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This paper addresses the problem of stability of equilibria of mechanical systems in nonpotential force fields. In the linear approximation, any nonpotential force is uniquely represented as the sum of a potential force and a circulatory force. The instability of equilibrium is established for the nonlinear problem in the case when the linearized system has a pair of zero eigenvalues. The mechanism of stability loss is due to the appearance of two asymptotic trajectories, one that enters an equilibrium point and one that exits it. A similar result takes place when additional gyroscopic forces and viscous friction forces are taken into account. Asymptotic solutions can be obtained in the form of series in inverse powers of time whose coefficients are polynomials in logarithms of time.

The properties of the Gibbs ensembles of Hamiltonian systems describing the motion along geodesics on a compact configuration manifold are discussed. We introduce weakly ergodic systems for which the time average of functions on the configuration space is constant almost everywhere. Usual ergodic systems are, of course, weakly ergodic, but the converse is not true. A range of questions concerning the equalization of the density and the temperature of a Gibbs ensemble as time increases indefinitely are considered. In addition, the weak ergodicity of a billiard in a rectangular parallelepiped with a partition wall is established.

The results of experimental investigations of the effect of distributed suction through a perforated section of a symmetric wing profile on the spatial development of disturbances in the boundary layer are presented. It is shown that this method of control makes it possible to reduce by ten times the intensity of natural pulsations of the boundary-layer velocity, while the disturbance intensity decreases by about 20 times, when an acoustic field is applied at the subharmonic frequency of the fundamental wave. The suction also significantly affects the mean flow up to eliminating the separation of the boundary layer near the trailing edge of the wing instead of contributing only a decrease in the intensity of high-frequency pulsations for natural and forced disturbances.

The stability of linear mechanical systems in nonpotential force fields is studied. Under the presence of circulatory forces, the system is nonconservative, yet it can always be presented in a Hamiltonian form. A point where the potential energy has its maximum is an unstable equilibrium, regardless of the presence of circulatory forces.

The paper presents the transfer function of the information-measuring system based on phase displacement sensors, which determines the relationship between the linear displacement value and the output voltage, or current value. An analysis of the transfer function of the measuring device is carried out by changing the phase shift of output voltage relative to the reference voltage. The obtained results can be used to control information-measuring systems. Factors affecting the initial phase of the output signal, such as sensor electrical parameters and the position of the magnetic shunt on the information line, are determined. The paper also considers the transfer function of an unbalanced sensor, in which circuit resistance of the cosine winding is not equal to circuit resistance of the sine winding. This difference complicates the equation expressing the dependence of the output voltage phase on the shunt displacement. We used mathematical modeling, in which real physical processes were described by the corresponding analytical dependencies. Amplitude-frequency and phase-frequency characteristics of the information-measuring system were obtained. They allow us to analyze errors and dynamic properties of the system, to select the optimal generator frequency for obtaining the maximum voltage at the system output, and to increase the system resistance to external factors. Recommended values for the information-measuring system are given.

New data on under-ice currents in Lake Baikal have been obtained with the use of high-accuracy instruments. The obtained data were used to analyze the time and space variations of under-ice currents, to calculate the coefficients of horizontal turbulent exchange and turbulent energy dissipation rates, and to derive spectral estimates of the obtained results. The estimated structures of the under-ice boundary layer near the Angara opening and far from it (up to 5 km) were compared.

This paper presents the results of a theoretical study of a fiber-optic delay line with an electro-optical modulator. On the basis of this line, radio-photon generators and frequency converters of microwave signals are performed. The results of experimental studies of such radio-photon devices are also given in this paper.

The problem of the reversibility of the trajectories of a reversible mechanical system with a non-compact configuration space is discussed. To identify the conditions of reversibility in systems with a non-negative potential energy, an invariant Gibbs measure is used. Despite the non-compactness, the Gibbs measure of the entire phase space can be finite, which guarantees reversibility of almost all phase trajectories. Sufficient conditions for reversibility of trajectories of systems with a homogeneous, non-negative potential energy are indicated. As a consequence, reversibility of almost all phase trajectories of the Yang–Mills Hamiltonian with three degrees of freedom is established.

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.

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.

