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DOI 10.1007/s10958-015-2212-0 Journal of Mathematical Sciences, Vol. 204, No. 6, February, 2015

SESSIONS OF THE WORKSHOP OF THE MATHEMATICS AND MECHANICS DEPARTMENT OF LOMONOSOV MOSCOW STATE UNIVERSITY, "URGENT PROBLEMS OF GEOMETRY AND MECHANICS" NAMED AFTER V. V. TROFIMOV D. V. Georgievskii and M. V. Shamolin UDC 51(09)

(For the b eginning of the list of sessions see J. Math. Sci., 154, No. 4, 462­495 (2008); 161, No. 5, 603­614 (2009); 165, No. 6, 607­615 (2010); 187, No. 3, 269­271 (2012).) Session 221 (February 12, 2010) D. V. Georgievskii. On p ossible statements of the dynamical problem in terms of stresses in the isotropic theory of elasticity. In the case where loadings are given on the whole surface of a deformable rigid b ody, it is convenient to state and solve (by numerical and analytical methods) the b oundary-value problem for determining the stress-strain state in terms of stresses. We discuss some features and the efficiency of four statements of the dynamical problem in the linear isotropic theory of elasticity. Session 222 (February 19, 2010) V. I. Gorbachev. Longitudinal oscillations of ro ds with variable parameters. We consider the problem of longitudinal oscillations of a nonhomogeneous rod with variable crosssection (the original problem). The oscillation process is describ ed by a hyp erb olic second-order differential equation with coefficients dep ending on the coordinate. In the case where the coefficients are discontinuous, the equation is understood in the generalized sense. Along with the original problem, we also consider the accompanying problem for a rod with constant parameters (Young's modulus, density, cross-section) and with the same input data as for the original problem. The solution of the accompanying problem is much easier than the solution of the original problem, and in many cases it is p ossible to obtain its exact analytical solution. For the solution of the original problem, we obtain an integral representation through the solution of the accompanying problem and Green's function of the original problem. If the accompanying problem has a smooth solution, the integral relation implies a representation of the solution of the original problem as a series with resp ect to the coordinate and time derivatives of the solution of the accompanying problem. Moreover, the coefficients of the derivatives dep end only on the coordinates; they are weighted moments of Green's function for one of the two coordinate variables. For these coefficients, we construct a recurrent system of ordinary differential equations. We discuss in detail a sp ecial case of the general dynamic problem of determining the eigenfrequencies of longitudinal oscillations of a rod with variable parameters. We obtain exact frequency equations for various options for fixing the ends of the rod.
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 88, Geometry and Mechanics, 2013. 1072­3374/15/2046­0715 c 2015 Springer Science+Business Media New York 715


Session 223 (February 26, 2010) T. I. Garyaeva and D. V. Georgievskii. On the first b oundary-value problem of the theory of elasticity for a cylindrical layer with strongly differing characteristic dimensions. We analyze the principal terms of general asymptotic expansions of solutions to the first b oundaryvalue problem in the three-dimensional elasticity theory for displacements (quasi-statics, compressibility) for a cylindrical layer. The natural small parameter of the problem is the ratio of the thickness of the layer to the length of the generatrix. Session 224 (March 19, 2010) I. A. Buyakov. Features of deformation of a spirally armored shell. Session 225 (April 2, 2010) N. A. Belov and V. A. Kadymov. On a b oundary-value problem for a a thin plastic layer. The rep ort is devoted to the study and solution of Ilyushin's b oundary-value problem on spreading of a thin plastic layer b etween approaching rigid plates. It is known that a plastic medium in such flows can b e adequately describ ed by a model of a viscous liquid. A solution obtained earlier in the approximation of the ideal-fluid model does not satisfy a dynamic condition at the free b oundary. However, based on this solution, one can also obtain the evolution equation for the b oundary. This work is devoted to the analytical study of the b oundary-value problem. In a neighb orhood of the b oundary layer, we obtain a solution that satisfies all b oundary conditions. If the flow of an ideal fluid is directed along the normal to the b oundary, then the velocity of the flow in the b oundary layer has nonzero tangent comp onent. We also show that for a sufficiently smooth b oundary, the evolution equation for the b oundary coincides with an equation obtained earlier. Session 226 (April 9, 2010) D. V. Georgievskii. On the work of the 81st Annual Meeting of the Society for Applied Mathematics and Mechanics, March 20­27, 2010, Karlsruhe, Germany. Session 227 in the framework of the Conference of young scientists of the Faculty of Mechanics and Mathematics of the M. V. Lomonosov Moscow State University and the International conference of students, postgraduates, and young scientists "Lomonosov­2010" (April 27, 2010). 1. M. V. Vassilieva. Numerical simulation of single-phase filtration on multiprocessor systems. 2. T. V. Yakovleva. Control of complex nonlinear vibrations of flexible b eams. Session 228 (7 V. A. Danilov. Calculation of currents. We prop ose new problems in the th 716 May 2010) time-dep endent characteristics of axially symmetric elastoviscoplastic formulations and numerical and analytical methods for solving b oundary-value eory of elastoviscoplasticity of the start, acceleration, and braking to complete


