S., Buslaev A. P., Kozlov V. V., Tatashev A. G., Yashina M. V., Generalized transport-logistic problem as class of dynamical systems, Mathematical Models and Computer Simulations, 2015, vol. 27, no. 12, pp. 65–87
The Dynamics of Systems with Servoconstraints. II
Regular and Chaotic Dynamics, 2015, vol. 20, no. 4, pp. 401-427
Abstract
pdf (861.95 Kb)
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
The Dynamics of Systems with Servoconstraints. I
Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 205-224
Abstract
pdf (810.79 Kb)
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
The dynamics of systems with servoconstraints. II
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 3, pp. 579-611
Abstract
pdf (560.42 Kb)
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, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 3, pp. 579-611
The dynamics of systems with servoconstraints. I
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 2, pp. 353-376
Abstract
pdf (505.17 Kb)
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, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 2, pp. 353-376
Principles of dynamics and servo-constraints
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 1, pp. 169-178
Abstract
pdf (316.31 Kb)
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.
A dynamical communication system on a network
Journal of Computational and Applied Mathematics, 2015, vol. 275, pp. 247–261
Abstract
A dynamical system is introduced and investigated. The system contains N vertices. The vertices send messages at discrete time instants according to a given rule. A conflict of two vertices takes place if the vertices try to send messages to each other at the same instant. Each vertex sends a message to another vertex at every step if no conflict takes place. In case of a conflict, only one of the two competing vertices sends a message. Deterministic and stochastic conflict resolution rules are considered. We investigate the average number of messages sent by a vertex per a time unit, called the productivity of this vertex, the total productivity of the system and other characteristics. The productivity of vertices depends on the initial state of the system, and the criterion of efficiency is the expected average productivity of vertices provided all possible initial states of the system are equiprobable. An ergodic version of the system is also considered in which any particle moves with approximately equal to 1 probability provided there is no conflict.
Kozlov V. V., Buslaev A. P., Tatashev A. G., A dynamical communication system on a network, Journal of Computational and Applied Mathematics, 2015, vol. 275, pp. 247–261
Monotonic walks on a necklace and a coloured dynamic vector
International Journal of Computer Mathematics, 2015, vol. 92, no. 9, pp. 1910–1920
Abstract
Stochastic and deterministic versions of a discrete dynamical system on a necklace are investigated. This network consists of a sequence of contours NSWE with nodes, i.e. the nodes are common points at W and E. There are two cells and a particle on each contour. Each time instance, the particle occupies a cell and, at every time unit, comes to the next cell in the same direction. The particles of the neighbouring contours move in accordance with rules of stochastic or deterministic type. The behaviour of the model with the rule of the first type is stochastic only at the beginning and after a time interval becomes a pure deterministic system. The system with the second rule comes to a steady mode, which depends on the initial state. The average velocity of particles and characteristics of the system are studied.
Keywords:
dynamical system, networks, monotonic walks of particles, characteristics of dynamical systems, synergy, collapse
Citation:
Kozlov V. V., Buslaev A. P., Tatashev A. G., Monotonic walks on a necklace and a coloured dynamic vector, International Journal of Computer Mathematics, 2015, vol. 92, no. 9, pp. 1910–1920
On real-valued oscillations of a bipendulum
Applied Mathematics Letters, 2015, vol. 46, pp. 44–49
Abstract
Theoretical and computational aspects of special case of logistical-routing problem are considered. Fluctuation of two particles on a grid connected by a channel also is considered. Velocity rate and sufficient conditions of system self-regulation are obtained.
Keywords:
Transport–logistic problem; Dynamical system; Markov’s process
Citation:
Kozlov V. V., Buslaev A. P., Tatashev A. G., On real-valued oscillations of a bipendulum, Applied Mathematics Letters, 2015, vol. 46, pp. 44–49
Coarsening in ergodic theory
Russian Journal of Mathematical Physics, 2015, vol. 22, no. 2, pp. 184–187
Abstract
This paper deals with the coarsening operation in dynamical systems where the phase space with a finite invariant measure is partitioned into measurable pieces and the summable function transferred by the phase flow is averaged over these pieces at each instant of time. Letting the time tend to infinity and then refining the partition, we arrive at a modernization of the von Neumann ergodic theorem, which is useful for the purposes of nonequilibrium statistical mechanics. In particular, for fine-grained partitions, we obtain the law of increment of coarse entropy for systems approaching the state of statistical equilibrium.
In memory of A. M. Il’in
Citation:
Kozlov V. V., Coarsening in ergodic theory, Russian Journal of Mathematical Physics, 2015, vol. 22, no. 2, pp. 184–187
Liouville's equation as a Schrödinger equation
Izvestiya: Mathematics, 2014, vol. 78, no. 4, pp. 744–757
Abstract
We show that every non-negative solution of Liouville's equation for an
arbitrary (possibly non-Hamiltonian) dynamical system admits a factorization
$\psi\psi^*$, where $\psi$ satisfies a Schrödinger equation of special
form. The corresponding quantum system is obtained by Weyl quantization
of a Hamiltonian system whose Hamiltonian is linear in the momenta.
We discuss the structure of the spectrum of the special Schrödinger
equation on a multidimensional torus and show that the eigenfunctions may
have finite smoothness in the analytic case. Our generalized solutions
of the Schrödinger equation are natural examples of non-selfadjoint
extensions of Hermitian differential operators. We give conditions for
the existence of a smooth invariant measure of a dynamical system. They
are expressed in terms of stability conditions for the conjugate
equations of variations.
The dynamics of systems with large gyroscopic forces and the realization of constraints
Journal of Applied Mathematics and Mechanics, 2014, vol. 78, no. 3, pp. 213–219
Abstract
Lagrangian systems with a large multiplier N on the gyroscopic terms are considered. Simplified equations of motion of general form with holonomic constraints are obtained in the first approximation with respect to the small parameter ɛ = 1/N. The structure of the solutions of the precessional equations is examined.
Citation:
Kozlov V. V., The dynamics of systems with large gyroscopic forces and the realization of constraints, Journal of Applied Mathematics and Mechanics, 2014, vol. 78, no. 3, pp. 213–219
International mathematical conferences and schools for young researchers in Suzdal
Russian Mathematical Surveys, 2014, vol. 69, no. 1, pp. 183–185
Abstract
Citation:
Davydov A. A., Kozlov V. V., Chernousko F. L., International mathematical conferences and schools for young researchers in Suzdal, Russian Mathematical Surveys, 2014, vol. 69, no. 1, pp. 183–185
On Rational Integrals of Geodesic Flows
Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 601-606
Abstract
pdf (145.78 Kb)
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.
Remarks on Integrable Systems
Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 145-161
Abstract
pdf (186.79 Kb)
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.
On Rational Integrals of Geodesic Flows
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, pp. 439-445
Abstract
pdf (302.05 Kb)
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.
Conservation laws of generalized billiards that are polynomial in momenta
Russian Journal of Mathematical Physics, 2014, vol. 21, no. 2, pp. 226–241
Abstract
This paper deals with dynamics particles moving on a Euclidean n-dimensional torus or in an n-dimensional parallelepiped box in a force field whose potential is proportional to the characteristic function of the region D with a regular boundary. After reaching this region, the trajectory of the particle is refracted according to the law which resembles the Snell -Descartes law from geometrical optics. When the energies are small, the particle does not reach the region D and elastically bounces off its boundary. In this case, we obtain a dynamical system of billiard type (which was intensely studied with respect to strictly convex regions). In addition, the paper discusses the problem of the existence of nontrivial first integrals that are polynomials in momenta with summable coefficients and are functionally independent with the energy integral. Conditions for the geometry of the boundary of the region D under which the problem does not admit nontrivial polynomial first integrals are found. Examples of nonconvex regions are given; for these regions the corresponding dynamical system is obviously nonergodic for fixed energy values (including small ones), however, it does not admit polynomial conservation laws independent of the energy integral.
Citation:
Kozlov V. V., Conservation laws of generalized billiards that are polynomial in momenta, Russian Journal of Mathematical Physics, 2014, vol. 21, no. 2, pp. 226–241
Behavior of pendulums on a regular polygon
Journal of Communication and Computer, 2014, vol. 11, no. 1, pp. 30–38
Abstract
pdf (294.98 Kb)
Stochastic and deterministic versions of a discrete dynamical network system are investigated. This network consists of a sequence of contours NSWE with nodes, which are common points at N, W, S and E. There are four cells and a particle on each contour. At each time instance, the particle occupies a cell, and attempts to occupy the next cell in the same direction. Particles of neighboring contours move in accordance with some rules. Both deterministic and stochastic rules are considered. The behavior of the model with the first rule is stochastic only at the beginning, and after a time interval the system becomes to pure deterministic. The system with the second rule comes to a steady state, which depends on the initial state. The average velocity of particles and other characteristics of the system are studied.
Keywords:
Dynamical system, networks, monotonic walks of particles, characteristics of a dynamical system, synergy
Citation:
Kozlov V. V., Buslaev A. P., Tatashev A. G., Behavior of pendulums on a regular polygon, Journal of Communication and Computer, 2014, vol. 11, no. 1, pp. 30–38
On a system of nonlinear differential equations for the model of totally connected traffic
Journal of Concrete and Applicable Mathematics, 2014, vol. 12, no. 1-2, pp. 86–93
Abstract
pdf (580.72 Kb)
In the paper the qualitative properties solutions of the system nonlinear equations, describing one-way movement of particles chain on a line with follower velocity defined by some function of distance from the leader, are researched. In the case when the given function is the velocity of the first particle (leader) in the chain, the model is called a model of leader. If the given function is the velocity of the last particle (outsider), the model is called a model of “shepherd”.
The sufficient conditions for the existence of the chain with the given constraints on the velocity and acceleration are obtained.
Keywords:
systems of nonlinear ordinary differential equations; follow-theleader model; interpretation for traffic
Citation:
Buslaev A. P., Kozlov V. V., On a system of nonlinear differential equations for the model of totally connected traffic, Journal of Concrete and Applicable Mathematics, 2014, vol. 12, no. 1-2, pp. 86–93
The classical Bohl argument theorem of a conditionally periodic function is generalized. Conditionally periodic motions on a torus are replaced by the solutions of a nonlinear system of differential equations with invariant measure. Cases in which this system is assumed ergodic or strictly ergodic are considered.
Keywords:
Bohl's argument theorem; conditionally periodic motion on the n-dimensional torus; (strictly) ergodic system of differential equations; uniformly distributed function; Birkhoff-Khinchine ergodic theorem
Citation:
Kozlov V. V., On Bohl's Argument Theorem, Mathematical Notes, 2013, vol. 93, no. 1, pp. 83–89
The behaviour of cyclic variables in integrable systems
Journal of Applied Mathematics and Mechanics, 2013, vol. 77, no. 2, pp. 128–136
Abstract
pdf (246.21 Kb)
A general theorem on the behaviour of the angular variables of integrable dynamical systems as functions of time is established. Problems on the motion of the nodal line of a Kovalevskaya top and of a three-dimensional rigid body in a fluid are considered in integrable cases as examples. This range of topics is discussed for systems of a more general form obtained from completely integrable systems after changing the time.
Citation:
Kozlov V. V., The behaviour of cyclic variables in integrable systems, Journal of Applied Mathematics and Mechanics, 2013, vol. 77, no. 2, pp. 128–136
Traffic modeling: monotonic total-connected random walk on a network
Mathematical Models and Computer Simulations, 2013, vol. 25, no. 8, pp. 3–21
Abstract
pdf (449.44 Kb)
Monotonic (particles move in the same direction) and total-connected (particles that occupy neighboring cells move synchronized) random ($p<1$) and deterministic ($p=1$) walks on closed networks, which consist of circles, are considered. An algorithm has been developed that allows to calculate the duration of the time interval after that all the particles will be contained in the unique cluster. It is proved that such the interval is finite in the considered model. Some statements are proved that allow to found the velocity of movement if deterministic movement occurs on the follows structures: two rings (two closed sequences of cells) that have a common cell; a closed sequence of rings each of that has two common cells with two the neighboring rings; a two-dimensional network structure in that each cell has common cells with four the neighboring rings; a similar infinite network.
