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We systematize some results on the study of the equations of spatial motion of dynamically symmetric fixed rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of a spatial motion of a free rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint, or the center of mass of the body moves rectilinearly and uniformly; this means that there exists a nonconservative couple of forces in the system

Proposed activity presents next stage of the study of the problem of the plane-parallel motion of a rigid body interacting with a resistant medium through the frontal plane part of its external surface. Under constructing of the force acting of medium, we use the information on the properties of medium streamline fl w around in quasistationarity conditions (for instance, on the homogeneous circular cylinder input into the water). The medium motion is not studied, and we consider such problem in which the characteristic time of the body motion with respect to its center of masses is comparable with the characteristic time of motion of the center of masses itself.

The so-called pendulum-like systems arise in dynamics of a rigid body in a non-conservative field, in the theory of oscillations, and in theoretical physics. In this article, the methods of analysis are described which allow us to generalize the previous results. Herewith, we deal with some qualitative questions of the theory of ordinary differential equations whose solution facilitates studying some dynamical systems. In result of investigating more general classes of systems, we show that these general systems possess the already known family of nonequivalent phase portraits. We also deal with the aspect of integrability.

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 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.

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.

Under study is the problem of spatial free deceleration of a rigid body in a resisting medium. It is assumed that a axisymmetric homogeneous body interacts with the medium only through the front part of its outer surface that has the shape of flat circular disk. Under the simplest assumptions about the impact forces from the medium, it is demonstrated that any oscillations of bounded amplitude are impossible. In this case, any precise analytical description of the force-momentum characteristics of the medium impact on the disk is missing. For this reason, we use the method of “embedding” the problem into a wider class of problems. This allows us to obtain some relatively complete qualitative description of the body motion. For the dynamical systems under consideration, it is possible to obtain particular solutions as well as the families of phase portraits in the three-dimensional space of quasi-velocities which consist of a countable set of the phase portraits that are trajectory-nonequivalent and have different nonlinear qualitative properties.

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.

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.

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.

Some qualitative analysis is carried out of the rectilinear and spatial problems concerning the motion of a rigid body in a resisting medium.Anonlinearmodel is constructed of impact of the mediumon the rigid body, which takes into account the dependence of the arm of force on the reduced angular velocity of the body. Moreover, the moment of this force itself is also a function of the angle of attack. As was shown by the processing the experimental data on the motion of homogeneous circular cylinders in water, these circumstances should be taken into account in the simulation. The analysis of the plane and spatial models of the interaction of a rigid body with a medium reveal the sufficient conditions of stability of the key regime of motion, i.e., the translational rectilinear deceleration. It is also shown that, under certain conditions, both stable or unstable auto-oscillating regimes can be presented in the system.

The author constructs a nonlinear mathematical model of a plane-parallel impact of the medium on a rigid body with a front part of the outer surface shaped as a circular cone. Multivariable analysis of the dynamic equations of motion was performed. A new family of phase patterns on the phase cylinder of quasi-velocities has been obtained. This family consists of infinitely numerous topologically inequivalent phase patterns. The sufficient conditions for the stability of an important mode of motion, i.e., rectilinear translational drag, have been obtained, as well as conditions for the presence of the self-oscillatory modes in the system.

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.

A mathematical model of the influence of a medium on a rigid body with some part of its external surface being flat is considered with due allowance for an additional dependence of the moment of the medium action force on the angular velocity of the body. A full system of equations of motion is given under quasi-steady conditions; the dynamic part of this system forms an independent third-order system, and an independent second-order subsystem is split from the full system. A new family of phase portraits on a phase cylinder of quasi-velocities is obtained. It is demonstrated that the results obtained allow one to design hollow circular cylinders (“shell cases”), which can ensure necessary stability in conducting additional full-scale experiments.

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.

A mechanical system whose phase space is the tangent bundle of a two-dimensional sphere is studied. The potential nonconservative systems describing a geodesic flow are classified. A multiparameter family of systems possessing a complete set of transcendental first integrals expressed in terms of finite combinations of elementary functions is found. Some examples illustrating the spatial dynamics of a rigid body interacting with a medium are discussed.

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.