stopping of axisymmetric flows with subsequent discharge. We take into account the accumulation of residual stresses and strains. For viscoplastic flows, we find conditions for the app earance and motion of the b oundaries of an elastic core. The Taylor­Couette flow b etween two coaxial cylinders is studied in detail. Session 229 (4 June 2010) M. U. Nikabadze. Some problems of microp olar elasticity theory. We consider some problems of microp olar elasticity theory. In particular, equations of motion (equilibrium) in stress tensors and moment stresses in the three- and two-dimensional theories are reduced to statements on the absence of volume load and inertial forces. We obtain a formula that expresses the stress tensor through the tensor of moment stresses and introduce tensor-functions of stresses. The equations of motion in terms of displacements and rotations are written in op erator form. For an arbitrary anisotropic material, we introduce four differential tensor-op erators and obtain expressions for the corresp onding differential tensor-op erators for different cases of anisotropy. We discuss isotropic, transversally isotropic, and orthotropic materials. Moreover, we introduce the matrix differential tensor-op erator whose differential subtensor-op erators are the tensor-op erators introduced ab ove; this matrix differential tensor-op erator allows one to write the equations of motion in the form of a single matrix differential tensor-op erator equation. The case of an isotropic material with a center of symmetry is studied in detail. We obtain expressions for differential tensor-op erators of cofactors and determinants of all differential tensor-op erators introduced ab ove, except for one tensor-op erator whose determinant vanishes. The expressions for the corresp onding matrix differential tensor-op erator of the cofactors and the determinant are also obtained in the case where the interior tensor of inertia of the material is an arbitrary tensor of rank 2 in principal axes. Further, we introduce differential tensor-op erators and matrix differential tensor-op erators, which, if applied to the corresp onding equations, allow one to split the system of equations and obtain separate equations for unknown vector-functions (vectors of displacement and rotation). Unlike the Galerkin representation, the b oundary conditions of the b oundary-value problems are preserved. We also obtain representations of solutions of the Galerkin problem. Problems similar to those describ ed ab ove are also considered for the classical elasticity theory as particular cases. We present three methods of obtaining formulas of the general complex representation in the planar microp olar elasticity theory taking into account volume loads in nonisotermic processes. In two methods, formulas are represented through two analytic functions of a complex variable and the general solution of the inhomogeneous Helmholtz equation and in the third method through three analytic functions of a complex variable. Session 230 (18 June 2010) V. E. Palosh. Stability of nonconservative mechanical systems. We examine the stability of equilibria for four nonlinear problems. First problem: a double p endulum loaded with tracing and conservative forces. The joints of the p endulum p ossess viscoelastic prop erties. The stability problem is solved for critical cases of one zero root, two imaginary roots, and one zero and two imaginary roots. Second problem: a homogeneous rod moving under the action of a constant tracing force applied to one of its ends. The emergence of the destabilization paradox is shown. Third problem: stabilization of the motion of a dynamically symmetric satellite by using exterior moments. In the linear case, we examine the p ossibility of the satellite with resp ect to the center of 717


mass by using constant moments. In the nonlinear case, we consider the critical case of stability of two pairs of imaginary roots and find the conditions of asymptotic stability. Fourth problem: a solar sail. In the linear statement, we examine the Lyapunov stability of the equilibrium of a solar sail. In the nonlinear statement, we show that the proof of Lyapunov stability is imp ossible. We examine the stability of the zero equilibrium for a variety of initial conditions. Session 231 (Septemb er 10, 2010) D. V. Georgievskii. Eigenvalue problem for the generalized Orr­Sommerfeld equation in the theory of hydrodynamical stability. Based on the method of integral relations, we analytically examine the stability of a numb er of one-dimensional plane-parallel stationary Newtonian flows. The mathematical statement is reduced to eigenvalue problems for the Orr­Sommerfeld equation. As b oundary conditions, we consider vanishing of all comp onents of the p erturbation of the velocity on b oth b oundaries of the layer (in this case, we have the classical Orr­Sommerfeld problem); vanishing of all comp onents of the p erturbation of the velocity on one b oundary and vanishing of the p erturbation of the tangent comp onent of the stress vector and the normal comp onent of the velocity on the other b oundary; vanishing of all comp onents of the p erturbation of the velocity on one b oundary and the requirement that the other b oundary is free. Boundary condition for the last case contain the sp ectral parameter. For kinematic conditions, we improve lower estimates of the critical Reynolds numb er. In other cases, we develop the method of integral relations, which leads to new stability results. Session 232 (Septemb er 17, 2010) S. V. Bogomolov. Problem on sto chastic diffusion mo dels in gas dynamics. The accuracy and efficiency of numerical algorithms in gas dynamics can b e improved by constructing a hierarchy of mathematical models based on micro-macro representations. The approach is usually based on the Boltzmann equation whose dimensionless form contains the coefficient 1/Kn of the collision integral, where the parameter Kn (the Knudsen numb er) dep ends on the space variable x. In view of the current high demands placed on the quality of computing technologies, the whole domain in which calculations are p erformed is partitioned into sub domains with different prop erties. If Kn is of the order of unity, then it is a sub domain in which the Boltzmann equation is needed. In domains where Kn is moderately small, one can use the Kolmogorov­Fokker­Planck equation whose coefficients are determined by the collision model and can b e calculated in explicit form under some simplifying assumptions. This nonlinear equation with resp ect to the seven-dimensional distribution function in the phase space is simpler than the Boltzmann equation; instead of the collision integral it contains the transfer op erator with diffusion in the velocity space, which can b e called the model collision integral. For moderate values of Kn, one can also obtain a macroscopic description: the equations of stochastic quasi-gas dynamics that are related to the Kolmogorov­Fokker­Planck equation by the coefficients obtained by averaging in space and time. For very small Kn these equations are similar to the Navier­ Stokes equations. A micro-macro "bridge" describ ed in the language of deterministic equations can b e constructed by using the theory of stochastic processes starting from the system of stochastic differential equations that describ e the gas for moderate and small values of the Knudsen numb er. We obtain a set of stochastic models that generate a sequence of Monte-Carlo methods that are promising from the 718