Keywords:
stochastic models; random walk; traffic flows
Citation:
Bugaev A. S., Buslaev A. P., Kozlov V. V., Tatashev A. G., Yashina M. V., Traffic modeling: monotonic total-connected random walk on a network , Mathematical Models and Computer Simulations, 2013, vol. 25, no. 8, pp. 3–21
The Euler–Jacobi–Lie Integrability Theorem
Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 329-343
Abstract
pdf (377.18 Kb)
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
Notes on integrable systems
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 459-478
Abstract
pdf (375.2 Kb)
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, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 459-478
The Euler–Jacobi–Lie integrability theorem
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 229-245
Abstract
pdf (377.18 Kb)
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, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 229-245
Polynomial in momenta invariants of Hamilton's equations, divergent series and generalized functions
Nonlinearity, 2013, vol. 26, no. 3, pp. 745–756
Abstract
We consider the problem of the motion of two identical interacting particles on a circle in a potential force field. We study conditions for the existence of an additional first integral in the form of a polynomial of degree 4 in momenta with periodic coefficients. These conditions are reduced to the solvability of a certain nonlinear functional-differential equation for the potential of the external force and the interaction potential. If the potentials are smooth, this equation has only trivial solutions: at least one of the potentials is a constant function. We classify generalized solutions of this equation when the interaction potential is a distribution while the external potential is a smooth function. Formal Fourier series for the interaction potential can be summed up by the (C, 2)-Cesaro method to periodic functions with singularities.
Citation:
Kozlov V. V., Polynomial in momenta invariants of Hamilton's equations, divergent series and generalized functions, Nonlinearity, 2013, vol. 26, no. 3, pp. 745–756
Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space
Izvestiya: Mathematics, 2012, vol. 76, no. 5, pp. 907–921
Abstract
pdf (519.88 Kb)
We consider problems related to the well-known conjecture on the degrees of irreducible polynomial integrals of a reversible Hamiltonian system with two degrees of freedom and toral position space. The main object of study is a special system arising in the analysis of irreducible polynomial integrals of degree 4. In a particular case we have the problem of the motion of two interacting particles on a circle in given potential fields. We prove that if the three potentials are smooth non-constant functions, then this problem has no non-trivial polynomial integrals of arbitrarily high degree. We prove the conjecture completely for systems with a polynomial first integral of degree 4 in the momenta.
Keywords:
irreducible integrals, systems with impacts, spectrum of a potential
Citation:
Denisova N. V., Kozlov V. V., Treschev D. V., Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space, Izvestiya: Mathematics, 2012, vol. 76, no. 5, pp. 907–921
Invariant manifolds of Hamilton's equations
Journal of Applied Mathematics and Mechanics, 2012, vol. 76, no. 4, pp. 378–387
Abstract
pdf (254.88 Kb)
The invariance conditions of smooth manifolds of Hamilton's equations are represented in the form of multidimensional Lamb's equations from the dynamics of an ideal fluid. In the stationary case these conditions do not depend on the method used to parameterize the invariant manifold. One consequence of Lamb's equations is an equation of a vortex, which is invariant to replacements of the time-dependent variables. A proof of the periodicity conditions of solutions of autonomous Hamilton's equations with n degrees of freedom and compact energy manifolds that admit of 2n – 3 additional first integrals is given as an application of the theory developed.
Citation:
Kozlov V. V., Invariant manifolds of Hamilton's equations, Journal of Applied Mathematics and Mechanics, 2012, vol. 76, no. 4, pp. 378–387
The statistical mechanics of a class of dissipative systems
Journal of Applied Mathematics and Mechanics, 2012, vol. 76, no. 1, pp. 15–24
Abstract
The statistical mechanics of dynamical systems on which only isotropic viscous friction forces act is developed. A non-stationary analogue of the Gibbs canonical distribution, which enables each such system to be made to correspond to a certain thermodynamic system that satisfies the first and second laws of thermodynamics, is introduced. The evolution of non-Gibbs probability distributions with time is also considered.
Citation:
Kozlov V. V., The statistical mechanics of a class of dissipative systems, Journal of Applied Mathematics and Mechanics, 2012, vol. 76, no. 1, pp. 15–24
An Extended Hamilton–Jacobi Method
Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 580-596
Abstract
pdf (216.54 Kb)
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.
On Invariant Manifolds of Nonholonomic Systems
Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 131-141
Abstract
pdf (239.09 Kb)
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.
An extended Hamilton–Jacobi method
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 549-568
Abstract
pdf (400.68 Kb)
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.
On invariant manifolds of nonholonomic systems
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 57-69
Abstract
pdf (329.1 Kb)
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.
Metropolis traffic modeling: from intelligent monitoring through physical representation to mathematical problems
Proc. International Conference on Computational and Mathematical Methods in Science and Engineering, 2012, vol. 1, pp. 750–756
Abstract
Citation:
Kozlov V. V., Buslaev A. P., Metropolis traffic modeling: from intelligent monitoring through physical representation to mathematical problems, Proc. International Conference on Computational and Mathematical Methods in Science and Engineering, 2012, vol. 1, pp. 750–756
Wigner measures on infinite-dimensional spaces and the Bogolyubov equations for quantum systems
Doklady Mathematics, 2011, vol. 84, no. 1, pp. 571–575
Abstract
Citation:
Kozlov V. V., Wigner measures on infinite-dimensional spaces and the Bogolyubov equations for quantum systems, Doklady Mathematics, 2011, vol. 84, no. 1, pp. 571–575
Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard
Proceedings of the Steklov Institute of Mathematics, 2011, vol. 273, pp. 196–213
Abstract
A linearized problem of stability of simple periodic motions with elastic reflections is considered: a particle moves along a straight-line segment that is orthogonal to the boundary of a billiard at its endpoints. In this problem issues from mechanics (variational principles), linear algebra (spectral properties of products of symmetric operators), and geometry (focal points, caustics, etc.) are naturally intertwined. Multidimensional variants of Hill’s formula, which relates the dynamic and geometric properties of a periodic trajectory, are discussed. Stability conditions are expressed in terms of the geometric properties of the boundary of a billiard. In particular, it turns out that a nondegenerate two-link trajectory of maximum length is always unstable. The degree of instability (the number of multipliers outside the unit disk) is estimated. The estimates are expressed in terms of the geometry of the caustic and the Morse indices of the length function of this trajectory.
Citation:
Kozlov V. V., Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 273, pp. 196–213
The equations of motion of a collisionless continuum
Journal of Applied Mathematics and Mechanics, 2011, vol. 75, no. 6, pp. 619–630
Abstract
pdf (689.16 Kb)
The equations of motion of a collisionless continuum are derived within an Eulerian approach. They differ from the classical equations of motion of an ideal gas, which take into account heat conduction phenomena. Several problems related to the weak convergence of the solutions of the equations of motion of a continuum when there is an unbounded increase in time are discussed. The problem of the correctness of the operation of truncating the exact infinite chain of equations of a collisionless gas is examined.
Citation:
Kozlov V. V., The equations of motion of a collisionless continuum, Journal of Applied Mathematics and Mechanics, 2011, vol. 75, no. 6, pp. 619–630
The Vlasov Kinetic Equation, Dynamics of Continuum and Turbulence
Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 602-622
Abstract
pdf (323.98 Kb)
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.
Statistical Irreversibility of the Kac Reversible Circular Model
Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 536-549
Abstract
pdf (202.6 Kb)
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.
The Lorentz force and its generalizations
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, pp. 627-634
Abstract
pdf (359.22 Kb)
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.
Statistical irreversibility of the Kac reversible circular model
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 1, pp. 101-117
Abstract
pdf (419.07 Kb)
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.
Kozlov V. V., Statistical irreversibility of the Kac reversible circular model, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 1, pp. 101-117
Some mathematical and information aspects of traffic modeling
T-Comm: Telecommunications and Transport, 2011, no. 4, pp. 29–31
Abstract
pdf (809.09 Kb)
Transport, communications and mathematics (in wider understanding - natural sciences) it becomes more and more obvious to despite different "age" of these three components. If the mathematics in Russia (USSR) was always, at least, since that moment as to Russia were invited L.Eyler (1976) and D. Bernulli (1725) that a traffic (traffic) as the appreciable phenomenon and an acute problem, appeared in the early nineties after opening of borders and a mass import of cars. With transition to market economy and small business mobility of the population, need for multipurpose cars and load of a street road network sharply increased.
Citation:
Bugaev A. S., Buslaev A. P., Kozlov V. V., Yashina M. V., Some mathematical and information aspects of traffic modeling, T-Comm: Telecommunications and Transport, 2011, no. 4, pp. 29–31
Bogoliubov type equations via infinite-dimensional equations for measures
Quantum bio-informatics IV, QP–PQ: Quantum Probability and White Noise Analysis, 2011, vol. 28, pp. 321–337
Abstract
The following sections are included:
Introduction
Symplectic locally convex spaces and Hamilton's equations.
Liouville's equations with respect to measures.
Systems of equations with respect to finite-dimensional distributions of probabilities.
Bogolyubov's systems of equations.
Wigner measures.
Generalization of Poincaré's model.
References
Citation:
Kozlov V. V., Smolyanov O. G., Bogoliubov type equations via infinite-dimensional equations for measures, Quantum bio-informatics IV, QP–PQ: Quantum Probability and White Noise Analysis, 2011, vol. 28, pp. 321–337
Distributed problems of monitoring and modern approaches to traffic modeling
2011 14th IEEE International IEEE Conference on Intelligent Transportation Systems (ITSC), 14th International IEEE Conference on Intelligent Transportation Systems-ITSC, 2011, pp. 477–481
Abstract
The paper discusses some mathematical models of traffic flow. We have introduced the concept of a stationary r-connected traffic flow on k-lane road as a development of the hydrodynamic approach and cellular automata method. A client-server based software “SSSR”-system, using smart-phone programming, for evaluating a distance of safety in continuous traffic was developed. A series of experiments were carried out using the SSSR-system, the results showing good agreement with those obtained by Greenshields in 1933. Other problems of traffic monitoring and control by the programmed SSSR-system are discussed. We also introduce a few open problems.
Citation:
Bugaev A. S., Buslaev A. P., Kozlov V. V., Yashina M. V., Distributed problems of monitoring and modern approaches to traffic modeling, 2011 14th IEEE International IEEE Conference on Intelligent Transportation Systems (ITSC), 14th International IEEE Conference on Intelligent Transportation Systems-ITSC, 2011, pp. 477–481
Infinite-dimensional Liouville equations with respect to measures
Doklady Mathematics, 2010, vol. 81, no. 3, pp. 476–480
Abstract
Citation:
Kozlov V. V., Smolyanov O. G., Infinite-dimensional Liouville equations with respect to measures, Doklady Mathematics, 2010, vol. 81, no. 3, pp. 476–480
On the variational principles of mechanics
Journal of Applied Mathematics and Mechanics, 2010, vol. 74, no. 5, pp. 505–512
Abstract
Variational principles, generalizing the classical d’Alembert–Lagrange, Hölder, and Hamilton–Ostrogradskii principles, are established. After the addition of anisotropic dissipative forces and taking the limit, when the coefficient of viscous friction tends to infinity, these variational principles transform into the classical principles, which describe the motion of systems with constraints. New variational relations are established for searching for the periodic trajectories of the reversible equations of dynamics.
Citation:
Kozlov V. V., On the variational principles of mechanics, Journal of Applied Mathematics and Mechanics, 2010, vol. 74, no. 5, pp. 505–512
Remarks on the degree of instability
Journal of Applied Mathematics and Mechanics, 2010, vol. 74, no. 1, pp. 10–12
Abstract
pdf (242.92 Kb)
Linear systems of differential equations allowing of functions in quadratic forms that do not increase along trajectories with time are considered. The relations between the indices of inertia of these forms and the degrees of instability of equilibrium states are indicated. These assertions generalize known results from the oscillation theory of linear systems with dissipation, and reveal the mechanism of loss of stability when non-increasing quadratic forms lose the property of a minimum.
