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This paper is devoted to the study of the motion of a platform maintained on an aircraft in a predetermined position by a system of gyroscopes, which does follows oscillations of the aircraft. Such systems are used for determining the angular position of the aircraft. We briefly review key results on this issue without detailed proofs.

We establish the integrability of homogeneous (in some variables) dynamical systems with dissipation in the case of an arbitrary odd order and thereby generalize the results earlier obtained by the author in particular cases of such systems.

In this paper, we examine the problem on the spatial free deceleration of a rigid body in a resistive medium under the assumption that the interaction of the homogeneous axisymmetric body with the medium is concentrated on the frontal part of the surface, which has the shape of a flat circular disk. In earlier works of the author, under the simplest assumptions on interaction forces, the impossibility of oscillations with bounded amplitude was proved. Note that an exact analytic description of forces and moments of the body-medium interaction is unknown, so we use the method of “embedding” of the problem into a wider class of problems; this allows one to obtain a sufficiently complete qualitative description of the motion of the body. For dynamical systems considered, we obtain particular solutions and families of phase portraits of quasi-velocities in the three-dimensional space that consist of countable sets of nonequivalent portraits with different nonlinear qualitative properties.

In this paper, we prove the integrability of certain classes of dynamical systems on the tangent bundles of four-dimensional manifolds (systems with four degrees of freedom). The force field considered possessed so-called variable dissipation; they are generalizations of fields studied earlier. This paper continues earlier works of the author devoted to systems on the tangent bundles of two- and three-dimensional manifolds.

We consider the problem of differential diagnostics of the functional status of control objects that have a module structure and a finite set of possible malfunctions and reduce this problem to two separate problems that can be solved separately: the inspection problem (ascertainment of malfunction criteria in the system) and the diagnostic problem (search of malfunctions). We prove the feasibility of solution of the problem of differential diagnostics in the cases of precise and inaccurate trajectory measurements at the level of mathematical models.

In this paper, we prove the explicit integrability of certain classes of dynamical systems on the tangent bundles of 2- and 3-dimensional spheres in the case where the forces are fields with so-called variable dissipation.

We establish the integrability for a class of homogeneous (with respect to a part of variables) dynamic systems of odd (seventh and ninth) order, admitting separation of a system on the tangent bundle of smooth manifolds. Bibliography: 13 titles.

We consider consecutive aggregation procedures for individual preferences 𝔠 ∈ ℭr (A) on a set of alternatives A, |A| ≥ 3: on each step, the participants are subject to intermediate collective decisions on some subsets B of the set A and transform their a priori preferences according to an adaptation function 𝒜. The sequence of intermediate decisions is determined by a lot J, i.e., an increasing (with respect to inclusion) sequence of subsets B of the set of alternatives. An explicit classification is given for the clones of local aggregation functions, each clone consisting of all aggregation functions that dynamically preserve a symmetric set 𝔇 ⊆ ℭr (A) with respect to a symmetric set of lots 𝒥. On the basis of this classification, it is shown that a clone ℱ of local aggregation functions that preserves the set ℜ2 (A) of rational preferences with respect to a symmetric set 𝒥 contains nondictatorial aggregation functions if and only if 𝒥 is a set of maximal lots, in which case the clone ℱ is generated by the majority function. On the basis of each local aggregation function f, lot J, and an adaptation function 𝒜, one constructs a nonlocal (in general) aggregation function fJ,A that imitates a consecutive aggregation procesure. If f dynamically preserves a set 𝔇 ⊆ ℭr (A) with respect to a set of lots 𝒥, then the aggregation function fJ,A preserves the set 𝔇 for each lot J ∈ 𝒥. If 𝔇 = ℜ2(A), then the adaptation function can be chosen in such a way that in any profile c ∈ (ℜ2(A))n, the Condorcet winner (if it exists) would coincide with the maximal element with respect to the preferences fJ,A (c) for each maximal lot J and f that dynamically preserves the set of rational preferences with respect to the set of maximal lots.