p oint of view of sup ercalculations. Our approach differs from other methods of construction of quasigas-dynamical equations by underlying hyp otheses that allow one to obtain a simpler description of a gas as compared to a kinetic description. Session 233 (Octob er 1, 2010) K. V. Kvachev. Lyapunov­Movchan method in a dynamical problem of elasticity theory. We consider a rectangular plate of constant thickness made of a homogeneous, elastic, isotropic material governed by Hooke's law. Two parallel edges of the plate are pivotally mounted, one edge is fixed, and the fourth edge is free. A sup ersonic flow of a gas flows around the plate; this flow is parallel to the pivotally mounted edges and is directed from the fixed edge to the free edge. We obtain stability conditions for oscillations of the plate and find the critical sp eed by using the Lyapunov­ Movchan stability theorem. Session 234: a joint session with the educational and scientific seminar of the department of Applied Mathematics of the Bauman Moscow State Technical University (Octob er 8, 2010) M. U. Nikabadze. On the statement and metho ds of solution of b oundary-value problems in the theory of thin b o dies. We consider a parametrization of the domain of a thin b ody with two small sizes in the case where an arbitrary line is chosen as the base line, and the classical parametrization of the domain of a thin b ody, i.e., when the median line is taken as the base line. We present three-dimensional statements of problems for a thin elastic b ody with two small sizes for considered parametrizations of the domain of the b ody. Based on these statements and using the moment theory with resp ect to the system of Legendre p olynomials, we obtain the corresp onding statements of problems in moments. We p erform the expansion of mechanical functions with resp ect to the system of Legendre p olynomials in one or in two transversal coordinates. Further, using the method of normalized moments for stress tensors and moment stressed and simplified method of the reduction of an infinite system of equations in moments to a finite system, we obtain the statements of problems of the zero, first, and second approximations. We also establish conditions under which equations of the first approximation imply the equations of the classical theories (Euler­Bernoulli and Timoshenko). Session 235 (Octob er 15, 2010) M. V. Shamolin. Survey of integrable cases in the dynamics of a four-dimensional rigid b ody in a nonconservative field. We review complete results on the study of the equations of motion of a dynamically symmetric four-dimensional (4D) rigid b ody in a nonconservative field that are currently available. The form of these equations is b orrowed from the dynamics of realistic two-dimensional (2D) and three-dimensional (3D) rigid b odies interacting with a resisting medium according to jet flow laws such that the b ody is under the action of a nonconservative force couple that forces the center of mass of the b ody to move rectilinearly and uniformly. In the rep ort, we discuss some integrable cases of the problem of motion of a b ody in a resisting medium that fills four-dimensional space, under the action of a tracing force that allows one to reduce the order of the general system of dynamical equations of motion. Earlier, the author proved the complete integrability of the equations of plane-parallel motion of a b ody in a resisting medium under the conditions of jet flow where the system of dynamical equations 719


p ossesses a first integral which is a transcendental function (in the sense of complex analysis, i.e., as a function of a complex variable having essential singularities) of quasi-velocities. In those works, it was assumed that the interaction of the b ody with the medium is concentrated on a part of the surface of the b ody that has the form of a (one-dimensional) plate. Later, the planar problem was generalized to the spatial (three-dimensional) case, in which the system of dynamical equations also has a complete set of transcendental first integrals. Here we already assume that the interaction of the b ody with the medium is concentrated on a part of the surface of the b ody that has the form of a planar (two-dimensional) disk. In the present work, we generalize some known results concerning integration of the equations of motion of two-dimensional and three-dimensional rigid b odies under the action of a nonconservative moment and examine the equations of motion of a dynamically symmetric four-dimensional rigid b ody. The structure of these equations is preserved after the generalization to higher-dimensional cases. Session 236 (Novemb er 12, 2010) B. N. Khimchenko. Hamiltonian systems and formula for calculation of the Laplacian of the eikonal. The necessity of the calculation of the Laplacian of the eikonal app ears in a series of problems of mathematical physics, for example, in transp ort equations. Integration of such equations leads to the construction of asymptotic expansions of solutions of differential equation with resp ect to a small parameter. We show that the conventional formula for calculation of the Laplacian of the eikonal is invalid and prop ose a new scheme for calculation using the matrix Riccati equation. Session 237 (Novemb er 19, 2010) I. A. Buyakov and G. E. Tashchilova. On the carrying capacity of compressed anisotropic tubular ro ds. Using the momentless theory of anisotropic shells, we study the carrying capacity of tubular anisotropic rods under axial compression. We obtain the dep endence of the carrying capacity of tubular rods on the complex of elastic characteristics of anisotropic material. Session 238 (Decemb er 3, 2010) N. N. Shamarov. Maslov­Poisson measure method. We present a method of solution of evolutionary equations whose particular case is equivalent to od Maslov's method for the solution of a certain class of Schr¨ inger equations by using the functional integral over a complex-valued countably additive measure of Poisson typ e. This measure is defined on the space of tra jectories in the dual space of physical coordinates and the corresp onding integral can b e called the integral by tra jectories in the momentum space. While the original Maslov's method is based on an exp onential series of Dyson typ e, the method discussed is based on product formulas of Chernov and Trotter typ e for approximations of one-parameter op erator semigroups. This method is applicable to equations with matrix coefficients (in general, not commuting), for example, the Schrodinger equation with matrix effective p otential, the famous Dirac ¨ equation for a relativistic electron, and the classical heat equation with matrix effective source (sink) of heat. Moreover, the method can b e adapted to analogs of the heat equation in which the "space variable" runs through a space over the field of -adic numb ers (not necessarily finite-dimensional), and the role of the Laplace op erator is played by the Vladimirov op erator (or its infinite-dimensional analog). 720