Citation:
Kozlov V. V., Remarks on the degree of instability, Journal of Applied Mathematics and Mechanics, 2010, vol. 74, no. 1, pp. 10–12
Lagrangian mechanics and dry friction
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, pp. 855-868
Abstract
pdf (265.11 Kb)
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.
The Vlasov kinetic equation, dynamics of continuum and turbulence
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 489-512
Abstract
pdf (276.41 Kb)
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, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 489-512
On the mechanism of stability loss
Differential Equations, 2009, vol. 45, no. 4, pp. 510–519
Abstract
We consider linear systems of differential equations admitting functions in the form of quadratic forms that do not increase along trajectories in the course of time. We find new relations between the inertia indices of these forms and the instability degrees of the equilibria. These assertions generalize well-known results in the oscillation theory of linear systems with dissipation and clarify the mechanism of stability loss, whereby nonincreasing quadratic forms lose the property of minimum.
Citation:
Kozlov V. V., On the mechanism of stability loss, Differential Equations, 2009, vol. 45, no. 4, pp. 510–519
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
Abstract
pdf (172.17 Kb)
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.
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
Kinetics of collisionless gas: equalization of temperature, growth of the coarse-grained entropy and the Gibbs paradox
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 377-383
Abstract
pdf (208.37 Kb)
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.
Kozlov V. V., Kinetics of collisionless gas: equalization of temperature, growth of the coarse-grained entropy and the Gibbs paradox, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 377-383
Gyroscopic stabilization of degenerate equilibria and the topology of real algebraic varieties
Doklady Mathematics, 2008, vol. 77, no. 3, pp. 412–415
Abstract
Citation:
Kozlov V. V., Gyroscopic stabilization of degenerate equilibria and the topology of real algebraic varieties, Doklady Mathematics, 2008, vol. 77, no. 3, pp. 412–415
On the instability of equilibria of conservative systems under typical degenerations
Differential Equations, 2008, vol. 44, no. 8, pp. 1064–1071
Abstract
We study systems of differential equations admitting first integrals with degenerate critical points. We find conditions for the instability of equilibria for the cases in which the first integral loses the minimum property. Results of general nature are used in the proof of the impossibility of gyroscopic stabilization of equilibria in conservative mechanical systems under simple typical bifurcations.
Citation:
Kozlov V. V., On the instability of equilibria of conservative systems under typical degenerations, Differential Equations, 2008, vol. 44, no. 8, pp. 1064–1071
Topology of Real Algebraic Curves
Functional Analysis and Its Applications, 2008, vol. 42, no. 2, pp. 98–102
Abstract
pdf (117.67 Kb)
The problem on the existence of an additional first integral of the equations of geodesics on noncompact algebraic surfaces is considered. This problem was discussed as early as by Riemann and Darboux. We indicate coarse obstructions to integrability, which are related to the topology of the real algebraic curve obtained as the line of intersection of such a surface with a sphere of large radius. Some yet unsolved problems are discussed.
Keywords:
geodesic flow, analytic first integral, geodesic convexity, M-curve
Citation:
Kozlov V. V., Topology of Real Algebraic Curves, Functional Analysis and Its Applications, 2008, vol. 42, no. 2, pp. 98–102
Gauss Principle and Realization of Constraints
Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 431-434
Abstract
pdf (144.15 Kb)
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.
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
Abstract
pdf (192.61 Kb)
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.
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
Lagrange’s Identity and Its Generalizations
Regular and Chaotic Dynamics, 2008, vol. 13, no. 2, pp. 71-80
Abstract
pdf (144.77 Kb)
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.
Gauss Principle and Realization of Constraints
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 3, pp. 281-285
Abstract
pdf (78.44 Kb)
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.
Lagrange’s identity and its generalizations
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 2, pp. 157-168
Abstract
pdf (128.42 Kb)
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.
Fine-grained and coarse-grained entropy in problems of statistical mechanics
Theoretical and Mathematical Physics, 2007, vol. 151, no. 1, pp. 539–555
Abstract
pdf (502.32 Kb)
We consider dynamical systems with a phase space Γ that preserve a measure μ. A partition of Γ into parts of finite μ-measure generates the coarse-grained entropy, a functional that is defined on the space of probability measures on Γ and generalizes the usual (ordinary or fine-grained) Gibbs entropy. We study the approximation properties of the coarse-grained entropy under refinement of the partition and also the properties of the coarse-grained entropy as a function of time.
Kozlov V. V., Treschev D. V., Fine-grained and coarse-grained entropy in problems of statistical mechanics, Theoretical and Mathematical Physics, 2007, vol. 151, no. 1, pp. 539–555
Euler and mathematical methods in mechanics (on the 300th anniversary of the birth of Leonhard Euler)
Russian Mathematical Surveys, 2007, vol. 62, no. 4, pp. 639–661
Abstract
pdf (425.6 Kb)
This article concerns the life of Leonhard Euler and his achievements in theoretical mechanics. A number of topics are discussed related to the development of Euler’s ideas and methods: divergent series and asymptotics of solutions of non-linear differential equations; the hydrodynamics of a perfect fluid and Hamiltonian systems; vortex theory for systems on Lie groups with left-invariant kinetic energy; energy criteria of stability; Euler’s problem of two gravitating centres in curved spaces.
Citation:
Kozlov V. V., Euler and mathematical methods in mechanics (on the 300th anniversary of the birth of Leonhard Euler), Russian Mathematical Surveys, 2007, vol. 62, no. 4, pp. 639–661
Asymptotic stability and associated problems of dynamics of falling rigid body
Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, pp. 531-565
Abstract
pdf (1.81 Mb)
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
Asymptotic stability and associated problems of dynamics of falling rigid body
Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 3, pp. 255-296
Abstract
pdf (1.62 Mb)
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, Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 3, pp. 255-296
Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat
Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 2, pp. 123-140
Abstract
pdf (263.66 Kb)
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.
Kozlov V. V., Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat, Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 2, pp. 123-140
Information entropy in problems of classical and quantum statistical mechanics
Doklady Mathematics, 2006, vol. 74, no. 3, pp. 910–913
Abstract
Citation:
Kozlov V. V., Smolyanov O. G., Information entropy in problems of classical and quantum statistical mechanics, Doklady Mathematics, 2006, vol. 74, no. 3, pp. 910–913
Square integrable solutions to the Klein-Gordon equation on a manifold
Doklady Mathematics, 2006, vol. 73, no. 3, pp. 441–444
Abstract
pdf (292.17 Kb)
Citation:
Volovich I. V., Kozlov V. V., Square integrable solutions to the Klein-Gordon equation on a manifold, Doklady Mathematics, 2006, vol. 73, no. 3, pp. 441–444
Wigner function and diffusion in collisionfree media of quantum particles
Theory of Probability and its Applications, 2006, vol. 51, no. 1, pp. 168–181
Abstract
pdf (952.66 Kb)
A quantum Poincaré model (realizing behavior of ideal gas of noninteracting quantum Bolztman particles) is introduced. We use the fact that the evolution of the Wigner function corresponding to a quantum system with a quadratic Hamiltonian coincides with the evolution of a probability distribution on a phase space of the Hamiltonian system, the quantization of which gives the quantum system under consideration.
Kozlov V. V., Smolyanov O. G., Wigner function and diffusion in collisionfree media of quantum particles, Theory of Probability and its Applications, 2006, vol. 51, no. 1, pp. 168–181
Vorticity equation of 2D-hydrodynamics, Vlasov steady-state kinetic equation and developed turbulence
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 4, pp. 425-434
Abstract
pdf (152.1 Kb)
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.
Kozlov V. V., Vorticity equation of 2D-hydrodynamics, Vlasov steady-state kinetic equation and developed turbulence, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 4, pp. 425-434
Finite action Klein–Gordon solutions on Lorentzian manifolds
International Journal of Geometric Methods in Modern Physics, 2006, vol. 3, no. 7, pp. 1349–1357
Abstract
pdf (163.15 Kb)
The eigenvalue problem for the square integrable solutions is studied usually for elliptic equations. In this paper we consider such a problem for the hyperbolic Klein–Gordon equation on Lorentzian manifolds. The investigation could help to answer the question why elementary particles have a discrete mass spectrum. An infinite family of square integrable solutions for the Klein–Gordon equation on the Friedman type manifolds is constructed. These solutions have a discrete mass spectrum and a finite action. In particular the solutions on de Sitter space are investigated.
Citation:
Kozlov V. V., Volovich I. V., Finite action Klein–Gordon solutions on Lorentzian manifolds, International Journal of Geometric Methods in Modern Physics, 2006, vol. 3, no. 7, pp. 1349–1357
Uniform distribution and Voronoi convergence
Sbornik: Mathematics, 2005, vol. 196, no. 10, pp. 1495–1502
Abstract
pdf (127.63 Kb)
There is a broad generalization of a uniformly distributed sequence according to Weyl where the frequency of elements of this sequence falling into an interval is defined by using a matrix summation method of a general form. In the present paper conditions for uniform distribution are found in the case where a regular Voronoi method is chosen as the summation method. The proofs are based on estimates of trigonometric sums of a certain special type. It is shown that the sequence of the fractional parts of the logarithms of positive integers is not uniformly distributed for any choice of a regular Voronoi method.
Citation:
Kozlov V. V., Madsen T., Uniform distribution and Voronoi convergence, Sbornik: Mathematics, 2005, vol. 196, no. 10, pp. 1495–1502
Weighted Means, Strict Ergodicity, and Uniform Distributions
Mathematical Notes, 2005, vol. 78, no. 3, pp. 329–337
Abstract
pdf (155.58 Kb)
We strengthen the well-known Oxtoby theorem for strictly ergodic transformations by replacing the standard Cesaro convergence by the weaker Riesz or Voronoi convergence with monotonically increasing or decreasing weight coefficients. This general result allows, in particular, to strengthen the classical Weyl theorem on the uniform distribution of fractional parts of polynomials with irrational coefficients.
On periodic trajectories of a billiard in a magnetic field
Journal of Applied Mathematics and Mechanics, 2005, vol. 69, no. 6, pp. 844–851
Abstract
pdf (384.03 Kb)
The problem of the existence of periodic trajectories of a charged particle in a magnetic field, when the particle moves inside a closed convex region and is elastically reflected from its boundary, is considered. The presence of an infinite number of different periodic trajectories at low magnetic field strengths is established using Poincaré's geometric theorem. The conditions for two-link trajectories to be stable in the case of a uniform magnetic field are obtained.
Citation:
Kozlov V. V., Polikarpov S. A., On periodic trajectories of a billiard in a magnetic field, Journal of Applied Mathematics and Mechanics, 2005, vol. 69, no. 6, pp. 844–851
Weighted averages, uniform distribution, and strict ergodicity
Russian Mathematical Surveys, 2005, vol. 60, no. 6, pp. 1121–1146
Abstract
pdf (293.33 Kb)
A circle of problems related to the application of the Riesz and Voronoi summation methods in ergodic theory, number theory, and probability theory is considered. The first digit paradox is discussed, strengthenings of the classical result of Weyl on the uniform distribution of the fractional parts of the values of a polynomial are indicated, and the possibility of sharpening the Birkhoff–Khinchin ergodic theorem is considered. In conclusion, some unsolved problems are listed.
Citation:
Kozlov V. V., Weighted averages, uniform distribution, and strict ergodicity, Russian Mathematical Surveys, 2005, vol. 60, no. 6, pp. 1121–1146
Remarks on a Lie theorem on the exact integrability of differential equations
Differential Equations, 2005, vol. 41, no. 4, pp. 588–590
Abstract
pdf (85.01 Kb)
Citation:
Kozlov V. V., Remarks on a Lie theorem on the exact integrability of differential equations, Differential Equations, 2005, vol. 41, no. 4, pp. 588–590
Restrictions of Quadratic Forms to Lagrangian Planes, Quadratic Matrix Equations, and Gyroscopic Stabilization
Functional Analysis and Its Applications, 2005, vol. 39, no. 4, pp. 271–283
Abstract
pdf (170.93 Kb)
We discuss the symplectic geometry of linear Hamiltonian systems with nondegenerate Hamiltonians. These systems can be reduced to linear second-order differential equations characteristic of linear oscillation theory. This reduction is related to the problem on the signatures of restrictions of quadratic forms to Lagrangian planes. We study vortex symplectic planes invariant with respect to linear Hamiltonian systems. These planes are determined by the solutions of quadratic matrix equations of a special form. New conditions for gyroscopic stabilization are found.