This paper considers Hamiltonian structures related to quantum mechanics. We do not discuss structures used in the quantization of classical Hamil� tonian or Lagrangian systems (which was first done by Heisenberg and Feynman); on the contrary, we are mainly interested in structures which make it possible to consider quantum systems as classical (although infinitedimensional in the most natural cases) Hamiltonian (or Lagrangian) systems. The introduc� tion of such structures can be called the dequantiza� tion of a quantum system. We emphasize that, whereas the quantization of a classical (Hamiltonian or Lagrangian) system sup� poses the introduction of fundamentally new mathe� matical structures (usually related to the passage from ordinary differential equations to partial differential equations), dequantization means only the passage to an equivalent description of the same quantum sys� tem. To obtain the result of two successive operations (usual quantization and subsequent dequantization), it suffices to know only properties of the initial classi� cal system; therefore, it can be said that the passage from the initial classical Hamiltonian (or Lagrangian) system to the Hamiltonian system being the result of the corresponding dequantization is yet another quan� tization method of the initial system, different from quantizations in the sense of Schrodinger, Heisenberg, and Feynman. We refer to this quantization as Hamil� tonian quantization (precise definitions are given below).

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.

Some problems from quantum statistical mechanics concerning the regular and chaotic behavior of quantum systems are discussed.

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.

Results of experimental studies on round and plane, macro- and microjets subjected to transverse acoustic field at low Reynolds numbers are presented. A new phenomenon associated with transformation of the round microjet to the plane one under acoustic forcing is revealed. Also, bifurcation of the round microjet is observed. It is shown that both plane macro- and microjets are prone to a sinusoidal instability. As is found, the plane microjet becomes twisted at its periphery in the direction of the oscillatory flow velocity induced by acoustic waves. Acoustic influence both upon the pseudo-plane and the planemicrojets results in their sinusoidal oscillations and bifurcation. New phenomena observed in the present experiments on the round and plane microjets are caused by a proportionality of the acoustic energy with that of the microjets.

26 In comparison with microjets, the investigation of free macrojets has attracted greater attention because of their wide use in various fields of science and engi� neering. However, interest in studying free microjets has recently increased considerably (1) because of, in particular, the development of MEMS technologies. The possibility of the potential use of microjets in var� ious processes appeared, for example, in microcool� ing, jet burning, nanopowder manufacture, etc. Spe� cial attention is given to the investigation of the acous� ticfield action on the microjet (2-5), which is of importance both for understanding the physics of the process and for the possibility of practical use of the phenomenon, for example, in aviation, astronautics, and the chemical industry. The purpose of these investigations consists in experimental study of the mechanism of developing round and plane macroand microjets under the acousticfield action and comparison of the obtained results with the results of recent works in this direction. In this study, we discussed the results of the experi� mental investigations of the mechanism of developing jet flows with small Reynolds numbers in a transverse acoustic field. We investigated the round macrojet (Re d = 5300, d = 20 mm) and microjet (Red = 20-60, d = 200-600 μm) and the plane macrojet (Reh = 3600, h = 14.5 mm) and microjet (Reh = 75, h = 200- 700 μm) depending on the velocity profile at the noz� zle end (from the shock one with the narrow velocity� gradient region and gradientfree jet core to the para� bolic one). Thermoanemometric measurements and the smoke visualization of flows were carried out. The patterns as a result of the smoke visualization of flows were obtained with the help of laser illumination of the jet synchronized with the acousticaction frequency, which enabled us to obtain new data on the mecha� nism of the development of jets.

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.

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.

144 Riblets are passive elements mounted on a smooth wall surface in a turbulent boundary layer, which can reduce friction drag up to 10%. These elements have the form of streamwise grooves with triangular or hemispherical cross section whose dimensions are comparable with those of a viscous sublayer. The systematic investigations of riblets as a means of reducing turbulent friction began in the late 1970s at the NASA Research Center in Langley [1‐4]. These investigations showed that the friction drag decreases when the dimensionless parameter of riblets s + = is approximately equal to 15. However, the drag increases at s + = 30. For the maximum drag reduction, the riblets must be oriented in the direction of the local velocity of the viscous sublayer. The net drag reduction is almost linearly proportional to the coverage of the entire streamlined surface by the riblets. The near-wall turbulent structure of flows on riblets was extensively studied using physical [3, 5, 6] and numerical experiments (DNS) [7, 8]. According to one of the hypotheses put forward in order to explain the drag reduction in the turbulent boundary layer on riblets, they modify coherent structures in the viscous sublayer. It was found that riblets operate as an obstacle for the transversal (spanwise) oscillations of streamwise vortices, which results in drag reduction [9]. In other words, riblets reduce the friction drag in the turbulent boundary layer by changing the sequence of the nearwall vortex dynamics by means of a passive spanwise su* ν