The modified method also leads to integrals by means of tra jectories different from Feynman integrals in the phase space for a solution of the classical Schr¨ inger equation. od Session 239 (Decemb er 10, 2010) A. V. Mokeev. On some problems of differential diagnostics. We discuss and solve two basic problems of differential diagnostics: the monitoring problem and the problem of fault diagnosis. The solutions of these problems are based on the mathematical model of the motion of the ob ject considered (in terms of dynamical control systems), including the domain of its initial conditions, the a priori list of p ossible faults, and the mathematical model of the motion of the ob ject under a corresp onding fault. We define the notion of a neighb orhood of a reference fault and, based on it, introduce a top ology on the space of reference faults (the so-called diagnosis space). The monitoring problem is solved by the construction of a monitoring surface; it can b e solved in b oth deterministic or statistical statements. We prop ose a method of construction of the monitoring surface by using the Monte-Carlo method. For the problem of fault diagnosis, we prove a diagnosis theorem and, as a consequence, prop ose two diagnosis algorithms. Session 240 (Decemb er 24, 2010) S. A. Dovbysh. Foundations of Ziglin's theory (nonintegrability of dynamical systems and monodromy groups of equations in variations). This rep ort is an introductory lecture for untrained readers. We discuss principal ideas and results of the theory develop ed by Ziglin, which states sufficient conditions of nonexistence of additional first integrals of dynamical systems in terms of prop erties of the normal equation in variations (NEV) along a known particular solution of the initial system (IS). We consider the following notions and results: construction of a homogeneous p olynomial (resp ectively, rational) first integral of the NEV from an analytic (resp ectively, meromorphic) first integral of the IS; a lemma that states that the existence of k functionally indep endent meromorphic first integrals of the IS implies the existence of k functionally (and, equivalently, algebraically) indep endent rational first integrals of the NEV; the monodromy group of the NEV and the relation b etween the integrability of the IS and the existence of rational first integrals of the monodromy group; necessary conditions of complete integrability of the Hamiltonian IS in terms of commutator prop erties of two elements of the monodromy group, one of which is nonresonant; a version of this result for systems with two degrees of freedom and its generalization to the case of a nondiagonalizable element. Session 241 in the framework of the International Scientific Symposium "Problems in Mechanics of Solids" devoted to the 100th Anniversary of A. A. Il'yushin (January 21, 2011) (January 21, 2011) D. V. Georgievskii and R. Wil le. Asymptotical integration in b oundary-value problems on a p erfect rigid plastic flow in a thin layer. The method of asymptotic integration for a series of b oundary-value problems on incompressible, p erfect, rigid, plastic flow in a thin plane layer under loading is develop ed by analytic methods. The material of the layer may occupy an arbitrary domain. We prop ose an algorithm for the construction of an asymptotic solution and consider the p ossibility of a p erfect, rigid, plastic flow along one of the coordinate lines. The results for some particular cases are discussed: the classic Prandtl problem and its axially symmetric analogs. 721


Session 242 (February 11, 2011) L. E. Evtushik. The Ostrogradsky theorem on the Hamiltonization of the Euler­Lagrange equations for Lagrangians with higher-order derivatives.

Session 243 (February 18, 2011) V. I. Van'ko. Cylindrical shell under an exterior pressure: nonclassical solution of the problem on large displacements.

Session 244 (February 25, 2011) I. A. Buyakov. The sudden stratification and bulging of a three-layer ro d under compression. We consider the problem of the exhaustion of the b earing capacity of a three-layer rod under compression due to the sudden detachment of its supp orting layers from the filler. We find the characteristic length of the lifting zone which must b e taken into account in the assignment of additional transversal constraints b etween the layers under insufficient adhesive or cohesive solidity of the adhesive b ond. Session 245 (March 11, 2011) D. V. Georgievskii. Gravitational stability of some two-layer vertically moving systems. We study the evolution of small initial p erturbations in a system consisting of a heavy layer of a Newtonian fluid that covers the half-space of an ideal fluid with another density. This system as a rigid b ody can move in the vertical direction by a given law. Linearizing the equations and b oundary conditions, we obtain the characteristic equation and consider the large-viscosity limit. Session 246 (March 18, 2011) E. S. Perelygina. Longitudinal b ending of an elastic-plastic rod under free diagram -. Session 247 (March 25, 2011) S. A. Dovbysh. Nonintegrability of dynamical systems and monodromy groups and Galois groups of linear ordinary differential equation. Some asp ects of Ziglin's theory (continuation). Session 248: a joint session with the educational and scientific seminar of the department of Applied Mathematics of the Bauman Moscow State Technical University (April 8, 2011) D. V. Georgievskii. Asymptotic analysis of the Prandtl problem in the dynamical statement. The dynamical statement of the problem of compression of a thin, ideal, rigid plastic layer by absolutely rigid plates that move towards each other with constant sp eeds contains two sp ecific dimensionless parameters. One of them, a small geometric parameter equal to the ratio of the thickness of the layer to its length explicitly dep ends on time; moreover, its order of smallness increases with time. The other dimensionless parameter, the reciprocal Euler numb er, is indep endent of time; we assume that it is much less than unity. Dep ending on the relation b etween these parameters, i.e., in 722