In many problems of dynamics, one has to deal with mechanical systems whose configurational spaces are two- or three-dimensional manifolds. For such a system, the phase space naturally coincides with the tangent bundle of the corresponding manifold. Thus, the problem of a flow past a (four-dimensional) pendulum on a (generalized) spherical hinge leads to a system on the tangent bundle of a two- or threedimensional sphere whose metric has a particular structure induced by an additional symmetry group. In such cases, dynamical systems have variable dissipation, and their complete list of first integrals consists of transcendental functions in the form of finite combinations of elementary functions. Another class of problems pertains to a point moving on a two- or three-dimensional surface with the metric induced by the encompassing Euclidean space. In this paper, we establish the integrability of some classes of dynamical systems on tangent bundles of two- and three-dimensional manifolds, in particular, systems involving fields of forces with variable dissipation and of a more general type than those considered previously.

We establish the integrability of some homogeneous (with respect to a part of variables) dynamical systems of odd (third and fifth) order. In the class of such systems, we extract a system on the tangent bundle of smooth manifolds. The force field is separated into the internal (conservative) force field and the external force field with alternating dissipation. The external force field is introduced by using a certain unimodular transformation and generalizes the cases studied by the author earlier.

We study the relative structural stability (relative roughness) of dynamical systems in some subspace of the space of all dynamical systems; moreover, the space of deformations of (dynamical) systems does not coincide with the space of all admissible deformations. We give some examples of relatively rough systems and relatively nonrough systems of different degree of nonroughness arising in rigid body dynamics and oscillation theory.

In many problems of multidimensional dynamics, systems appear whose state spaces are spheres of finite dimension. Clearly, phase spaces of such systems are tangent bundles of these spheres. In this paper, we examine nonconservative force fields in the dynamics of a multidimensional rigid body in which the system possesses a complete set of first integrals that can be expressed as finite combinations of elementary transcendental functions. We consider the case where the moment of nonconservative forces depends on the tensor of angular velocity.

We establish the integrability for some classes of dynamic systems on the tangent bundles of multidimensional manifolds. We consider the case where the force fields possess variable dissipation. An example of a four-dimensional manifold is discussed in detail.

We establish the integrability for some classes of dynamic systems on the tangent bundles of two– and three-dimensional manifolds (systems with two and three degrees of freedom). The force fields possess the so-called variable dissipation and generalize those considered in the previous publications of the author.

This paper contains a systematic exposition of some results on the equations of motion of a dynamically symmetric n-dimensional rigid body in a nonconservative field of forces. Similar bodies are considered in the dynamics of actual rigid bodies interacting with a resisting medium under the conditions of jet flow past the body with a nonconservative following force acting on the body in such a way that its characteristic point has a constant velocity, which means that the system has a nonintegrable servo-constraint.

In this review, we discuss new cases of integrable systems on the tangent bundles of finite-dimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in non-conservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

We consider a mathematical model of the influence of a medium on a rigid body with a specific shape of its surface. In this model, we take into account the additional dependence of the moment of the interaction force on the angular velocity of the body. We present a complete system of equations of motion under the quasi-stationarity conditions. The dynamical part of equations of motion forms an independent third-order system, which contains, in its turn, an independent secondorder subsystem. We ovtain a new family of phase portraits on the phase cylinder of quasi-velocities, which differs from families obtained earlier.

In this review, we discuss new cases of integrable systems on the tangent bundles of finitedimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in nonconservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

We establish the integrability of dynamical systems of some classes arising in multidimensional dynamics. The force fields under consideration possess the so-called variable dissipation with zero mean and generalize the studied earlier ones.

In this paper, we examine new cases of integrability of dynamical systems on the tangent bundle to a low-dimensional sphere, including flat dynamical systems that describe a rigid body in a nonconservative force field. The problems studied are described by dynamical systems with variable dissipation with zero mean. We detect cases of integrability of equations of motion in transcendental functions (in terms of classification of singularity) that are expressed through finite combinations of elementary functions.

In this paper, we consider a class of oscillatory mechanical systems described by nonlinear second-order differential equations that contain parameters with variable dissipation and possess the property of dynamic symmetry. We study properties of mathematical models of such systems depending on parameters. Special attention is paid to bifurcations and reconstructions of phase portraits and the appearance and disappearance of cycles that envelope the phase cylinder or lie entirely on its covering. We present a complete parametric analysis of system with position-viscous friction and four parameters, where the force action linearly depends on the speed. The space of parameters is split into domains in which the topological behavior of the system is preserved. For some classes of pendulum systems whose right-hand sides depend on smooth functions, a qualitative analysis is performed, i.e., the space of systems considered is split into domains of different behavior of trajectories on the phase cylinder of quasi-velocities. We also perform a systematic analysis of the problem on an aerodynamic pendulum and detect periodic modes unknown earlier; these modes correspond to cycles in the mathematical model.