Kozlov V. V., Restrictions of Quadratic Forms to Lagrangian Planes, Quadratic Matrix Equations, and Gyroscopic Stabilization, Functional Analysis and Its Applications, 2005, vol. 39, no. 4, pp. 271–283
Gibbs and Poincaré statistical equilibria in systems with slowly varying parameters
Doklady Mathematics, 2004, vol. 69, no. 2, pp. 278–281
Abstract
pdf (182.91 Kb)
Citation:
Kozlov V. V., Gibbs and Poincaré statistical equilibria in systems with slowly varying parameters, Doklady Mathematics, 2004, vol. 69, no. 2, pp. 278–281
Polynomial Conservation Laws in Quantum Systems
Theoretical and Mathematical Physics, 2004, vol. 140, no. 3, pp. 1283–1298
Abstract
pdf (219.79 Kb)
We consider systems with a finite number of degrees of freedom and potential energy that is a finite sum of exponentials with purely imaginary or real exponents. Such systems include the generalized Toda chains and systems with a toric configuration space. We consider the problem of describing all the quantum conservation laws, i.e. the differential operators that are polynomial in the derivatives and commute with the Hamiltonian operator. We prove that in the case where the potential energy spectrum is invariant under reflection with respect to the origin, such nontrivial operators exist only if the system under consideration decomposes into a direct sum of decoupled subsystems. In the general case (without the spectrum symmetry assumption), we prove that the existence of a complete set of independent conservation laws implies the complete integrability of the corresponding classical system.
Keywords:
Hamiltonian operator, polynomial differential operator, system with exponential interaction, potential spectrum
Citation:
Kozlov V. V., Treschev D. V., Polynomial Conservation Laws in Quantum Systems, Theoretical and Mathematical Physics, 2004, vol. 140, no. 3, pp. 1283–1298
Dynamics of variable systems, and Lie groups
Journal of Applied Mathematics and Mechanics, 2004, vol. 68, no. 6, pp. 803–808
Abstract
pdf (478.49 Kb)
Mechanical systems whose configuration space is a Lie group and whose Lagrangian is invariant to left translations on that group are considered. It is assumed that the mass geometry of the system may change under the action of only internal forces. The equations of motion admit of a complete set of Noether integrals which are linear in the velocities. For fixed values of these integrals, the equations of motion reduce to a non-autonomous system of first-order differential equations on the Lie group. Conditions under which the system can be brought from any initial position to another preassigned position by changing its mass geometry are discussed. The “falling cat” problem and the problem of the motion of a body of variable shape in an unlimited volume of ideal fluid are considered as examples.
Citation:
Kozlov V. V., Dynamics of variable systems, and Lie groups, Journal of Applied Mathematics and Mechanics, 2004, vol. 68, no. 6, pp. 803–808
Linear systems with a quadratic integral and the symplectic geometry of Artin spaces
Journal of Applied Mathematics and Mechanics, 2004, vol. 68, no. 3, pp. 329–340
Abstract
pdf (648.21 Kb)
New relations are established between the spectrum of a linear system and the indices of inertia of its quadratic integral. A detailed investigation is made of the case in which the positive and negative indices of inertia of the quadratic integral are identical. Conditions are found under which the singular planes will be Lagrangian relative to some natural symplectic structure. They are closely related to the conditions for strong stability of a linear system. The general results are applied to the classical problem of gyroscopic stabilization.
Citation:
Kozlov V. V., Linear systems with a quadratic integral and the symplectic geometry of Artin spaces, Journal of Applied Mathematics and Mechanics, 2004, vol. 68, no. 3, pp. 329–340
Billiards, invariant measures, and equilibrium thermodynamics. II
Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 91-100
Abstract
pdf (283.7 Kb)
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
Notes on diffusion in collisionless medium
Regular and Chaotic Dynamics, 2004, vol. 9, no. 1, pp. 29-34
Abstract
pdf (148.96 Kb)
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
On weighted mean values of weakly dependent random variables
Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2004, vol. 59, no. 5, pp. 36–39
Abstract
pdf (288.46 Kb)
Citation:
Kozlov V. V., Madsen T., Sorokin A. A., On weighted mean values of weakly dependent random variables, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2004, vol. 59, no. 5, pp. 36–39
The spectrum of a linear Hamiltonian system and symplectic geometry of a complex Artin space
Doklady Mathematics, 2003, vol. 68, no. 3, pp. 385–387
Abstract
pdf (133.43 Kb)
Citation:
Kozlov V. V., The spectrum of a linear Hamiltonian system and symplectic geometry of a complex Artin space, Doklady Mathematics, 2003, vol. 68, no. 3, pp. 385–387
Weak limits of probability distributions in systems with nonstationary perturbations
Doklady Mathematics, 2003, vol. 67, no. 2, pp. 283–285
Abstract
pdf (148.42 Kb)
Citation:
Kozlov V. V., Weak limits of probability distributions in systems with nonstationary perturbations, Doklady Mathematics, 2003, vol. 67, no. 2, pp. 283–285
Evolution of Measures in the Phase Space of Nonlinear Hamiltonian Systems
Theoretical and Mathematical Physics, 2003, vol. 136, no. 3, pp. 1325–1335
Abstract
pdf (168.45 Kb)
We establish the existence of weak limits of solutions (in the class $L_p$, $p\ge1$) of the Liouville equation for nondegenerate quasihomogeneous Hamilton equations. We find the limit probability distributions in the configuration space. We give conditions for a uniform distribution of Gibbs ensembles for geodesic flows on compact manifolds.
Kozlov V. V., Treschev D. V., Evolution of Measures in the Phase Space of Nonlinear Hamiltonian Systems, Theoretical and Mathematical Physics, 2003, vol. 136, no. 3, pp. 1325–1335
Weak Convergence of Solutions of the Liouville Equation for Nonlinear Hamiltonian Systems
Theoretical and Mathematical Physics, 2003, vol. 134, no. 3, pp. 339–350
Abstract
pdf (168.45 Kb)
We suggest sufficient conditions for the existence of weak limits of solutions of the Liouville equation as time increases indefinitely. The presence of the weak limit of the probability distribution density leads to a new interpretation of the second law of thermodynamics for entropy increase.
Kozlov V. V., Treschev D. V., Weak Convergence of Solutions of the Liouville Equation for Nonlinear Hamiltonian Systems, Theoretical and Mathematical Physics, 2003, vol. 134, no. 3, pp. 339–350
The motion in a perfect fluid of a body containing a moving point mass
Journal of Applied Mathematics and Mechanics, 2003, vol. 67, no. 4, pp. 553–564
Abstract
pdf (271.99 Kb)
The motion of a system (a rigid body, symmetrical about three mutually perpendicular planes, plus a point mass situated inside the body) in an unbounded volume of a perfect fluid, which executes vortex-free motion and is at rest at infinity, is considered. The motion of the body occurs due to displacement of the point mass with respect to the body. Two cases are investigated: (a) there are no external forces, and (b) the system moves in a uniform gravity field. An analytical investigation of the dynamic equations under conditions when the point performs a specified plane periodic motion inside the body showed that in case (a) the system can be displaced as far as desired from the initial position. In case (b) it is proved that, due to the permanent addition of energy of the corresponding relative motion of the point, the body may float upwards. On the other hand, if the velocity of relative motion of the point is limited, the body will sink. The results of numerical calculations, when the point mass performs random walks along the sides of a plane square grid rigidly connected with the body, are presented.
Citation:
Kozlov V. V., Onishchenko D. A., The motion in a perfect fluid of a body containing a moving point mass, Journal of Applied Mathematics and Mechanics, 2003, vol. 67, no. 4, pp. 553–564
Rationality conditions for the ratio of elliptic integrals and the great Poncelet theorem
Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2003, vol. 58, no. 4, pp. 1–7
Abstract
pdf (508.66 Kb)
Citation:
Kozlov V. V., Rationality conditions for the ratio of elliptic integrals and the great Poncelet theorem, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2003, vol. 58, no. 4, pp. 1–7
Nonintegrability of a system of interacting particles with Dyson potential
Fundamental and applied problems in vortex theory, Ser. Komp'yut. Mat. Fiz. Biol., Izhevsk: Institute of Computer Science, 2003, pp. 387–391
Abstract
Citation:
Borisov A. V., Kozlov V. V., Nonintegrability of a system of interacting particles with Dyson potential, Fundamental and applied problems in vortex theory, Ser. Komp'yut. Mat. Fiz. Biol., Izhevsk: Institute of Computer Science, 2003, pp. 387–391
On the stochastization of plane-parallel flows of an ideal fluid
Fundamental and applied problems in vortex theory, Ser. Komp'yut. Mat. Fiz. Biol., Izhevsk: Institute of Computer Science, 2003, pp. 303–307
Abstract
Citation:
Kozlov V. V., On the stochastization of plane-parallel flows of an ideal fluid, Fundamental and applied problems in vortex theory, Ser. Komp'yut. Mat. Fiz. Biol., Izhevsk: Institute of Computer Science, 2003, pp. 303–307
Motion of a body with undeformable shell and variable mass geometry in an unbounded perfect fluid
Journal of Dynamics and Differential Equations, 2003, vol. 15, no. 2–3, pp. 553–570
Abstract
Keywords:
undeformable shell, uniform force field, variable mass geometry
Citation:
Kozlov V. V., Onishchenko D. A., Motion of a body with undeformable shell and variable mass geometry in an unbounded perfect fluid, Journal of Dynamics and Differential Equations, 2003, vol. 15, no. 2–3, pp. 553–570
On new forms of the ergodic theorem
Journal of Dynamical and Control Systems, 2003, vol. 9, no. 3, pp. 449–453
Abstract
pdf (95.44 Kb)
We present generalizations of the classical Birkhoff and von Neumann ergodic theorems, where the time average is replaced by a more general average, including some density.
Keywords:
ergodic theorems, summation methods
Citation:
Kozlov V. V., Treschev D. V., On new forms of the ergodic theorem, Journal of Dynamical and Control Systems, 2003, vol. 9, no. 3, pp. 449–453
Summation of divergent series and ergodic theorems
Journal of Mathematical Sciences, 2003, vol. 114, no. 4, pp. 1473–1490
Abstract
pdf (406.3 Kb)
In this article questions on the possibility of sharpening classic ergodic theorems is considered. To sharpen these theorems the author uses methods of summation of divergent sequences and series. The main topic is connected with the individual ergodic Birkhoff–Khinchin theorem. The theorem is studied in connection with the Riesz and Voronoi summation methods. These methods are weaker than those of the Cesaro method of arithmetic means. It is shown that already for the Bernoulli transformation of the unit interval, meaningful problems arise. These problems are interesting in connection with the possibility of extension of the strong law of large numbers. The questions of suitable summation factors and of the solution of homological equations by means of divergent series is also discussed.
Citation:
Kozlov V. V., Summation of divergent series and ergodic theorems, Journal of Mathematical Sciences, 2003, vol. 114, no. 4, pp. 1473–1490
On the motion of a body with a rigid hull and changing geometry of masses in an ideal fluid
Doklady Mathematics, 2002, vol. 47, no. 2, pp. 132–135
Abstract
pdf (335.6 Kb)
Citation:
Kozlov V. V., Ramodanov S. M., On the motion of a body with a rigid hull and changing geometry of masses in an ideal fluid, Doklady Mathematics, 2002, vol. 47, no. 2, pp. 132–135
Steady motions of a continuum, resonances, and Lagrangian turbulence
Journal of Applied Mathematics and Mechanics, 2002, vol. 66, no. 6, pp. 897–904
Abstract
pdf (1.04 Mb)
A method which enables one to establish a non-regularity property of the motion of fluid particles (known as chaotic advection or Lagrangian turbulence) for typical steady flows is developed. The method is based on expanding solutions of the equations of motion of a continuous medium in powers of a small parameter and using the conditions for the destruction of invariant resonant tori when perturbations are added. It is shown that the velocity field, defined as the solution of the Burgers equations, generates a generally non-regular dynamical system. For an ideal barotropic fluid in an irrotational force field, the method proposed yields a well-known necessary condition for chaotization: the velocity field is collinear with its curl. Special attention is given to investigating the chaotization of typical steady flows of a heat-conducting perfect gas.