different time intervals, we construct solutions in the form of expansions by integer p owers of using the procedure of asymptotic integration. Session 249 (April 15, 2011) M. M. Kantor. Modeling of deformation of thin b odies with two small sizes. We consider a parametrization of the domain of a thin b ody with two small sizes in the case where an arbitrary line is chosen as the base line, and the classical parametrization of the domain of a thin b ody, i.e., when the median line is taken as the base line. We present three-dimensional statements of problems for a thin elastic b ody with two small sizes for these parametrizations of the domain of the b ody. We develop the theory of moments with resp ect to Legendre p olynomials that allows one to calculate the moment of order (m, n) of any expression. Based on these statements and using the theory of moments with resp ect to the system of Legendre p olynomials, we obtain the corresp onding statements of the problem in moments. We p erform an expansion of mechanical functions with resp ect to the system of Legendre p olynomials in one or in two transversal coordinates. We construct various approximations from zeroth to fifth order for the classical and microp olar theories. To satisfy the b oundary conditions on the front surfaces, we use the method of normalized moments of the stress tensors and moment stresses and the method of correcting terms. We write programs that allow one to p erform a numerical simulation of test problems and compare the obtained numerical results with classical solutions, including the solution obtained by the method of finite elements. The comparison is p erformed for the problem for a microp olar two-dimensional domain, the problem of the action of concentrated forces on a two-dimensional domain, and the problem for a two-layer two-dimensional domain. Session 250: anniversary session (April 29, 2011). Session 251 in the framework of the XV International Conference "Dynamical Systems: Modelling and Stability" (May 27, 2011). 1. S. A. Agafonov and I. A. Kostyushko. On the reduction of a nonautonomous linear system and an application. We consider a nonautonomous linear system of ordinary differential equations. We reduce it to an autonomous system and then study it by well-known methods. 2. V. I. Van'ko. Cylindrical shell under an exterior pressure: nonclassical solution of the problem on large displacements. In previous works, the author considered the problem on large displacements of p oints of the median surface of an infinitely long, circular, cylindrical shell under the action of an exterior hydrostatic pressure. The author has succeeded in following the deformation process until to complete flattening by using a kinematic scheme. In this work, the influence of the parameters of characteristic length and thickness and the b oundary-value condition on the deformation process is examined. 3. D. V. Georgievskii. On generalized Orr­Sommerfeld problems in continuum mechanics. We study the generalized Orr­Sommerfeld equation in the linearized theory of hydrodynamical stability. Methods of integral relations that have b een intensively develop ed in the past decades for materials with complicated defining relation are also effective for this sp ectral problem. These methods allow one to complete sufficient estimates of the stability of a process without knowing any exact or approximate 723


solution of the linearized problem at each time instant. Generalizations of the Squire theorem for sp ectral problems are also obtained. 4. M. V. Shamolin. Comparison of completely integrable cases in the dynamics of 2D, 3D, and 4D rigid b odies in nonconservative fields. The complete integrability of the equations of motion of a four-dimensional rigid b ody has b een examined in a large numb er of works. The author has succeeded in generalizing the equations of motion of low-dimensional (two- and three-dimensional) rigid b odies in nonconservative fields to the motion of a four-dimensional rigid b ody in a similar field. As a result, the author obtained several new integrable cases in the problem of motion of a b ody in a resisting medium that fills four-dimensional space under the action of some tracing force that allows one to reduce the order of the general system of dynamical equations of motion. In the present work, we generalize some previously known results on integration of the equations of motion of two- and three-dimensional rigid b odies under the action of a nonconservative force moment and study the equations of motion of a dynamically symmetric four-dimensional rigid b ody in one of the two p ossible cases, dep ending on the principal moments of inertia. The structure of such equations of motion is preserved in some sense when they are generalized to higher-dimensional cases. Session 252: Towards the XX All-Russia Congress on Fundamental Problems of Theoretical and Applied Mechanics (June 10, 2011). Session 253 (Septemb er 9, 2011) V. S. Yushutin. Stability of a deformable channel in the case where a nonlinearly viscous medium with p ower hardening law flows in it. We consider a dynamical model of a system consisting of a cylindrical deformable vessel and a nonlinearly viscous medium with p ower hardening law that flows in the vessel. The flow and the deformation are assumed to b e axisymmetric. We examine the stationary solutions and their stability with resp ect to small p erturbations. In the space of dimensionless parameters of the problem, we construct stability domains. Session 254 (Septemb er 16, 2011) O. A. Ryabova. Mathematical modeling of stress-strain states of b o dies with rigid circular inclusions under finite planar deformations. We develop a method of construction of mathematical models that under finite planar deformations, describ e the stress-stain states of infinite nonlinearly elastic and viscoelastic b odies with rigid circular inclusions that app ear after loading. Session 255 (Septemb er 30, 2011) K. V. Kvachev. Statement of the dynamical problem on the stability of oscillations of a cylindrical shell in a sup ersonic gas flow. We consider a cylindrical shell of constant thickness made of a homogeneous, elastic, isotropic material governed by Hooke's law. One edge of the shell is fixed and the other is free. A sup ersonic flow of a gas streamlines the plate; this flow is parallel to the pivotally mounted edges and is directed from the fixed edge to the free edge. We present the statement of the problem under the assumption that deformations are small and the "piston" theory is applicable. 724


Session 256 (Octob er 14, 2011) I. A. Buyakov. Generalized edge effect in thin-wall laminate b o dies.

Session 257 (Octob er 21, 2011) V. M. Ovsyannikov. Using higher-order terms of the finite-difference continuity Euler equation in phenomena of mixing and generation of sound.

Session 258 (Octob er 28, 2011) D. V. Kuznetsova and I. N. Sibgatul lin. Intermittency on the background of quasi-p erio dic regimes in p enetrative convection. We consider convection in a planar layer of water in a temp erature interval that includes the maximum density p oint 4 C. We describ e qualitative features of p eriodic, two-p eriodic, and quasip eriodic regimes. We examine the stability of stationary and p eriodic regimes on large horizontal scales. When the stability of quasi-p eriodic regimes is lost, intermittency occurs. Session 259 (Novemb er 11, 2011) M. V. Shamolin. Systems with variable dissipation: approaches, methods, and applications. This work is a survey of integrable cases in the dynamics of two-, three-, and four-dimensional rigid b odies in nonconservative fields. The problems are describ ed by dynamical systems with variable dissipation with zero mean. The problem of the search for complete sets of transcendental first integrals of systems with dissipation is quite urgent, and many pap ers are devoted to it. We consider a new class of dynamical systems that have a p eriodic coordinate. Owing to the fact that such systems have nontrivial symmetry groups, we prove that such systems p ossess variable dissipation, which means that on the average for a p eriod with resp ect to the p eriodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space either energy pumping or dissipation can occur. Dynamical systems that app ear in the dynamics of a rigid b ody are analyzed and several new cases of complete integrability of the equations of motions in transcendental functions that can b e expressed through finite combinations of elementary functions are detected. Some generalizations of integrability conditions for more general classes of nonconservative dynamical systems in the dynamics of a four-dimensional rigid b ody are obtained. Session 260 (Novemb er 18, 2011) I. Kh. Sabitov. On an ill-p osed b oundary-value problem of Markushevich.