In this paper, we examine the existence of transcendental first integrals for some classes of systems with symmetries. We obtain sufficient conditions of existence of first integrals of second-order nonautonomous homogeneous systems that are transcendental functions (in the sense of the theory of elementary functions and in the sense of complex analysis) expressed as finite combinations of elementary functions.

In this paper, we present a qualitative analysis of plane-parallel and spatial problems on the motion of realistic rigid bodies in a resistive medium and construct a nonlinear model of the influence of the medium on the rigid body.

We consider a mathematical model of a plane-parallel action of a medium on a rigid body whose surface has a part which is a circular cone. We present a complete system of equations of motion under the quasi-stationarity conditions. The dynamical part of equations of motion form an independent system that possesses an independent second-order subsystem on a two-dimensional cylinder. We obtain an infinite family of phase portraits on the phase cylinder of quasi-velocities corresponding to the presence in the system of only a nonconservative pair of forces.

Many problems of multidimensional dynamics involve systems for which the spaces of states are spheres of finite dimension and the spaces of phases are the tangent bundles of such spheres. We study conservative systems and present nonconservative force fields such that the systems involving such forces possess a complete collection of first integrals that are expressed through a finite combination of elementary functions and, in general, are transcendental functions of their variables. Bibliography: 32 titles.

This paper is a survey of integrable cases in the dynamics of a five-dimensional rigid body under the action of a nonconservative force field. We review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean. The problem of the search for complete sets of transcendental first integrals of systems with dissipation is quite topical; a large number of works are devoted to it. We introduce a new class of dynamical systems that have a periodic coordinate. Due to the existence of nontrivial symmetry groups of such systems, we can prove that these systems possess variable dissipation with zero mean, which means that on the average for a period with respect to the periodic 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. Based on the facts obtained, we analyze dynamical systems that appear in the dynamics of a five-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions that can be expressed through a finite combination of elementary functions.

This paper is a review of some previous and new results on integrable cases in the dynamics of a three-dimensional rigid body in a nonconservative field of forces. These problems are stated in terms of dynamical systems with the so-called zero-mean variable dissipation. Finding a complete set of transcendental first integrals for systems with dissipation is a very interesting problem that has been studied in many publications. We introduce a new class of dynamical systems with a periodic coordinate. Since such systems possess some nontrivial groups of symmetries, it can be shown that they have variable dissipation whose mean value over the period of the periodic coordinate vanishes, although in various regions of the phase space there may be energy supply or scattering. The results obtained allow us to examine some dynamical systems associated with the motion of rigid bodies and find some cases in which the equations of motion can be integrated in terms of transcendental functions that can be expressed as finite combinations of elementary functions.

This paper is a survey of integrable cases in dynamics of two-, three-, and four-dimensional rigid bodies under the action of a nonconservative force field. We review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean.

In this work, we consider some problems of the qualitative theory of ordinary differential equations; the study of dissipative systems, as well as variable dissipation system considered below, which, in particular, arise in the dynamics of a rigid body interacting with a medium and in the oscillation theory, depends on solutions of these problems. We consider such problems as existence and uniqueness problems for trajectories having infinitely remote points as limit sets for systems on the plane, elements of qualitative theory of monotone vector fields, and also existence problems for families of long-period and Poisson stable trajectories. In conclusion, we study the possibility of extending the Poincar´e two-dimensional topographical system and the comparison system to the many-dimensional case.

This paper studies conditions for local (in time) solvability of a qualitatively new singularllimit problem, the free (unknown) boundary problem appearing recently. In fact, there are not so many different free boundary problems, which corresponds to not so large a variety of principally different phase transitions of the first and second kinds. Therefore, the appearance of principally new problems elicits interest. This paper studies structural features of a certain problem on the basis of a certain method developed previously, precisely, the localization method [1, 3, 9].

An important role in studying the classical Cahn–Hilliard problem [5] is played by its singular-limit problem, the so-called Melin–Sikerk free boundary problem, which, at present allows one to only numerically describe the instability of the crystallization process. The purpose of this work is to prepare the material for deducing the singular-limit problem for the essentially asymmetric model [8, 21].