Citation:
Denisova N. V., Kozlov V. V., Steady motions of a continuum, resonances, and Lagrangian turbulence, Journal of Applied Mathematics and Mechanics, 2002, vol. 66, no. 6, pp. 897–904
Summation of divergent series and ergodic theorems
Trudy Seminara imeni I. G. Petrovskogo, 2002, vol. 22, pp. 142–168
Abstract
pdf (972.52 Kb)
In this article questions on the possibility of sharpening classic ergodic theorems is considered. To sharpen these theorems the author uses methods of summation of divergent sequences and series. The main topic is connected with the individual ergodic Birkhoff–Khinchin theorem. The theorem is studied in connection with the Riesz and Voronoi summation methods. These methods are weaker than those of the Cesaro method of arithmetic means. It is shown that already for the Bernoulli transformation of the unit interval, meaningful problems arise. These problems are interesting in connection with the possibility of extension of the strong law of large numbers. The questions of suitable summation factors and of the solution of homological equations by means of divergent series is also discussed.
Citation:
Kozlov V. V., Summation of divergent series and ergodic theorems, Trudy Seminara imeni I. G. Petrovskogo, 2002, vol. 22, pp. 142–168
On the Integration Theory of Equations of Nonholonomic Mechanics
Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 161-176
Abstract
pdf (456.43 Kb)
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
On Justification of Gibbs Distribution
Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, pp. 1-10
Abstract
pdf (324.27 Kb)
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
On effect of “emerging” of a heavy rigid body in a fluid
Mechanics of Solids, 2002, vol. 37, no. 1, pp. 54-59
Abstract
pdf (315.81 Kb)
The fall of a heavy rigid body in unbounded volume of an ideal fluid is considered. The fluid performs an irrotational motion and rests at infinity. It is assumed that the wider side of the body is horizontal at the initial time instant at which a velocity is communicated to the body in the horizontal direction. Then at the next time instants the body starts descending. However, if the associated mass of the body in the transverse direction is sufficiently large, the body then sharply comes to the surface with his narrower side ahead and rises to the height exceeding that at the initial time instant. The analysis of the surfacing effect involves the expansion of the solutions of Kirchhoff's equations into series in powers of time and the evaluation of the coefficients of these series by means of Cauchy majorants.
Citation:
Deryabin M. V., Kozlov V. V., On effect of “emerging” of a heavy rigid body in a fluid, Mechanics of Solids, 2002, vol. 37, no. 1, pp. 54-59
On the theory of integration of the equations of nonholonomic mechanics
Nonholonomic dynamical systems, Ser. Komp'yut. Mat. Fiz. Biol., Izhevsk: Institute of Computer Science, 2002, pp. 149–172
Abstract
Citation:
Kozlov V. V., On the theory of integration of the equations of nonholonomic mechanics, Nonholonomic dynamical systems, Ser. Komp'yut. Mat. Fiz. Biol., Izhevsk: Institute of Computer Science, 2002, pp. 149–172
On theorems of dynamics
Nonholonomic dynamical systems, Ser. Komp'yut. Mat. Fiz. Biol., Izhevsk: Institute of Computer Science, 2002, pp. 51–58
Abstract
Citation:
Kozlov V. V., Kolesnikov N. N., On theorems of dynamics, Nonholonomic dynamical systems, Ser. Komp'yut. Mat. Fiz. Biol., Izhevsk: Institute of Computer Science, 2002, pp. 51–58
Gyroscopic stabilization and parametric resonance
Journal of Applied Mathematics and Mechanics, 2001, vol. 65, no. 5, pp. 715–721
Abstract
pdf (438.42 Kb)
The problem of the conditions for the gyroscopic stabilization of unstable equilibria using gyroscopic forces with a degenerate matrix is considered. Systems with an odd number of degrees of freedom are an important example. The gyroscopic forces can generally be removed using a non-autonomous orthogonal transformation. The equations of motion then become a system of Sturm—Liouville type equations with a time-dependent potential. The conditions imposed on the skew-symmetric matrix of the gyroscopic forces for which the new potential depends periodically on time are indicated. These conditions are necessarily satisfied for non-zero matrices of the gyroscopic forces of minimum rank equal to two. Hence, the problem of gyroscopic stabilization reduces, in a number of cases, to investigating the stability of the equilibrium positions of systems with a periodic potential. The use of parametric-resonance theory enables new constructive conditions to be obtained for the stability of the equilibria of mechanical systems acted upon by additional degenerate gyroscopic forces. These conditions have the form of the conditions for an extremum of certain functions which depend solely on the position of the system. Particular attention is devoted to the stability conditions for large gyroscopic forces. It is shown, using examples, that the conditions of gyroscopic stabilization obtained are only sufficient. However, if the potential energy in the equilibrium position has a maximum and the matrix of the gyroscopic forces are non-degenerate, they are close to the necessary stability conditions.
Citation:
Kozlov V. V., Gyroscopic stabilization and parametric resonance, Journal of Applied Mathematics and Mechanics, 2001, vol. 65, no. 5, pp. 715–721
On the motion of a variable body in an ideal fluid
Journal of Applied Mathematics and Mechanics, 2001, vol. 65, no. 4, pp. 579–587
Abstract
pdf (626.32 Kb)
The dynamics of a deformable body in an unbounded volume of an ideal fluid, which performs irrotational motion and is at rest at infinity, is investigated. It is assumed that a change in the geometry of the masses and shape of the body occurs due to the action of internal forces and that the displacements of the particles of the body are known functions of time in a certain moving frame of reference. The equations of motion of the moving trihedron are represented in the form of Kirchhoff's equations. The conservation laws when there are no external forces are indicated. Using these laws, the equations of motion are reduced to a non-autonomic system of first-order differential equations in the group of displacements of the configurational space. In the case of plane-parallel motion of the body, these equations are explicitly integrated in quadratures. A special case, when the boundary of the body does not change, is considered. It is established that, in the case of non-equal added masses, due to the change in the geometry of the body masses, the body can move from any position into any other position.
Citation:
Kozlov V. V., Ramodanov S. M., On the motion of a variable body in an ideal fluid, Journal of Applied Mathematics and Mechanics, 2001, vol. 65, no. 4, pp. 579–587
Kinetics of Collisionless Continuous Medium
Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 235-251
Abstract
pdf (468.67 Kb)
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
The first Lyapunov method for strongly non-linear systems of differential equations
Resenhas, 2001, vol. 5, no. 1, pp. 1–24
Abstract
Citation:
Kozlov V. V., Furta S. D., The first Lyapunov method for strongly non-linear systems of differential equations, Resenhas, 2001, vol. 5, no. 1, pp. 1–24
Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space
Sbornik: Mathematics, 2000, vol. 191, no. 2, pp. 189–208
Abstract
pdf (339 Kb)
The problem considered here is that of finding conditions ensuring that a reversible Hamiltonian system has integrals polynomial in momenta. The kinetic energy is a zero-curvature Riemannian metric and the potential a smooth function on a two-dimensional torus. It is known that the existence of integrals of degrees $1$ and $2$ is related to the existence of cyclic coordinates and the separation of variables. The following conjecture is also well known: if there exists an integral of degree $n$ independent of the energy integral, then there exists an additional integral of degree $1$ or $2$. In the present paper this result is established for $n = 3$ (which generalizes a theorem of Byalyi), and for $n = 4$, $5$, and $6$ this is proved under some additional assumptions about the spectrum of the potential.
Citation:
Denisova N. V., Kozlov V. V., Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space, Sbornik: Mathematics, 2000, vol. 191, no. 2, pp. 189–208
Two-link billiard trajectories: extremal properties and stability
Journal of Applied Mathematics and Mechanics, 2000, vol. 64, no. 6, pp. 903–907
Abstract
pdf (342.81 Kb)
Two-link periodic trajectories of a plane convex billiard, when a point mass moves along a segment which is orthogonal to the boundary of the billiard at its end points, are considered. It is established that, if the caustic of the boundary lies within the billiard, then, in a typical situation, there is an even number of two-link trajectories and half of them are hyperbolic (and, consequently, unstable) and the other half are of elliptic type. An example is given of a billiard for which the caustic intersects the boundary and all of the two-link trajectories are hyperbolic. The analysis of the stability is based on an analysis of the extremum of a function of the length of a segment of a convex billiard which is orthogonal to the boundary at one of its ends.
Citation:
Kozlov V. V., Two-link billiard trajectories: extremal properties and stability, Journal of Applied Mathematics and Mechanics, 2000, vol. 64, no. 6, pp. 903–907
Billiards, Invariant Measures, and Equilibrium Thermodynamics
Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 129-138
Abstract
pdf (207.72 Kb)
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
The Newton and Ivory theorems of attraction in spaces of constant curvature
Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2000, vol. 55, no. 5, pp. 16–20
Abstract
pdf (343.59 Kb)
Citation:
Kozlov V. V., The Newton and Ivory theorems of attraction in spaces of constant curvature, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2000, vol. 55, no. 5, pp. 16–20
Integrable analogue of the Gauss principle
Facta Universitatis. Series Mechanics, Automatic Control and Robotics, 2000, vol. 2, no. 10, pp. 1055–1060
Abstract
pdf (223.44 Kb)
The famous Gauss principle states that an actual motion is the one among conceivable motions that deviates least from the released motion. Herz based his forceless dynamics on this principle [1]. Gauss called the deviations of the conceivable motions from the released one the constraint. An explicit expression of the constraint in generalized coordinates was obtained first by Lipshitz [2]. In this paper, two new theorems are pointed out.
Citation:
Kozlov V. V., Integrable analogue of the Gauss principle, Facta Universitatis. Series Mechanics, Automatic Control and Robotics, 2000, vol. 2, no. 10, pp. 1055–1060
Nonintegrability of a System of Interacting Particles with the Dyson Potential
Doklady Physics, 1999, vol. 59, no. 3, pp. 485-486
Abstract
pdf (162.58 Kb)
Citation:
Borisov A. V., Kozlov V. V., Nonintegrability of a System of Interacting Particles with the Dyson Potential, Doklady Physics, 1999, vol. 59, no. 3, pp. 485-486
On the instability of isolated equilibria of dynamical systems with an invariant measure in an odd-dimensional space
Mathematical Notes, 1999, vol. 65, no. 5, pp. 565–570
Abstract
pdf (396.9 Kb)
We discuss the conjecture asserting that isolated equilibrium states of autonomous systems admitting invariant measures are unstable in spaces of odd dimension. This conjecture is proved for systems for which quasihomogeneous truncations with isolated singularities can be found. We consider a counterexample in the class of systems with infinitely differentiable right-hand sides and zero Maclaurin series at the equilibrium state.
Keywords:
dynamical system, invariant measure, homogeneity, equilibrium, stability in the sense of Lyapunov, quasihomogeneous truncation, invariant torus
Citation:
Kozlov V. V., Treschev D. V., On the instability of isolated equilibria of dynamical systems with an invariant measure in an odd-dimensional space, Mathematical Notes, 1999, vol. 65, no. 5, pp. 565–570
A condition for the freezing of a direction field, small denominators, and the chaotization of steady flows of a viscous fluid
Journal of Applied Mathematics and Mechanics, 1999, vol. 63, no. 2, pp. 229–235
Abstract
pdf (544.53 Kb)
A classical theorem of Helmholtz states that vortex lines are frozen into a flow of barotropic ideal fluid in a potential force field. This result leads to the following general problem: it is required to find conditions under which a given dynamical system admits of a direction field frozen into its phase flow. By the rectification theorem for trajectories, a whole family of frozen direction fields always exists locally. It turns out that the problem of the existence of non-trivial frozen direction fields defined in the whole phase space is closely related to the well-known problem of small denominators. Results of a general nature are applied to Hamiltonian systems, and also to steady flows of a viscous fluid.