It is well known [1] that the laminar‐turbulent transition at a low turbulence level of the free flow is associated with the development of instability waves, the so-called Tollmien‐Schlichting waves. When a twodimensional Tollmien‐Schlichting wave reaches a certain amplitude at the nonlinear stage of its development, it undergoes three-dimensional distortion and, as a result, characteristic three-dimensional Λ structures arise [1]. Owing to certain features of the appearance and development of these structures, they are not only typical for the classical laminar‐turbulent transition, but are also inevitable attributes of a transition to more complex flows, e.g., flows modulated with longitudinal streaky structures, such as Hertler vortices, transverseflow vortices on sliding wings, etc., as well as flows in the viscous sublayer of a turbulent boundary layer. In these cases, they arise in particular due to the secondary high-frequency instability of such flows and may be manifested not only as Λ structures, but also in the form of horseshoe vortices ( Ω structures), hairpin vortices, etc. A characteristic feature of the development of such structures, e.g., on a sliding wing, is the disappearance of one of the counter-rotating vortices due to the transverse flow, whereas the development of a classical Λ structure can be observed on a straight wing [1]. The high-frequency secondary instability of transition and turbulent near-wall flows in the presence of streaky structures is often attributed to so-called sinusoidal and varicose instability. Both instability modes were investigated under controlled conditions at the linear and initial stages of nonlinear development. When the transverse size of the streaky structure was larger than the thickness of the shear layer, growth of varicose instability was observed. At the same time, when the transverse size of the streaky structure was comparable to or smaller than the thickness of the shear layer, it became more instable with respect to antisymmetric (sinusoidal) modes than to symmetric (varicose) modes. The experiment reported in [2] clearly shows that the growth of the symmetric mode leads to the formation of hairpin vortices, which are a pair of counterrotating longitudinal vortices that are connected by a head, i.e., a Λ vortex, while an antisymmetric mode is developed to a train of quasi-longitudinal vortices with alternating-sign vorticity. Unfortunately, the experiments reported in [2] concerned only the initial stage of the nonlinear development of disturbances, and spatial resolution was insufficiently high to reveal the structure of the flow in more detail. In this paper, we report on our experimental investigations of the nonlinear stage of the varicose and sinusoidal instability of the streaky structure in the Blasius boundary layer. In contrast to the experiment reported in [2], the study is more detailed (thermal anemometer measurements of the longitudinal velocity component and velocity pulsations in space ( xyz ) at 5 × 10 4 points) in order to reveal the features of the dynamics of the appearance, development, and internal structure of coherent formations up to the later stages of their nonlinear development. The experiments were carried out under controlled conditions in a low-turbulent wind tunnel. A plane plate 14 mm in thickness, 1000 mm in width, and 2000 mm in length was placed in parallel in the operation part of the tunnel. The streaky structure was generated by means of a cylindrical roughness element, which had a height of 1.1 mm and a diameter of 5.8 mm and was placed in the center of the plate at a distance of x 0 = 438 mm from the fore. The velocity of the flow was equal to U ∞ = 7.8 m/s, and the turbulence level was no higher than 0.04%. In the absence of the roughness element, the laminar boundary layer was developed without any waves and the velocity profile was close to the Blasius profile. A roughness-element height of h = 1.1 mm is close to the thickness of the displacement of

Stochastic and deterministic versions of a discrete dynamical system on a necklace network are investigated. This network contains several contours. There are three cells and a particle on each contour. The particle occupies one of the cells and, at each step, it makes an attempt to move to the next cell in the direction of movement. As well as on neighboring contours the particles move in accordance with rules of stochastic or deterministic type. We prove that the behavior of the model with a rule of the first type is stochastic only at the beginning, and after a time interval the behavior becomes purely deterministic. The system with a rule of the first type reaches a stationary mode which depends on the initial state. The average velocity of particles and other characteristics of the dynamical systems are studied.

We briefly describe the recent development of the optical evanescent-field amplifiers including the solid-state integrated-optic and fiber-optic devices. Application of these amplifiers to the optical communication systems is discussed.