The problem presented below is a singular-limit problem of the extension of the Cahn-Hilliard model obtained via introducing the asymmetry of the surface tension tensor under one of the truncations (approximations) of the inner energy [2, 5, 6, 7, 8, 10, 12, 13].

A certain one-phase problem with free boundary is studied. The local (in time) solvability of this problem is proved; moreover, the general method elaborated is applied in a more concrete case. For this purpose, a new change of variables and the parametrization of the boundary are introduced, and the problem studied is reduced to a problem in a constant domain.

We present complete results devoted to the study of the equations of motion of a dynamically symmetric four-dimensional rigid body in a nonconservative force field. The form of the body is taken from the dynamics of real two- or three-dimensional rigid bodies interacting with a resisting medium according to the streamline flow around laws under which a non-conservative pair of forces acts on the body and forces the body center of mass to move rectilinearly and uniformly.

This paper shows that for mechanical systems, the dimension of whose base space is larger than time (there also exist spatial coordinates), the system of equations defining the evolution of the system must be a hyperbolic system of pseudodifferential equations.

This work is devoted to the search for integrability cases of a certain class of complex, linear second-order, nonautonomous ordinary differential equations.

Pseudodifferential operators occurring in real equations of continuous-medium mechanics are discussed.

This work is devoted to the development of qualitative methods in the theory of nonconservative systems that arise, e.g., in such fields of science as the dynamics of a rigid body interacting with a resisting medium, oscillation theory, etc. This material can arouse the interest of specialists in the qualitative theory of ordinary differential equations, in rigid body dynamics, as well as in fluid and gas dynamics since the work uses the properties of motion of a rigid body in a medium under streamline flow-around conditions.

The results of the presented work are due to the study of the applied problem of the rigid body motion in a resisting medium; see [210, 211], where complete lists of transcendental first integrals expressed through a finite combination of elementary functions were obtained. This circumstance allowed the author to perform a complete analysis of all phase trajectories and highlight those properties of them which exhibit the roughness and preserve for systems of a more general form. The complete integrability of those systems is related to symmetries of a latent type. Therefore, it is of interest to study sufficiently wide classes of dynamical systems having analogous latent symmetries.

Session 179 in the framework of the scientific-educational program for schoolboys and schoolgirls in mechanics and robotics and courses of advanced qualification, SUNTs MGU (February 8, 2008) D. V. Georgievskii and M. V. Shamolin. Π-Theorem of Dimension Theory (Devoted to the 100th Anniversary of Its Proof). About one hundred years ago, there appeared the proof of one of the most prominent and universal assertions in mechanics and physics, the Π-theorem of dimension theory, which allows one to deduce the dependences of some quantities on others only from dimension arguments, not solving initialboundary-value problems and even having no mathematical models of phenomena. As is revealed, it is difficult to present a concrete date for the 100th anniversary, since in the world’s literature devoted to mechanics, during the entire XXth century, there was no united opinion with respect to the authority of the Π-theorem proof. In the report, we speak about physical quantity dimensions related to combinations of measurement units, the standard scales served for the measurement. The measurement units are partitioned into the main and derived units. We introduce classes of measurements of unit systems, which differ from each other only by some quantity but not the nature of the main units. We formulate lemmas on powerdimension expressions and unary choice of dimension. Also, we define a basis for the dimensionality elimination as the maximal tuple of dimensionally independent quantities through whose dimensions the dimensions of all other quantities in the problem considered are expressed in a power way. The formulation of the Π-theorem is given purely mathematically; this means that in nature, all physical laws relating some quantities with others are described by generalized homogeneous functions. As an illustration, we perform an analysis of dimensions in the following three problems: the perturbation propagation under a strong point blow-up condition in the atmosphere, the flow around an immovable ball by a viscous fluid flow and the finding of the small oscillation for the mathematical pendulum. The dimensionless Reynolds, Froude, and Strouhal numbers are introduced. The problems of scale modeling and criteria for the satisfaction of similarity in natural and model systems are presented.

We give a definition of the group of colors and study the questions: what group can be a group of colors of some algebra and what is the difference between the group of colors and the semigroup of graduation.