The author returns to constructing the nonlinear mathematical model of a plane-parallel impact of a medium on a rigid body with allowance for the dependence of the arm of force on the reduced angular velocity of the body (a kind of Strouhal number). In this case, the arm of the force is also a function of the angle of attack. As was shown by the processing of the experimental data on the motion of homogeneous circular cylinders in water, these circumstances should be taken into account in the simulation. The analysis of a plane model of the interaction of a rigid body with a medium revealed new cases of the complete integrability in elementary functions, which made it possible to find qualitative analogies between the motion of free bodies in a resisting medium and vibrations of partly fixed bodies in a flow of an oncoming medium. Phase patterns obtained in the analysis of the nonlinear model of the impact of the medium are compared with real Karman’s vortex streets.

A mathematical model of the effect of a medium on a homogeneous rigid body whose outer surface includes a circumferential cone is considered. The complete system of equations of motion under quasistationarity conditions is given. In the dynamic part forming an independent third-order system, an independent second-order subsystem is distinguished. A new two-parameter family of phase portraits on the phase cylinder of quasivelocities is obtained.

A complete analysis of the phase trajectories representing the motion of a rigid body in a resisting medium in quasistationary conditions is carried out. More general systems characterized by hidden nontrivial symmetries are studied

The dynamic part of equations of motion of a dynamically symmetric 4D rigid body, where the force field is concentrated on that part of the body that has the form of a two-dimensional disc, is investigated. In this case, the tensor of the angular velocity of such a body is six-dimensional, while the velocity of the center of mass is four-dimensional. Under certain conditions, a complete list of first integrals, which are expressed through elementary functions, is obtained.

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.

The dynamic part of equations of motion of a dynamically symmetric 4D-rigid body, where the force field is accumulated on that part of the body surface that is shaped as a three-dimensional ball, is investigated. In this case, the tensor of the angular velocity of such a body is six-dimensional, while the velocity of the center of mass is four-dimensional. Under certain conditions, the complete list of first integrals, which are expressed through elementary functions, is obtained.

The problem of plane-parallel motion of a uniform symmetric rigid body interacting with a medium only through a flat region of its outer surface is studied. The force field is constructed on the basis of information on the properties of jet flow under quasistationarity conditions. The motion of the medium is not studied. The problem of rigid body dynamics is considered for the case when the characteristic time of motion of the body relative to its center of mass is comparable with the characteristic time of motion of this mass center.

The paper studies the spatial motion of a rigid body in a resisting medium under the action of a follower force that causes the center of mass to move rectilinearly and uniformly. The body, which is axisymmetric and homogeneous, interacts with the medium by its frontal area that has the form of a flat circular disk. Since there is no exact analytic description of the forces and torques exerted by the medium on the disk, the problem is “immersed” in a wider class of problems. Partial solutions and phase portraits in the three-dimensional space of quasivelocities are obtained for the dynamic systems under consideration. The transcendental first integrals of the dynamic part of the equations of motion are listed

The results of this work appeared due to investigation of a certain problem on the motion of a rigid bodyin a resistant medium [1, 2], where it was necessary todeal with first integrals of dynamic systems with nonstandard properties. Specifically, the integrals wereneither analytical nor smooth, and for certain sets,they were even discontinuous. The last circumstancesallowed us to completely analyze all phase trajectoriesand to indicate their properties, which possessed“roughness” and were retained for the systems of amore general form with certain nontrivial encapsulatedtype symmetries. Therefore, it is of interest toinvestigate sufficiently wider classes of systems withsimilar properties, specifically, those taken from thedynamics of a rigid body interacting with a medium.New cases of integrability in the problem of spatialmotion of the rigid body in the resistant medium willbe presented.SYSTEMS WITH SYMMETRIES AND VARIABLE DISSIPATIONWITH A ZERO MEANWe studied the systems of ordinary differentialequations having at least one periodic phase coordinate. The systems under study possess symmetry properties such that their phase volume is retained on average for the period by the periodic coordinate. Forexample, the pendulum system with a smooth andperiodic righthand partα

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.