Citation:
Kozlov V. V., A condition for the freezing of a direction field, small denominators, and the chaotization of steady flows of a viscous fluid, Journal of Applied Mathematics and Mechanics, 1999, vol. 63, no. 2, pp. 229–235
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
Abstract
pdf (242 Kb)
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
Hydrodynamic Theory of a Class of Finite-Dimensional Dissipative Systems
Proceedings of the Steklov Institute of Mathematics, 1998, vol. 223, pp. 178–184
Abstract
pdf (231.63 Kb)
Finite-dimensional systems with viscous friction are studied in the case when the Rayleigh function modeling this friction is proportional to the kinetic energy. Hydrodynamic analogies are presented for these systems.
Citation:
Kozlov V. V., Hydrodynamic Theory of a Class of Finite-Dimensional Dissipative Systems, Proceedings of the Steklov Institute of Mathematics, 1998, vol. 223, pp. 178–184
The stability of uniformly accelerated motions
Journal of Applied Mathematics and Mechanics, 1998, vol. 62, no. 5, pp. 671–675
Abstract
pdf (353.15 Kb)
Differential equations with quadratic right-hand sides and additional constant terms are considered. Important examples are self-driven gyroscopes and the problem of the motion of a rigid body in an unbounded volume of ideal fluid subject to a force and a torque which are constant in an attached frame of reference. Under certain simple conditions, these equations have solutions that increase linearly with time. In problems of dynamics they describe uniformly accelerated motions of mechanical systems. The stability of such motions is investigated in the first approximation and using bundles of integrals. The general results are used to investigate the stability of uniformly accelerated screw motions of a rigid body in a fluid.
Citation:
Kozlov V. V., The stability of uniformly accelerated motions, Journal of Applied Mathematics and Mechanics, 1998, vol. 62, no. 5, pp. 671–675
Branching of solutions and polynomial integrals of the equations of dynamics
Journal of Applied Mathematics and Mechanics, 1998, vol. 62, no. 1, pp. 1–8
Abstract
pdf (620.19 Kb)
The conditions for branching of the solutions of the equations of motion of natural mechanical systems in the complex time plane are obtained. The relation between the structure of the branching and the number of independent momentum-polynomial first integrals is investigated. Results of a general form are illustrated by examples from dynamics.
Citation:
Kozlov V. V., Branching of solutions and polynomial integrals of the equations of dynamics, Journal of Applied Mathematics and Mechanics, 1998, vol. 62, no. 1, pp. 1–8
On periodic solutions of Duffing's equations
Proceedings of Scientific Seminar at A. A. Blagonravov Institute of Engineering, 1998, pp. 75–88
Abstract
pdf (283.81 Kb)
Citation:
Kozlov V. V., On periodic solutions of Duffing's equations, Proceedings of Scientific Seminar at A. A. Blagonravov Institute of Engineering, 1998, pp. 75–88
Stabilization of the unstable equilibria of charges by intense magnetic fields
Journal of Applied Mathematics and Mechanics, 1997, vol. 61, no. 3, pp. 377–384
Abstract
pdf (576.42 Kb)
The problem of stabilizing unstable (by Earnshaw's theorem) equilibria of a free charge in an electrostatic field by adding a steady magnetic field is considered. The additional Lorentz force that thereby arises has a gyroscopic form. An example of the possibility of stabilization in a rigorous relativistic formulation of the problem is given. Criteria for the stabilization of unstable equilibria of linearized systems are obtained. The conditions for charge stability in intense magnetic fields are investigated and estimates of the stabilization probability are given. Some multidimensional analogues of these results are presented. In particular, the problem of gyroscopic stabilization when the matrix of the gyroscopic forces is degenerate is considered. Some extremal criteria of the stability of the equilibrium positions are given.
Citation:
Kozlov V. V., Stabilization of the unstable equilibria of charges by intense magnetic fields, Journal of Applied Mathematics and Mechanics, 1997, vol. 61, no. 3, pp. 377–384
Averaging in a neighborhood of stable invariant tori
Regular and Chaotic Dynamics, 1997, vol. 2, no. 3-4, pp. 41-46
Abstract
pdf (564.5 Kb)
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, no. 3-4, pp. 41-46
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
Abstract
pdf (811.01 Kb)
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
The study is reported of a diffusion in a model of degenerate Hamiltonian systems. The Hamiltonian under consideration is the sum of a linear function of action variables and a periodic function of angle variables. Under certain choices of these functions the diffusion of action variables exists. In the case of two degrees of freedom during the process of diffusion, the vector of the action variables returns many times near its initial value. In the case of three degrees of freedom the choice of Hamiltonian allows one to obtain a diffusion rate faster than any prescribed one.
Citation:
Kozlov V. V., Moshchevitin N. G., Diffusion in Hamiltonian systems, Chaos, 1997, vol. 8, no. 1, pp. 245–247
Topology of domains of possible motions of integrable systems
Sbornik: Mathematics, 1996, vol. 187, no. 5, pp. 679–684
Abstract
pdf (298.87 Kb)
A study is made of analytic invertible systems with two degrees of freedom on a fixed three-dimensional manifold of level of the energy integral. It is assumed that the manifold in question is compact and has no singular points (equilibria of the initial system). The natural projection of the energy manifold onto the two-dimensional configuration space is called the domain of possible motion. In the orientable case it is sphere with $k$ holes and $p$ attached handles. It is well known that for $k=0$ and $p\geqslant 2$, the system possesses no non-constant analytic integrals on the corresponding level of the energy integral. The situation in the case of domains of possible motions with a boundary turns out to be very different. The main result can be stated as follows: there are examples of analytically integrable systems with arbitrary values of $p$ and $k\geqslant 1$.
Citation:
Kozlov V. V., Ten V. V., Topology of domains of possible motions of integrable systems, Sbornik: Mathematics, 1996, vol. 187, no. 5, pp. 679–684
An extension of the Hamilton-Jacobi method
Journal of Applied Mathematics and Mechanics, 1996, vol. 60, no. 6, pp. 911–920
Abstract
pdf (585.47 Kb)
A method of solving the canonical Hamilton equations, based on a search for invariant manifolds, which are uniquely projected onto position space, is proposed. These manifolds are specified by covector fields, which satisfy a system of first-order partial differential equations, similar in their properties to Lamb's equations in the dynamic of an ideal fluid. If the complete integral of Lamb's equations is known, then, with certain additional assumptions, one can integrate the initial Hamilton equations explicitly. This method reduces to the well-known Hamilton-Jacobi method for gradient fields. Some new conditions for Hamilton's equations to be accurately integrable are indicated. The general results are applied to the problem of the motion of a variable body.
Citation:
Kozlov V. V., An extension of the Hamilton-Jacobi method, Journal of Applied Mathematics and Mechanics, 1996, vol. 60, no. 6, pp. 911–920
Lyapunov's first method for strongly non-linear systems
Journal of Applied Mathematics and Mechanics, 1996, vol. 60, no. 1, pp. 7–18
Abstract
Lyapunov's classical First Method is developed for strongly non-linear systems. Techniques for “truncating” strongly non-linear systems that possess a well-defined group of symmetries are described. Given such a group, it is possible, under fairly general assumptions, to determine, by purely algebraic methods, particular solutions of the truncated systems with prescribed asymptotic expansions. It is shown that these solutions can be extended to solutions of the full system by using certain series. Sufficient conditions for the existence of parametric families of solutions of the full system that possess certain asymptotic properties are also derived. The theory is illustrated by a wide range of examples. A new proof is given of one of the inversions of the Lagrange-Dirichlet theorem on the stability of equilibrium. It is shown that the method developed here may also be used to construct collision trajectories in problems of celestial mechanics in real time.
Citation:
Kozlov V. V., Furta S. D., Lyapunov's first method for strongly non-linear systems, Journal of Applied Mathematics and Mechanics, 1996, vol. 60, no. 1, pp. 7–18
Symmetries and Regular Behavior of Hamilton's Systems
Regular and Chaotic Dynamics, 1996, vol. 1, no. 1, pp. 3-14
Abstract
pdf (708.49 Kb)
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
Analytical properties of solutions for Euler-Poincaré equations on solvable Lie algebras
Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1996, vol. 51, no. 3, pp. 12–16
Abstract
pdf (268.16 Kb)
The paper deals with the ramification conditions for complex time-dependent solutions to Euler-Poincaré equations. A class of solvable Lie algebras is identified for which the Euler-Poincaré equations have multivalued solutions for any inertia tensor.
Izmailova O. V., Kozlov V. V., Analytical properties of solutions for Euler-Poincaré equations on solvable Lie algebras, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1996, vol. 51, no. 3, pp. 12–16
Symmetries and regular behavior of Hamiltonian systems
Chaos, 1996, vol. 6, no. 1, pp. 1–5
Abstract
pdf (453.44 Kb)
The behavior of the phase trajectories of the Hamilton equations is commonly classified as regular and chaotic. Regularity is usually related to the condition for complete integrability, i.e., a Hamiltonian system with n degrees of freedom has n independent integrals in involution. If at the same time the simultaneous integral manifolds are compact, the solutions of the Hamilton equations are quasiperiodic. In particular, the entropy of the Hamiltonian phase flow of a completely integrable system is zero. It is found that there is a broader class of Hamiltonian systems that do not show signs of chaotic behavior. These are systems that allow n commuting ‘‘Lagrangian’’ vector fields, i.e., the symplectic 2‐form on each pair of such fields is zero. They include, in particular, Hamiltonian systems with multivalued integrals.
Citation:
Kozlov V. V., Symmetries and regular behavior of Hamiltonian systems, Chaos, 1996, vol. 6, no. 1, pp. 1–5
On the theory of systems with unilateral constraints
Journal of Applied Mathematics and Mechanics, 1995, vol. 59, no. 4, pp. 505–512
Abstract
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The realization of a unilateral constraint is considered in a situation in which the stiffness and coefficient of viscosity and the added masses tend to infinity simultaneously in a consistent manner. The main result is that limiting motions exist, which are identical on the boundary with the motions of a holonomic system with fewer degrees of freedom. However, a special effect, not present in the classical model, occurs here, namely, a delay in the time at which the constraint is released.
Citation:
Deryabin M. V., Kozlov V. V., On the theory of systems with unilateral constraints, Journal of Applied Mathematics and Mechanics, 1995, vol. 59, no. 4, pp. 505–512
Some integrable generalizations of the Jacobi problem on geodesics on an ellipsoid
Journal of Applied Mathematics and Mechanics, 1995, vol. 59, no. 1, pp. 1–7
Abstract
pdf (432.65 Kb)
The problem of a point moving on the surface of an $n$-dimensional ellipsoid in a conservative field of force is considered. It is shown that if the potential energy terms are inversely proportional to the squares of the distances to the $(n−1)$-dimensional planes of symmetry of the ellipsoid, the problem can be explicitly integrated by using separation variables in elliptic Jacobi coordinates. It has $n$ independent commuting integrals that are quadratic functions of the momenta. If $n=2$, an additional integral can be found explicitly by using redundant coordinates. In the limit, when the least semi-axis approaches zero, one obtains a new integrable billiards problem inside the ellipse. Extensions of these results to a space of constant non-zero curvature are discussed.
Citation:
Kozlov V. V., Some integrable generalizations of the Jacobi problem on geodesics on an ellipsoid, Journal of Applied Mathematics and Mechanics, 1995, vol. 59, no. 1, pp. 1–7
On solutions with generalized power asymptotics to systems of differential equations
Mathematical Notes, 1995, vol. 58, no. 6, pp. 1286–1293
Abstract
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In the paper we study methods for constructing particular solutions with nonexponential asymptotic behavior to a system of ordinary differential equations with infinitely differentiable right-hand sides. We construct the corresponding formal solutions in the form of generalized power series whose first terms are particular solutions to the so-called truncated system. We prove that these series are asymptotic expansions of real solutions to the complete system. We discuss the complex nature of the functions that are represented by these series in the analytic case.