The physics of mixing in jets is of considerableinterest from the standpoint of both fundamental sci-ence and practical application. The intensity and homo-geneity of mixing significantly affect the combustionefficiency, heat transfer coefficient, formation of waste,and jet noise (see, e.g., [1]).Laminar jets have inflectional mean velocity pro-files. This property leads to the formation of Kelvin–Helmholtz vortices, which represent the main instabil-ity of such shear layers. The initial stage in the develop-ment of these vortices is usually well described by lin-ear stability theory (see, e.g., review [2]). The next,nonlinear stage is characterized by amplitude saturationand vortex pairing due to the resonance of disturbanceswith subharmonics and superharmonics. Further devel-opment of nonlinear structures is often accompanied bythe occurrence of longitudinal, or streamwise, vorticalstructures. Their formation is usually attributed to theso-called secondary three-dimensional instability of theKelvin–Helmholtz vortices [3, 4]. Experiments showthat the dynamics of these structures play an importantrole in the mixing process in the far wake of a jet [5].Other longitudinal disturbances that can often bedeveloped in shear layers are formed downstream ofnozzle surface irregularities [6]. These are regions ofquasi-stationary three-dimensional deformations,mainly of the longitudinal velocity in the shear flow,and have a characteristic form of “streaks” in the visu-alization pictures [7]. Their occurrence is not due to thesecondary instability of the Kelvin–Helmholtz vortices.These structures arise as a result of stabilization effectsupon the development of compact three-dimensionaldisturbances of the normal velocity component evenwith small amplitude [7]. These longitudinal structuresintensely interact with other flow disturbances, e.g.,instability waves. This interaction usually acceleratesflow turbulization [8]. This feature of the streaky struc-tures makes them a promising means for improving themixing and control over the flow in jets.There is an extensive literature on the investigationof longitudinal structures in wall flows (see, e.g., theoverview in [7]). However, investigations into the phys-ics of similar structures in jets began only recently. Suc-cessful excitation and observation of natural longitudi-nal structures and their interaction with Kelvin–Helm-holtz vortices in circular jets [9] offer the possibility ofstudying controlled longitudinal structures in planejets. To this end, a simple but efficient smoke visualiza-tion technique using a pulsed laser sheet synchronizedwith the Kelvin–Helmholtz vortex shedding wasapplied in this study.The plane jet generator and visualization schematicsare presented in Fig. 1. The flow was produced by aDISA wind tunnel that is designed for the calibration ofsensors of a hot-wire anemometer and ensures a stableair flow rate. Smoke was supplied to the test sectionthrough a tube connected to a smoke generator. The

We consider an infinitely smooth system of differential equations for which the origin x = 0 is a critical point: $$\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}),\;\mathbf{x}(\mathbf{0}) = \mathbf{0},\;\mathbf{x} \in {\mathbb{R}}^{n},$$ (1.1)

In the preceding chapter we investigated asymptotic solutions of smooth systems of differential equations as t → ± 0, or as t → ± ∞, where it wasn’t stipulated that the desired solutions be located in a neighborhood of a critical point.

Here, m is the mass of gas particles, g is gravitational acceleration, k is the Boltzmann constant, and τ is absolute temperature. In the case of thermal equilibrium, the temperature is assumed to be independent of the atmosphere altitude. As was noticed by Sommerfeld [1], “meteorologists sometimes protested against this statement.” Formula (1) is easily derived from the equilibrium condition for an ideal liquid

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.

The study of the oscillations of a system in the neighborhood of an equilibrium position or a periodic motion usual begins with linearization. The linearized system can be integrated. After this is done, the main properties of the oscillations in the original system can frequently be determined using the theory of normal forms of Poincare-Birkhoff. This theory is an analog of perturbation theory (Ch. 5, §2). Here the linearized system plays the role of the unperturbed system with respect to the original one. In this chapter we describe the basic elements of such an approach.

Suppose that two points (r 1, m 1) and (r 2, m 2) mutually interact with potential energy U(|r 1 − r 2|), so that the equations of motion have the form \({m_1}{\ddot r_1} = - \frac{{\partial U}}{{\partial {r_1}}},\;{m_2}{\ddot r_2} = - \frac{{\partial U}}{{\partial {r_2}}}.\)