The paper discusses a nonlinear model that describes the interaction of a rigid body with a medium and takes into account (based on experimental data on the motion of circular cylinders in water) the dependence of the arm of the force on the normalized angular velocity of the body and the dependence of the moment of the force on the angle of attack. An analysis of plane and spatial models (in the presence or absence of an additional follower force) leads to sufficient stability conditions for translational motion, as one of the key types of motions. Either stable or unstable self-oscillation can be observed under certain conditions

Planar and spatial non-linear models of the action of a medium on a rigid body which take into account the dependence of the arm of the force on the reduced angular velocity of the body are constructed, when the moment of the force is also a function of the angle of attack. New cases of complete integrability in terms of elementary functions are found, enabling of qualitative similarities to be discovered between the motions of free bodies in a resistive medium and the oscillations of bodies that are partially fixed and are immersed in a flow of the medium. It is shown that if the additional damping action of the medium on the body that appears in the system is significant, the attainment of stability of the rectilinear translational deceleration of the body is possible when it moves with finite angles of attack. The question of the roughness of the description of this phenomenon is of current interest: a finer property of relative roughness is discovered when reduced dynamical systems are investigated. ©2008.

The results of this paper are due to some previous studies of the application problem of motion of a rigid body in a resisting medium. During these studies, a transcendental integral expressed in terms of elementary functions was obtained for a particular case. This fact allowed one to completely analyze all phase trajectories and to indicate those properties that were “rough” and were retained for some more general systems. The integrability of such a system is related to latent symmetries. Therefore, the study of sufficiently wide classes of dynamic systems with similar symmetries is of interest.

The paper addresses the stability of solutions of ordinary differential equations of particular type for different statements and assumptions. The equations are interpreted as models of motion of a rigid body under the action of the ambient medium

Due to its complexity, the problem of the motion of a rigid body in an unbounded medium requires the introduction of certain simplifying restrictions. The main aim in this connection is to introduce hypotheses that would make it possible to study the motion of the rigid body separately from the motion of the medium in which the body is embedded. On the one hand, a similar approach was realized in the classical Kirchhoff problem on the motion of a body in an unbounded ideal incompressible fluid that undergoes an irrotational motion and is at rest at infinity [1]. On the other hand, it is obvious that the above-mentioned Kirchhoff problem does not exhaust the possibilities of this kind of simulation.

The complete integrability of the plane problem of the motion of a rigid body in a resisting medium under jet flow conditions is shown, when one first integral, which is transcendental function of the quasi-velocities (in the sense of the theory of functions of a complex variable having essentially singular points), exists in the system of dynamic equations. It is assumed that all the interaction of the medium with the body is concentrated in that part of the surface of the body which has the form of a (one-dimensional) plate. The plane problem is extended to the three-dimensional case; then a complete set of first integrals exists in the system of dynamic equations: one analytic, one meromorphic and one transcendental. It is assumed here that all the interaction of the medium with the body is concentrated in that part of the surface of the body which has the form of a plane (two-dimensional) disc. An attempt is also made to construct an extension of the “low-dimensional” cases to the dynamics of a so-called four-dimensional rigid body, interacting with a medium which is concentrated in that part of the (three-dimensional) surface of the body which has the form of a (three-dimensional) sphere. The vector of the angular velocity of the motion of such a body in this case is six-dimensional, while the velocity of the centre of mass is four-dimensional.

The purpose of the paper is to elaborate qualitative methods for studying the dynamics of rigid bodies interacting with a resisting medium under quasistationarity conditions. This material refers to the qualitative theory of ordinary differential equations, as well as to the dynamics of rigid bodies. In this paper, we use the properties of motion of a body in a medium under conditions of a jet flow past this body. We study the plane and three-dimensional model problems of motion of a body in a resisting medium. New families of phase portraits of variable dissipation systems on twoand three-dimensional manifolds are obtained, and their absolute or relative roughness (i.e., structural stability) is demonstrated. The Jacobi integrable cases of equations of motion of rigid bodies are found, including those in the problem of motion of a spherical pendulum placed in the incoming flow of a medium.