Citation:
Kozlov V. V., Furta S. D., On solutions with generalized power asymptotics to systems of differential equations, Mathematical Notes, 1995, vol. 58, no. 6, pp. 1286–1293
Integral invariants of the Hamilton equations
Mathematical Notes, 1995, vol. 58, no. 3, pp. 938–947
Abstract
pdf (599.62 Kb)
Conditions are found for the existence of integral invariants of Hamiltonian systems. For two-degrees-of-freedom systems these conditions are intimately related to the existence of nontrivial symmetry fields and multivalued integrals. Any integral invariant of a geodesic flow on an analytic surface of genus greater than 1 is shown to be a constant multiple of the Poincaré-Cartan invariant. Poincaré's conjecture that there are no additional integral invariants in the restricted three-body problem is proved.
Citation:
Kozlov V. V., Integral invariants of the Hamilton equations, Mathematical Notes, 1995, vol. 58, no. 3, pp. 938–947
The Maupertuis principle and geodesic flows on the sphere arising from integrable cases in the dynamics of a rigid body
Russian Mathematical Surveys, 1995, vol. 50, no. 3, pp. 473–501
Abstract
pdf (1.52 Mb)
Citation:
Bolsinov A. V., Kozlov V. V., Fomenko A. T., The Maupertuis principle and geodesic flows on the sphere arising from integrable cases in the dynamics of a rigid body, Russian Mathematical Surveys, 1995, vol. 50, no. 3, pp. 473–501
On multivalued integrals of Hamiltonian equations
Problems of nonlinear analysis in engineering systems, 1995, no. 1, pp. 30–34
Abstract
pdf (284.07 Kb)
Citation:
Kozlov V. V., On multivalued integrals of Hamiltonian equations, Problems of nonlinear analysis in engineering systems, 1995, no. 1, pp. 30–34
Symmetry fields of geodesic flows
Russian Journal of Mathematical Physics, 1995, vol. 3, no. 3, pp. 279–295
Abstract
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A symmetry field of the dynamical system defined by a vector field X is a vector field commuting with X. We study symmetry fields for systems on 3-dimensional manifolds with a smooth invariant measure, and for geodesic flows on surfaces. It turns out that the existence of a symmetry field implies the existence of a multivalued integral. Geodesic flows with nonobvious symmetry fields form a very special class of integrable flows.
Keywords:
symmetry fields; geodesic flow; integrable flows
Citation:
Bolotin S. V., Kozlov V. V., Symmetry fields of geodesic flows, Russian Journal of Mathematical Physics, 1995, vol. 3, no. 3, pp. 279–295
Problemata nova, ad quorum solutionem mathematici invitantur
Dynamical systems in classical mechanics, Amer. Math. Soc. Transl. Ser. 2, 1995, vol. 168, pp. 239–254
Abstract
pdf (826.77 Kb)
Citation:
Kozlov V. V., Problemata nova, ad quorum solutionem mathematici invitantur, Dynamical systems in classical mechanics, Amer. Math. Soc. Transl. Ser. 2, 1995, vol. 168, pp. 239–254
Hydrodynamics of noncommutative integration of Hamiltonian systems
Dynamical systems in classical mechanics, Amer. Math. Soc. Transl. Ser. 2, 1995, vol. 168, pp. 227–238
Abstract
pdf (578.71 Kb)
Citation:
Kozlov V. V., Hydrodynamics of noncommutative integration of Hamiltonian systems, Dynamical systems in classical mechanics, Amer. Math. Soc. Transl. Ser. 2, 1995, vol. 168, pp. 227–238
Various aspects of n-dimensional rigid body dynamics
Dynamical systems in classical mechanics, Amer. Math. Soc. Transl. Ser. 2, 1995, vol. 168, pp. 141–171
Abstract
pdf (1.34 Mb)
Citation:
Fedorov Y. N., Kozlov V. V., Various aspects of n-dimensional rigid body dynamics, Dynamical systems in classical mechanics, Amer. Math. Soc. Transl. Ser. 2, 1995, vol. 168, pp. 141–171
Polynomial integrals of geodesic flows on a two-dimensional torus
Sbornik: Mathematics, 1994, vol. 83, no. 2, pp. 469–481
Abstract
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he geodesic curves of a Riemannian metric on a surface are described by a Hamiltonian system with two degrees of freedom whose Hamiltonian is quadratic in the momenta. Because of the homogeneity, every integral of the geodesic problem is a function of integrals that are polynomial in the momenta. The geodesic flow on a surface of genus greater than one does not admit an additional nonconstant integral at all, but on the other hand there are numerous examples of metrics on a torus whose geodesic flows are completely integrable: there are polynomial integrals of degree $\leqslant2$ that are independent of the Hamiltonian. It appears that the degree of an additional 'irreducible' polynomial integral of a geodesic flow on a torus cannot exceed two. In the present paper this conjecture is proved for metrics which can arbitrarily closely approximate any metric on a two-dimensional torus.
Citation:
Kozlov V. V., Denisova N. V., Polynomial integrals of geodesic flows on a two-dimensional torus, Sbornik: Mathematics, 1994, vol. 83, no. 2, pp. 469–481
The asymptotic motions of systems with dissipation
Journal of Applied Mathematics and Mechanics, 1994, vol. 58, no. 5, pp. 787–792
Abstract
pdf (311.84 Kb)
Citation:
Kozlov V. V., The asymptotic motions of systems with dissipation, Journal of Applied Mathematics and Mechanics, 1994, vol. 58, no. 5, pp. 787–792
Integrable systems on the sphere with elastic interaction potentials
Mathematical Notes, 1994, vol. 56, no. 3, pp. 927–930
Abstract
pdf (239 Kb)
Citation:
Kozlov V. V., Fedorov Y. N., Integrable systems on the sphere with elastic interaction potentials, Mathematical Notes, 1994, vol. 56, no. 3, pp. 927–930
On tensor invariants of dynamical systems on three-dimensional manifolds
Teorijska i Primenjena Mehanika, 1994, no. 20, pp. 119–129
Abstract
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We consider dynamical systems on compact three-dimensional manifolds which have an invariant volume form. An important example is given by Hamilton equations of a system with two degrees of freedom restricted to three-dimensional lever surface of the energy integral. In this system we study the existence of tensor invariants (a first integral, a symmetry field, an invariant form) and give conditions of integrability by quadratures under the existence of a tensor invariant. We show that the infinite number of nondegenerate periodic trajectories and splitting of separatices obstruct the existence of nontrivial integral invariants analytical on the three-dimensional manifold.
Symmetries and topology of dynamical systems with two degrees of freedom
Hamiltonian mechanics. Integrability and chaotic behavior. NATO ASI Series. Series B. Physics, 1994, vol. 331, pp. 167–172
Abstract
This volume offers a state-of-the-art overview of Hamiltonian mechanics and its applications. Its contents include papers by distinguished Russian scientists not previously published in English. Topics covered in the refereed contributions include: Non-integrability criterion of Hamiltonian systems based on Ziglin’s theorem and its relation to the singular point analysis; Natural boundaries of normalizing transformations; The structure of chaos; Successive elimination of harmonics: a way to explore the resonant structure of a Hamiltonian system; The tendency toward ergodicity with increasing number of degrees of freedom in Hamiltonian systems; Numerical integration of Hamiltonian systems in the presence of additional integrals: application of the observer method.
Kozlov V. V., Symmetries and topology of dynamical systems with two degrees of freedom, Hamiltonian mechanics. Integrability and chaotic behavior. NATO ASI Series. Series B. Physics, 1994, vol. 331, pp. 167–172
Symmetries and the topology of dynamical systems with two degrees of freedom
Sbornik: Mathematics, 1993, vol. 80, no. 1, pp. 105–124
Abstract
pdf (891.83 Kb)
The problem of geodesic curves on a closed two-dimensional surface and some of its generalizations related with the addition of gyroscopic forces are considered. The authors study one-parameter groups of symmetries in the four-dimensional phase space that are generated by vector fields commuting with the original Hamiltonian vector field. Fields of symmetries of the first and second degree are studied. The presence of such symmetries is related to the existence of ignorable cyclic coordinates and separated variables. The influence of gyroscopic forces on the existence of fields of symmetries with polynomial components is studied.
Keywords:
geodesic curves; gyroscopic forces; one-parameter groups of symmetries; cyclic coordinates; symmetries with polynomial components
Citation:
Kozlov V. V., Denisova N. V., Symmetries and the topology of dynamical systems with two degrees of freedom, Sbornik: Mathematics, 1993, vol. 80, no. 1, pp. 105–124
On the degree of instability
Journal of Applied Mathematics and Mechanics, 1993, vol. 57, no. 5, pp. 771–776
Abstract
pdf (331.54 Kb)
The degree of instability of an equilibrium position in an autonomous dynamical system is defined as the number of eigenvalues of its linearization that lie in the right half-plane. Dissipative systems with Morse functions that do not increase along their trajectories are considered. The critical points of such functions are precisely the equilibrium positions. It will be shown that the degree of instability of a non-degenerate equilibrium position has the same parity as the index of the Morse function at that point. In particular, if the index is odd, the equilibrium is unstable. This result carries over to compact invariant manifolds of a dynamical system, provided they are non-degenerate, reducible and ergodic. An example is the problem of the stability of the steady motion of a heavy cylindrical rigid body in an unbounded volume of ideal liquid with non-zero circulation.
Citation:
Kozlov V. V., On the degree of instability, Journal of Applied Mathematics and Mechanics, 1993, vol. 57, no. 5, pp. 771–776
The Liouville property of invariant measures of completely integrable systems and the Monge–Ampère equation
Mathematical Notes, 1993, vol. 53, no. 4, pp. 389–393
Abstract
pdf (301.55 Kb)
Citation:
Kozlov V. V., The Liouville property of invariant measures of completely integrable systems and the Monge–Ampère equation, Mathematical Notes, 1993, vol. 53, no. 4, pp. 389–393
On the invariant measures of Euler-Poincaré equations on unimodular groups
Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1993, vol. 48, no. 2, pp. 45–50
Abstract
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The inertial motion of a mechanical system with a Lie group as its configuration space and left-invariant kinetic energy is studied. The equation of motion is restricted on a Lie group with the help of Noetherian integrals. The right-hand Haar measure of this group is proved to be an integral invariant of the resulting dynamical system.
Kozlov V. V., Yaroshchuk V. A., On the invariant measures of Euler-Poincaré equations on unimodular groups, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1993, vol. 48, no. 2, pp. 45–50
Dynamical systems determined by the Navier–Stokes equations
Russian Journal of Mathematical Physics, 1993, vol. 1, no. 1, pp. 57–69
Abstract
Time-independent differential equations describing the velocity field of a stationary flow of a viscous incompressible liquid in three-dimensional Euclidean space are considered. It is proved that, in the general case, this system does not admit any nontrivial first integrals, symmetry fields, and linear integral invariants. Certain examples of stationary flows with chaotic properties are given.
Linear-systems with a quadratic integral
Journal of Applied Mathematics and Mechanics, 1992, vol. 56, no. 6, pp. 803–809
Abstract
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It is shown that a linear system of $n$ differential equations with constant coefficients, at least one of whose integrals is a non-degenerate quadratic form, may be reduced to a canonical system of Hamiltonian equations. In particular, $n$ is even and the phase flow preserves the standard measure; if the index of the quadratic integral is odd, the trivial solution is unstable, and so on. For the case $n = 4$ the stability conditions are given a geometrical form. The general results are used to investigate small oscillations of non-holonomic systems, and also the problem of the stability of invariant manifolds of non-linear systems that have Morse functions as integrals.
Citation:
Kozlov V. V., Linear-systems with a quadratic integral, Journal of Applied Mathematics and Mechanics, 1992, vol. 56, no. 6, pp. 803–809
The problem of realizing constraints in dynamics
Journal of Applied Mathematics and Mechanics, 1992, vol. 56, no. 4, pp. 594–600
Abstract
pdf (475.25 Kb)
Problems associated with the limiting transition in the second-order Lagrange equations, when the coefficients of rigidity and viscosity and added masses tend to infinity are considered. Under certain conditions, the solutions of the initial equations approach those of the limiting problem with constraints. For integrable constraints, the limiting equations are identical with the usual equations with constraint multipliers. In the case of non-integrable constraints, the solutions depends closely on the way in which they are realized. The generalized models of the dynamics of systems with non-integrable constraints and the properties of the limiting equations of motion are discussed.