Session 261 dedicated to the memory E. V. Lobanov (Novemb er 25, 2011) D. V. Georgievskii. Asymptotics in the Prandtl problem in a dynamical statement.

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Session 262 (Decemb er 2, 2011) M. U. Nikabadze. A complete system of eigentensors for a p ositive-definite symmetric tensor of any even rank.

Session 263 (Decemb er 16, 2011) A. R. Ulukhanyan. On the general disp ersion equation for a homogeneous anisotropic microp olar medium. We obtain a general disp ersion equation for a homogeneous, anisotropic microp olar medium. We consider a homogeneous, microp olar, infinite medium p ossessing a center of symmetry. The equations of motion of the microp olar theory and the defining relations of the microp olar medium are written in comp onents. Substituting the defining relations in the equations of motion, we obtain equations for the homogeneous, anisotropic microp olar medium in terms of comp onents of translation and rotation vectors. Since the equations obtained are hyp erb olic, we find solutions in the form of two wave functions. As usual, equating the determinant of the system obtained to zero, we derive the general disp ersion equation, which allows one to find the sp eed of waves in the infinite, microp olar anisotropic medium. This also shows that in each direction, no more than six waves can propagate; to find them, one needs to solve biquadratic equations. For a microp olar, isotropic, elastic medium, disp ersion equations were obtained, for example, by V. Novatsky. M. Nikabadze proved that the mechanical prop erties of microp olar, elastic, transversally isotropic media and orthotropic media are describ ed by 20 and 30 indep endent comp onents, resp ectively. Knowing the numb er of indep endent comp onents, we can obtain similar equations for transversally isotropic and orthotropic media from the general disp ersion equation. We present general representations of solutions of hyp erb olic equations of fourth and sixth orders. Earlier, the author obtained a system of equations of motion for a microp olar, anisotropic, elastic medium of variable thickness with resp ect to the system of orthogonal Legendre p olynomials. Based on this system, in the case of the classical theory, for isotropic and transversally isotropic prismatic b odies of constant thickness, hyp erb olic equations of fourth and sixth orders in the first and second approximation, resp ectively, were obtained. By the Fourier method of separation of variables, hyp erb olic equations of fourth and sixth orders are reduced to elliptic equation of the same orders. By the Vekua method of representation of general solutions of elliptic equations of order 2n by n analytic functions, we present general representations of solutions of hyp erb olic equations of fourth and sixth orders. We also consider the problem on the b ending of a rectangular plate in the first approximation. Session 264 (March 2, 2012) V. Yu. Alekseev. Some metho ds of solution of dynamical problems in the mechanics of comp osites with p eriodic structure. Session 265 (March 23, 2012) D. V. Georgievskii. Adjusting exp eriments for finding material functions for tensor nonlinear defining relations. We describ e the scheme of an adjusting exp eriment for finding material functions of two invariants that enter in the defining relations of a tensor nonlinear incompressible medium. As a basic flow of such medium, we take a combination of radial spreading and two one-dimensional shifts in two mutually p erp endicular directions that are realized in a cylindrical layer. At the b oundaries of the 726


cylinders, leaking can occur, but the tangential comp onent of the velocity of the medium is equal to the velocity of the motion of the cylinders. We state the problem of the existence of a viscous p otential for sp ecific media. Session 266 (March 30, 2012) M. V. Shamolin. On the problem of a p endulum in a nonconservative case. This work is a survey of integrable cases for the equations of motion of a fixed p endulum in a nonconservative field. These problems are describ ed by dynamical systems with variable dissipation with zero mean. The problem of the search for a complete sets of transcendental first integrals of systems with dissipation is quite current and many pap ers are devoted to it. We consider a new class of dynamical systems that have a p eriodic coordinate. Owing to the fact that such systems have nontrivial symmetry groups, we prove that the systems p ossess variable dissipation, which means that on the average for a p eriod with resp ect to the p eriodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either energy pumping from outside or dissipation can occur. Dynamical systems that app ear in the dynamics of a rigid b ody are analyzed and several new cases of complete integrability of the equations of motions in terms of transcendental functions that can b e expressed through finite combinations of elementary functions are analyzed. Some generalizations of integrability conditions for more general classes of nonconservative dynamical systems in the dynamics of a four-dimensional rigid b ody are obtained. Session 267 (April 6, 2012) V. V. Vedeneev. Panel flutters under low sup ersonic sp eeds. In the theory of panel flutters (instabilities of elastic plates in a gas flow), two typ es of stability violation are known: flutter of fixed typ e and one-mode flutter. The first typ e app ears in the case of large sup ersonic sp eeds of the flow; it is studied in detail by the "piston theory," i.e., an approximate expression for the p erturbation of the pressure acting on the plate for large Mach numb ers . The agreement of the b oundaries of the flutter of fixed typ e with exp eriments for M > 1.7 is satisfactory. The one-mode flutter app ears for 1 < M < 2 and have not yet b een studied since the exact theory of p otential gas flows that is necessary for its study leads to complicated integro-differential equations. Moreover, in exp eriments, the one-mode flutter has not b een occurred. We present the results of recent theoretical and exp erimental studies of plate flutter for low sup ersonic sp eeds. Using the asymptotic global-instability method, we solve the problem on the flutter of a plate of large but finite size. We obtain criteria of fixed and one-mode flutters and examine the structure of the sp ectrum of the system. We also study the physical reasons of the amplification of oscillations under one-mode flutter. We study the influence of the b oundary layer on one-mode flutter. The stabilization or destabilization of p erturbations of the plates by a b oundary layer is determined by the profile of the layer and its thickness. We find that amplification of oscillations of the plate for a sp ecific shap e of its profile and thickness of the b oundary layer is p ossible. We numerically solve the problem of stability of a plate in a p otential flow for arbitrary sizes of the plate. The b oundary of the flutter (b oth one-mode and fixed) are compared with asymptotic results. We p erform an exp erimental study of panel flutter in the transonic range of sp eeds. The app earance of one-mode flutter in a realistic construction is identified for the first time. We theoretically examine nonlinear oscillations of the plate and the growth of the amplitude of a limit cycle of oscillations. We prove the p ossibility of simultaneous existence of different limit cycles. 727