A dynamic model of the interaction of a rigid body with a jet flow of a resistant medium is considered. This model allows us to obtain three-dimensional analogs of plane dynamic solutions for a solid interacting with the medium and to reveal new cases where the equations are Jacobi integrable. In such cases, the integrals are expressed in terms of elementary functions. The classical problems of a spherical pendulum in a flow and three-dimensional motion of a body with a servoconstraint are shown to be integrable. Mechanical and topological analogs of these problems are found

In this activity the qualitative analysis of spatial problems of the real rigid body motions in a resistant medium is fulfilled. A nonlinear model that describes the interaction of a rigid body with a medium and takes into account (based on experimental data on the motion of circular cylinders in water) the dependence of the arm of the force on the normalized angular velocity of the body and the dependence of the moment of the force on the angle of attack is constructed. An analysis of plane and spatial models (in the presence or absence of an additional tracking force) leads to sufficient stability conditions for translational motion, as one of the key types of motions. Either stable or unstable self-oscillation can be observed under certain conditions.

The results of this work appeared in the process of studying a certain problem on the rigid body motion in a medium with resistance [Chaplygin, 1933], [Chaplygin, 1976], [Shamolin-1, 2007], [Shamolin-2, 2007], where we needed to deal with first integrals having nonstandard properties. Precisely, they are not analytic, not smooth, and on certain sets, they can be even discontinuous. Moreover, they are expressed through a finite combination of elementary functions. However, the latter circumstances allowed us to carry out a complete analysis of all phase trajectories and show those their properties which have a ”roughness” and are preserved for systems of a more general form having certain symmetries of latent type. Therefore, it is interesting to study sufficiently wide classes of dynamical systems that have analogous properties and moreover, coming from the dynamics of a rigid body interacting with a medium.

Institute of Mechanics, Lomonosov Moscow State University, Michurinskii Ave., 1, 119899 Moscow, Russian FederationThe results of the presented work are due to the study of the applied problem of the rigid body motion in a resisting medium.Moreearlierthecomplete listsoftranscendentalfirstintegralsexpressed througha finitecombinationofelementary functionswere obtained. This circumstance allowed the author to perform a complete analysis of all phase trajectories and highlightthose properties of them which exhibit the roughness and preserve for systems of a more general form.

The purpose of the activity is to elaborate the qualitative methods for studying the dynamics of rigid bodies interacting with a resisting medium under quasistationarity conditions. This material refers equally to the qualitative theory of ordinary differential equations and the dynamics of rigid bodies. We use the properties of body’s motion in a medium under conditions of the jet flow past this body. We study the plane model problems of the motion of a body with the cone form of its shape in a resisting medium. The new families of phase portraits of variable dissipation systems are obtained, their absolute or relative roughness is demonstrated. The integrable cases of equations of motion of rigid bodies are found. c � 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

In is well–known due to its complexity, the problem of the motion of a rigid body in an unbounded medium requires the introduction of certain simplifying restrictions. The main aim in this connection is to introduce hypotheses that would make it possible to study the motion of the rigid body separately from the motion of the medium in which the body is embedded. On the one hand, a similar approach was realized in the classical Kirchhoff problem on the motion of a body in an unbounded ideal incompressible fluid that undergoes an irrotational motion and is at rest at infinity. On the other hand, it is obvious that the above–mentioned Kirchhoff problem does not exhaust the possibilities of this kind of simulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A vast number of papers are devoted to studying the complete integrability of equations of four-dimensional rigid-body motion. Although in studying low-dimensional equations of motion of quite concrete (two- and three-dimensional) rigid bodies in a nonconservative force field, the author arrived at the idea of generalizing the equations to the case of a four-dimensional rigid body in an analogous nonconservative force field. As a result of such a generalization, the author obtained the variety of cases of integrability in the problem of body motion in a resisting medium that fills the four-dimensional space in the presence of a certain tracing force that allows one to reduce the order of the general system of dynamical equations of motion in a methodical way.

This paper is a survey of integrable cases in dynamics of a multi-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 multi-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions, which can be expressed through a finite combination of elementary functions. As applications, we study dynamical equations of motion arising in the study of the plane and spatial dynamics of a rigid body interacting with a medium and also a possible generalization of the obtained methods to the study of general systems arising in the qualitative theory of ordinary differential equations, in the theory of dynamical systems, and also in oscillation theory.