Citation:
Kozlov V. V., The problem of realizing constraints in dynamics, Journal of Applied Mathematics and Mechanics, 1992, vol. 56, no. 4, pp. 594–600
Tensor invariants of quasihomogeneous systems of differential equations, and the Kovalevskaya–Lyapunov asymptotic method
Mathematical Notes, 1992, vol. 51, no. 2, pp. 138–142
Abstract
pdf (341.6 Kb)
Citation:
Kozlov V. V., Tensor invariants of quasihomogeneous systems of differential equations, and the Kovalevskaya–Lyapunov asymptotic method, Mathematical Notes, 1992, vol. 51, no. 2, pp. 138–142
On impulsive isoenergetic control
Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1992, vol. 47, no. 5, pp. 17–19
Abstract
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The authors consider possible trajectories on a manifold in the abstract coordinate space associated with a conservative mechanical system. They point out that if the total energy associated with a trajectory is less than the maximum possible potential energy, certain portions of the trajectory cannot be traversed, and consequently it may not always be possible to move directly from one point on the trajectory to another on it. Some theorems associated with this concept are enunciated, but the discussion depends heavily on concepts from other papers.
Keywords:
maximum potential energy; possible trajectories; abstract coordinate space; conservative mechanical system; total energy
Citation:
Kozlov V. V., Khmelevskaya A. Y., On impulsive isoenergetic control, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1992, vol. 47, no. 5, pp. 17–19
Kepler's problem in constant curvature spaces
Celestial Mechanics and Dynamical Astronomy, 1992, vol. 54, no. 4, pp. 393–399
Abstract
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In this article the generalization of the motion of a particle in a central field to the case of a constant curvature space is investigated. We found out that orbits on a constant curvature surface are closed in two cases: when the potential satisfies Iaplace-Beltrami equation and can be regarded as an analogue of the potential of the gravitational interaction, and in the case when the potential is the generalization of the potential of an elastic spring. Also the full integrability of the generalized two-centre problem on a constant curvature surface is discovered and it is shown that integrability remains even if elastic “forces” are added.
Keywords:
Central field, closed orbits, spheroconical coordinates
Citation:
Kozlov V. V., Harin A. O., Kepler's problem in constant curvature spaces, Celestial Mechanics and Dynamical Astronomy, 1992, vol. 54, no. 4, pp. 393–399
On the stability of periodic trajectories of a three-dimensional billiard
Journal of Applied Mathematics and Mechanics, 1991, vol. 55, no. 5, pp. 576–580
Abstract
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The stability of periodic trajectories of a material point moving between two convex walls with elastic reflections is investigated. The problem is closely bound related to wave propagation theory in the shortwave approximation [1], The simplest periodic trajectory is a section of a straight line orthogonal to the walls at its endpoints. The problem of the stability of a two-part trajectory was solved in [1] in two dimensions. It will be solved here in three dimensions using the method developed in [2].
Citation:
Kozlov V. V., Chigur I. I., On the stability of periodic trajectories of a three-dimensional billiard, Journal of Applied Mathematics and Mechanics, 1991, vol. 55, no. 5, pp. 576–580
On the Lyapunov problem on stability with respect to given state functions
Journal of Applied Mathematics and Mechanics, 1991, vol. 55, no. 4, pp. 442–445
Abstract
pdf (309.47 Kb)
The general formulation of the problem of the stability of motion with respect to prescribed functions of the coordinates and velocities is due to Lyapunov [1]. A special case is that of partial stability, i.e., stability with respect to some of the variables [2–4]. In this paper some ideas of Lyapunov's first method are applied to the general problem of the stability of the equilibrium of reversible systems. The study is based on an examination of the trajectories asymptotic to an equilibrium position: if the equilibrium is stable with respect to a function $Q$, this function is stable on the asymptotic trajectories. Asymptotic solutions are sought using a special kind of series. As an application, a relativistic version of Earnshaw's theorem on the instability of the equilibrium of a charge in a stationary electric field is proved.
Citation:
Vujičić V. A., Kozlov V. V., On the Lyapunov problem on stability with respect to given state functions, Journal of Applied Mathematics and Mechanics, 1991, vol. 55, no. 4, pp. 442–445
The stability of equilibrium positions in a nonstationary force field
Journal of Applied Mathematics and Mechanics, 1991, vol. 55, no. 1, pp. 8–13
Abstract
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The stability of equilibrium positions is investigated for mechanical systems in force fields with potentials of the form $p(t)V$, where $V$ is a function of the generalized coordinates. Systems of this form are frequently encountered in applications. It is shown that if the factor $p(t)$ increases monotonically to $+\infty$ as $t\to+\infty$, then stability conditions for equilibria can be formulated in the form of extremal properties of the function $V$. The general results are applied to the problem of the motion of a rigid body in an infinite volume of an ideal fluid.
Citation:
Kozlov V. V., The stability of equilibrium positions in a nonstationary force field, Journal of Applied Mathematics and Mechanics, 1991, vol. 55, no. 1, pp. 8–13
Stability of periodic trajectories and Chebyshev polynomials
Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1991, vol. 46, no. 5, pp. 4–9
Abstract
pdf (366.41 Kb)
Keywords:
Chebyshev polynomials; asymptotic solution; Birkhoff billards; stability; mechanical systems; works of Chebyshev and Lyapunov
Citation:
Kozlov V. V., Stability of periodic trajectories and Chebyshev polynomials, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1991, vol. 46, no. 5, pp. 4–9
On randomization of plane parallel flow of an ideal fluid
Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1991, vol. 46, no. 1, pp. 29–32
Abstract
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A sinusoidal perturbation of constant direction on a stationary velocity field has been shown to give rise to chaotic motion of an ideal fluid. The perturbed motion due to a pair of vortices of opposite intensities has been numerically computed.
Keywords:
sinusoidal perturbation; chaotic motion; pair of vortices of opposite intensities
Citation:
Kozlov V. V., On randomization of plane parallel flow of an ideal fluid, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1991, vol. 46, no. 1, pp. 29–32
Realization of holonomic constraints
Journal of Applied Mathematics and Mechanics, 1990, vol. 54, no. 5, pp. 705–708
Abstract
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The idea of realizing holonomic constraints by means of elastic forces was proposed by Lecornu, Klein and Prandtl [1] when dealing with the paradox of dry friction discovered by Painleve. The general theorem on the realization of holonomic constraints with the help of elastic forces directed towards the configurational manifold of a constrained system was proposed by Courant and was proved in [2]. The generalization of Courant's theorem was considered in [3–5] by studying the passage to the limit in the case when the velocity of the system at the initial instant is transverse to the manifold defined by the constraint equations. In [2–5] the assumption that the system in question in conservative is used to a considerable degree.
Citation:
Kozlov V. V., Neishtadt A. I., Realization of holonomic constraints, Journal of Applied Mathematics and Mechanics, 1990, vol. 54, no. 5, pp. 705–708
On the problem of fall of a rigid body in a resisting medium
Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1990, vol. 45, no. 1, pp. 30–36
Abstract
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A simplified mathematical model of motion of a rigid body in a resisting medium is considered, in which viscous friction forces defined by the Rayleigh dissipative function are taken into account along with the associated mass effect. Using the small parameter method, the existence of a stable auto-rotation is established, where, on the average, the center of mass of the body comes down with a constant velocity along an inclined line. The stability of stationary vertical descent of the rigid body is analyzed.
Keywords:
motion of a rigid body; resisting medium; viscous friction forces; Rayleigh dissipative function; stable auto-rotation
Citation:
Kozlov V. V., On the problem of fall of a rigid body in a resisting medium, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1990, vol. 45, no. 1, pp. 30–36
Relyativistskaya zadacha mnogikh tel i ee kvantovanie
v kn.: E. M. Nikishin, Izbrannye voprosy matematicheskogo analiza, Moskva–Tula, 1990, pp. 430–431
Abstract
Citation:
Kozlov V. V., Relyativistskaya zadacha mnogikh tel i ee kvantovanie, v kn.: E. M. Nikishin, Izbrannye voprosy matematicheskogo analiza, Moskva–Tula, 1990, pp. 430–431
Constructive approach to proof of the dynamics of systems with constraints
Theoretical Mechanics, Collection of scientific and methodological papers, no. 20, MPI, Moscow, 1990, pp. 8–15
Abstract
pdf (382.68 Kb)
Citation:
Kozlov V. V., Constructive approach to proof of the dynamics of systems with constraints, Theoretical Mechanics, Collection of scientific and methodological papers, no. 20, MPI, Moscow, 1990, pp. 8–15
Polynomial integrals of Hamiltonian systems with exponential interaction
Mathematics of the USSR - Izvestiya, 1989, vol. 34, no. 3, pp. 555–574
Abstract
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The problem on the complete integrability of Hamiltonian systems with exponential interaction is considered. These systems include, in particular, Toda chains and their generalizations. Conditions for the existence of a complete set of independent polynomial integrals are found. A complete classification of integrable systems is given by means of Dynkin diagrams. Certain new integrable chains are indicated.
Citation:
Kozlov V. V., Treschev D. V., Polynomial integrals of Hamiltonian systems with exponential interaction, Mathematics of the USSR - Izvestiya, 1989, vol. 34, no. 3, pp. 555–574
A problem of Kelvin
Journal of Applied Mathematics and Mechanics, 1989, vol. 53, no. 1, pp. 133–135
Abstract
pdf (208.35 Kb)
A problem of the stability of equilibrium of a system of interacting particles distributed within a bounded volume of Euclidean space is considered. Sufficient conditions for the instability and existence of the motions approaching the position of equilibrium without bounds, containing the Kelvin theorem /1/ as a special case, are obtained. The results are based on the general theory of instability of equilibrium in a force field with a subharmonic force function.
Citation:
Kozlov V. V., A problem of Kelvin, Journal of Applied Mathematics and Mechanics, 1989, vol. 53, no. 1, pp. 133–135
Polynomial integrals of dynamical systems with one-and-a-half degrees of freedom
Mathematical Notes, 1989, vol. 45, no. 4, pp. 296–300
Abstract
pdf (304.73 Kb)
Citation:
Kozlov V. V., Polynomial integrals of dynamical systems with one-and-a-half degrees of freedom, Mathematical Notes, 1989, vol. 45, no. 4, pp. 296–300
Poinsot's geometric representation in the dynamics of a multidimensional rigid body
Trudy Sem. Vektor. Tenzor. Anal., 1988, vol. 23, pp. 202–204
Abstract
The authors consider the inertial rotation of an n-dimensional rigid body about a fixed point. The dynamics of such a motion is fully described by the Euler-Poincaré equations in terms of the SO(n) algebra. Introducing the angular velocity vector (internal and external), angular momentum vector, kinetic energy, inertial characteristic etc., the Poinsot’s interpretation of this motion is obtained: the ellipsoid of inertia is rolling without sliding upon a fixed plane which is perpendicular to the angular momentum vector.
Citation:
Zenkov D. V., Kozlov V. V., Poinsot's geometric representation in the dynamics of a multidimensional rigid body, Trudy Sem. Vektor. Tenzor. Anal., 1988, vol. 23, pp. 202–204
Dynamics of systems with nonintegrable constraints. V: Freedom principle and ideal constraints condition
Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1988, vol. 43, no. 6, pp. 23–29
Abstract
Generalized mathematical models are considered of the motion of mechanical systems with nonintegrable constraints produced by a limiting process approaching infinity in anisotropic visous friction coefficient and attached mass. Provided an appropriate definition, the variations of kinematically admissible paths of such systems turn out to be extrema of the action functional. The validity is shown of the generalized principle of releasability and the idealness of constraints in an integral form. Constraint reactions in this case are no longer functions of the system states but are functionals of the system motions.
Keywords:
motion of mechanical systems with nonintegrable constraints; anisotropic visous friction; attached mass; generalized principle of releasability
Citation:
Kozlov V. V., Dynamics of systems with nonintegrable constraints. V: Freedom principle and ideal constraints condition, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1988, vol. 43, no. 6, pp. 23–29