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–Poincare? 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

The problem ofšintegrability conditions for systems ofšdifferential equations isšdiscussed. Darboux?s classical results onšthe integrability ofšlinear non-autonomous systems with anšincomplete set ofšparticular solutions are generalized. Special attention isšpaid tošlinear Hamiltonian systems. The paper discusses the general problem ofšintegrability ofšthe systems ofšautonomous differential equations inšanš$n$-dimensional space which permit the algebra ofšsymmetry fields ofšdimension $\geqslant n$. Using ašmethod due tošLiouville, this problem isšreduced tošinvestigating the integrability conditions for Hamiltonian systems with Hamiltonians linear inšthe momentums inšphase space ofšdimension that isštwice asšlarge. Inšconclusion, the integrability ofšanšautonomous system inšthree-dimensional space with two independent non-trivial symmetry fields isšproved. Itšshould bešemphasized that nošadditional conditions are imposed onšthese fields.

This paper addresses ašclass ofšproblems associated with the conditions for exact integrability ofšašsystem ofšordinary differential equations expressed inšterms ofšthe properties ofštensor invariants. The general theorem ofšintegrability ofšthe system ofš$n$šdifferential equations isšproved, which admits $n ? 2$šindependent symmetry fields and anšinvariant volume $n$-form (integral invariant). General results are applied tošthe study ofšsteady motions ofšašcontinuous medium with infinite conductivity.

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.

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.

Wešdevelop ašnew method for solving Hamilton?s canonical differential equations. The method isšbased onšthe search ofšinvariant vortex manifolds ofšspecial type. Inšthe case ofšLagrangian (potential) manifolds, wešarrive atšthe classical Hamilton?Jacobi method.

Invariant manifolds ofšequations governing the dynamics ofšconservative nonholonomic systems are investigated. These manifolds are assumed tošbešuniquely projected onto configuration space. The invariance conditions are represented inšthe form ofšgeneralized Lamb?s equations. Conditions are found under which the solutions tošthese equations admit ašhydrodynamical description typical ofšHamiltonian systems. Asšanšillustration, nonholonomic systems onšLie groups with ašleft-invariant metric and left-invariant (right-invariant) constraints are considered.

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.

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 structure ofšthe Lorentz force and the related analogy between electromagnetism and inertia are discussed. The problem ofšinvariant manifolds ofšthe equations ofšmotion for ašcharge inšanšelectromagnetic field and the conditions for these manifolds tošbešLagrangian are considered.

The Kac circular model isšašdiscrete dynamical system which has the property ofšrecurrence and reversibility. Within the framework ofšthis model M.Kac formulated necessary conditions for irreversibility over ?short? time intervals toštake place and demonstrated Boltzmann?s most important exploration methods and ideas, outlining their advantages and limitations. Wešstudy the circular model within the realm ofšthe theory ofšGibbs ensembles and offer ašnew approach tošašrigorous proof ofšthe ?zeroth? law ofšthermodynamics basing onšthe analysis ofšweak convergence ofšprobability distributions.

Ašgeneralization ofšAmantons? law ofšdry friction for constrained Lagrangian systems isšformulated. Under ašchange ofšgeneralized coordinates the components ofšthe dry-friction force transform according tošthe covariant rule and the force itself satisfies the Painlev? condition. Inšparticular, the pressure ofšthe system onšašconstraint isšindependent ofšthe anisotropic-friction tensor. Such anšapproach provides anšinsight into the Painlev? dry-friction paradoxes. Asšanšexample, the general formulas for the sliding friction force and torque and the rotation friction torque onšašbody contacting with ašsurface are obtained.

Wešconsider ašcontinuum ofšinteracting particles whose evolution isšgoverned byšthe Vlasov kinetic equation. Anšinfinite sequence ofšequations ofšmotion for this medium (inšthe Eulerian description) isšderived and its general properties are explored. Anšimportant example isšašcollisionless gas, which exhibits irreversible behavior. Though individual particles interact via ašpotential, the dynamics ofšthe continuum bears dissipative features. Applicability ofšthe Vlasov equations tošthe modeling ofšsmall-scale turbulence isšdiscussed.

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.

The Poincar? model for dynamics ofšašcollisionless gas inšašrectangular parallelepiped with mirror walls isšconsidered. The question onšsmoothing ofšthe density and the temperature ofšthis gas and conditions for the monotone growth ofšthe coarse-grained entropy are discussed. All these effects provide ašnew insight ofšthe classical paradox ofšmixing ofšgases.

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.

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.

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.

The paper generalizes the classical Gauss principle for non-constrained dynamical systems. For large anisotropic external forces ofšviscous friction our statement transforms into the common Gauss principle for systems with constraints.

The famous Lagrange identity expresses the second derivative ofšthe moment ofšinertia ofšašsystem ofšmaterial points through the kinetic energy and homogeneous potential energy. The paper presents various extensions ofšthis brilliant result tošthe caseš1) ofšconstrained mechanical systems, 2) when the potential energy isšquasi-homogeneous inšcoordinates andš3) ofšcontinuumof interacting particles governed byšthe well-known Vlasov kinetic equation.

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.

Wešconsider two problems from the rigid body dynamics and use new methods ofšstability and asymptotic behavior analysis for their solution. The first problem deals with motion ofšašrigid body inšanšunbounded volume ofšideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, isšconcerned with motion ofšašsleigh onšanšinclined plane. The equations ofšmotion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. Ašcomprehensive survey ofšreferences isšgiven and new problems connected with falling motion ofšheavy bodies inšfluid are proposed.

The paper develops anšapproach tošthe proof ofšthe ?zeroth? law ofšthermodynamics. The approach isšbased onšthe analysis ofšweak limits ofšsolutions tošthe Liouville equation asštime grows infinitely. Ašclass ofšlinear oscillating systems isšindicated for which the average energy becomes eventually uniformly distributed among the degrees ofšfreedom for any initial probability density functions. Anšexample ofšsuch systems are sympathetic pendulums. Conditions are found for nonlinear Hamiltonian systems with finite number ofšdegrees ofšfreedom tošconverge inšašweak sense tošthe state where the average energies ofšthe interacting subsystems are the same. Some issues related tošstatistical models ofšthe thermostat are discussed.

The issues discussed inšthis paper relate tošthe description ofšdeveloped two-dimensional turbulence, when the mean values ofšcharacteristics ofšsteady flow stabilize. More exactly, the problem ofšašweak limit ofšvortex distribution inštwo-dimensional flow ofšanšideal fluid atštime tending tošinfinity isšconsidered. Relations between the vorticity equation and the well-known Vlasov equation are discussed.

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.

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.

In this paper, we present results of simulations of a model of the Galton board for various degrees of elasticity of the ball-to-nail collision.

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.

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.

In this article we develop Poincar? 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.

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.

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.

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.

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.

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.