Session 268 (April 13, 2012) V. I. Van'ko. On the stability of elements of constructions. We study the b ehavior of the Shanley rod model and present principal statements and features of the longitudinal b ending of elasto-plastic elements. Using a numerical solution of the problem of b ending of a rod made of an elasto-plastic material (including accounting of creep) we conclude that the geometrically linear statement of the problem is correct (i.e., to a small increment of the force or time there corresp onds a p ositive increment of the deflection) until the longitudinal force exceeds the b ending stiffness of the most highly stressed (by the value of the intrinsic moment) cross-section (in dimensionless values). We present a nonclassical approach to the study of large displacements of p oints of the median surface of circular cylindrical shells (infinitely long or of finite length) under an exterior hydrostatic pressure. We consider shells made of linearly or nonlinearly elastic and linearly or nonlinearly viscous (steady creep) materials. We also present examples of numerical solutions of the problems considered comparing them to the results of asymptotic analysis. In the study of the b ehavior of an aerodynamical profile in an air flow (plane-parallel motion with three degrees of freedom), we obtain a sufficient condition of instability of equilibria in the Lyapunov sense. We give a generalization of this condition to the system of circular profiles and discuss p ossible applications of these results to the structural design of split-phase wires of high voltage p ower lines. Session 269 (April 20, 2012) D. V. Georgievskii. General solutions of systems in stresses that are not equivalent to the classical system of elasticity theory. We analyze general solutions of some weakened systems of equations in stresses in the isotropic theory of elasticity. These nonequivalent classical systems, in addition to equilibria equations, contain only three of six compatibility equations, diagonal or off-diagonal. We discuss the equivalence of the statements of quasi-static b oundary-value problems in terms of stresses of elasticity theory. Session 270 (April 27, 2012) M. U. Nikabadze. Anisotropy in the linear microp olar elasticity theory. In the linear microp olar elasticity theory of anisotropic b odies without centers of symmetry in the sense of elastic prop erties, we introduce tensor-columns of the stress tensors and moment stress tensors and also the deformation and b ending-torsion tensors. We also introduce the block-tensor matrix of tensors of elasticity moduli; it consists of four tensors of fourth rank. We give representations of the elastic deformation energy and defining relations (Hooke's law) using the introduced tensor-columns and the block-tensor matrix. We define p ositive-definite block-tensor matrices and prove the p ositive definiteness of the matrix of the tensor of elasticity moduli. We introduce the notions of an eigenvalue and an eigentensor-column of a block-tensor matrix and consider the problem of finding the eigenvalues and eigentensor-columns. In explicit form, we construct a complete system of eigentensors for a p ositive-definite symmetric tensor of any even rank and a complete system of eigentensor columns for a p ositive-definite symmetric block-tensor matrix consisting of tensors of the same even rank in a space of arbitrary finite dimension. Sp ecial cases are considered. 728


Session 271 (May 11, 2012) K. V. Kvachev. The Lyapunov­Movchan method in a problem on the stability of oscillations of a thin elastic cylindrical shell in a gas flow. By using the Lyapunov­Movchan method, we consider the dynamical stability of an elastic cylindrical shell of finite length that is rigidly clamp ed on one edge and is free on the other in a sup ersonic gas flow. Displacements of the shell are sub ordinated to some restrictions that allow one to integrate one of the equations of motion and construct the Lyapunov­Movchan functional. Applying the Lyapunov stability theorem, we obtain restrictions for the sp eed and mechanical and geometrical parameters of the system. The results obtained are compared with the results of the problem of stability of axisymmetric oscillations. Session 272 (May 18, 2012) V. S. Yushutin. Integral statement of visco elastic flows in a cylinder. We consider a longitudinal axisymmetric flow of a viscoplastic Shvedov­Bingham medium inside a cylinder. The motion is caused by a pressure drop varying in time. Such problems were studied earlier by numerical simulation. The problem contains two variables: the transversal coordinate and time. We prop ose to apply the semi-inverse method starting from the profile of the longitudinal velocity and integrate the equation of motion with resp ect to cross-section. Then the problem will dep end only on time and the corresp onding equations b ecome ordinary differential equations. In the framework of this integral statement, the problem can b e considered as a dynamical system with two variables: the velocity and the size of the stiffness core located at the channel axis. The stationary p osition of the dynamical system corresp onds to stationary Poiseuille flows and is a stable-node typ e p oint. We construct numerical solutions of the acceleration of the medium by various regimes of the pressure drop. If a pressure drop is absent, then the Shvedov­Bingham medium moves by inertia and, as is well known, stops in finite time. Based on the integral statement of the problem, we obtain estimates of the stop time and compare them with known upp er estimates of the stop time in the exact statement of the problem. Session 273 (June 1, 2012) V. A. Kadymov and N. A. Belov. On the spreading of a plastic layer consisting of various media b etween approaching rigid plates. Back in the mid-19th century, A. A. Il'yushin prop osed an effective two-dimensional, averaged (with resp ect to the thickness of the layer) mathematical model of a flow of a thin plastic layer b etween approaching rigid plates. In the framework of this model, he stated the b oundary-value problem for a linearly viscous fluid with resp ect to three unknown functions, two comp onents of the velocity of the flow and the contact pressure in a domain with moving b oundary. He also indicated the p ossibility of simplifying the statement of the problem to a flow of an "ideal fluid." Subsequently, all researchers restricted themselves to this simplified statement. In the present work, we consider flows of a plastic layer that cannot b e describ ed in the framework of the model of an ideal fluid, in particular, the spreading of a layer consisting of various media. For the correct description of such flows, one must use a model of a viscous fluid or the simpler theory of b oundary layers. 729


Session 274 (Octob er 19, 2012) D. V. Georgievskii and M. V. Shamolin. Urgent problems of geometry and mechanics: foundations, problems, methods, and applications. At the session of the seminar, a numb er of problems were suggested to undergraduate students, graduate students, and researchers. The following domains of research were prop osed: relative structural stability in nonconservative dynamical systems; problems on the existence of closed orbits on manifolds of various top ological typ es; the dynamics of a rigid b ody interacting with a resisting medium under quasi-stationary conditions; integrability and nonintegrability in the classical dynamics and dynamics of strongly nonconservative systems; problems of structural optimization; dynamics of multidimensional rigid b odies; characteristic classes (invariants) of Maslov­Arnold­Trofimov and generalized characteristic classes for dissipative systems; stability of processes in the mechanics of deformable rigid b odies with resp ect to some classes of p erturbations (general methods, approaches, and criteria); stability of flows of materials with complicated rheology and intrinsic structure (comp osites); stability of rods, plates, shells, and other typical elements of engineering constructions; visco-elasticplastic flows (stationary and nonstationary) in nonclassical domains; theory of defining relations in biomechanics; the sensitivity of solutions of problems with resp ect to p erturbations of material functions; numerical modeling of the violation of stability and p ossible physical interpretations. Session 275 (Octob er 26, 2012) K. V. Kvachev. The Lyapunov­Movchan metho d in dynamical problems of elastic stability. This rep ort is a presentation of a Ph.D. thesis. The thesis consists of four chapters. The first chapter is a survey of the Lyapunov­Movchan method. The other three chapters are devoted to the mathematical development of this method and applications to problems of aeroelastic stability. In the second chapter, problems on the stability of oscillations of plates of various configurations in a sup ersonic gas flow are considered. The third and fourth chapters are devoted to problems on the stability of oscillations of cylindrical shells in a sup ersonic gas flow. For each problem considered, we obtain sufficient stability conditions in terms of an upp er estimate of the critical sp eed. In problems on the stability of oscillations of cylindrical shells, additional restrictions on the change of mechanical and geometrical parameters of the system app ear. Session 276 (Novemb er 9, 2012) R. R. Gadelev. On the application of concentration tensors for an inhomogeneous half-space.

Session 277 (Novemb er 16, 2012) E. S. Perelygina. Notes on elasto-plastic b ending.

Session 278 (Novemb er 23, 2012) L. E. Evtushik. On the Hamiltonization of Euler­Lagrange equations with higher derivatives.

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Session 279: a joint session with the seminar "Qualitative theory of differential equations" (Novemb er 30, 2012) A. F. Pronevich. Integrals of systems of exact differential equations.

Session 280 (Decemb er 14, 2012) M. V. Shamolin. Integrable cases in the dynamics of a multi-dimensional rigid b o dy in a nonconservative field. This work is a survey of integrable cases in the dynamics of two-, three-, and four-dimensional rigid b odies in nonconservative fields. These problems are describ ed by dynamical systems with variable dissipation with zero mean. We study nonconservative systems for which the usual methods of studying Hamiltonian systems are not applicable. Thus, for such systems, we must "directly" integrate the main equation of dynamics. We prop ose a more universal exp osition of new and previously obtained cases of complete integrability of the equations of motions in terms of transcendental functions in the dynamics of two-, three-, and four-dimensional rigid b odies in nonconservative fields. Of course, in the general case, the construction of a theory of integration of nonconservative systems (even of low dimension) is a quite difficult task. In a numb er of cases, where the systems considered have additional symmetries, we succeed in finding first integrals through finite combinations of elementary functions. We obtain a series of complete integrable nonconservative dynamical systems with nontrivial symmetries. Moreover, in almost all cases, all first integrals are expressed through finite combinations of elementary functions; these first integrals are transcendental functions of their variables. In this case, transcendence is understood in the sense of complex analysis, when the analytic continuation of a function into the complex plane has essentially singular p oints. This fact is caused by the existence of attracting and rep elling limit sets in the system (for example, attracting and rep elling focuses). Session 281 (Decemb er 21, 2012) D. V. Georgievskii. Symmetrization of the tensor op erator of compatibility equations in stresses in the anisotropic elasticity theory. We find the general form of the term that symmetrizes the differential rank-4 tensor-op erator of the compatibility equations in stresses in the anisotropic elasticity theory. For an arbitrary anisotropy, it includes two arbitrary parameters with the dimension of elastic compliances. The symmetrized compatibility equations themselves contain only one of these parameters. Session 282 (Decemb er 28, 2012) M. V. Smolentsev. On the sp ectra of frequencies of zeros for solutions of third-order linear differential equations.

D. V. Georgievskii Institute of Mechanics of the M. V. Lomonosov Moscow State University, Moscow, Russia E-mail: georgiev@mech.math.msu.su M. V. Shamolin Institute of Mechanics of the M. V. Lomonosov Moscow State University, Moscow, Russia E-mail: shamolin@imec.msu.ru 731