ON INTEGRABILITY IN ELEMENTARY FUNCTIONS ... - Springer Link

Journal of Mathematical Sciences, Vol. 161, No. 5, 2009

ON INTEGRABILITY IN ELEMENTARY FUNCTIONS OF CERTAIN CLASSES OF NONCONSERVATIVE DYNAMICAL SYSTEMS M. V. Shamolin

UDC 517.925; 531.01; 531.552

Abstract. 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. As is known, the concept of integrability is sufficiently broad and undeterminate in general. In its construction, it is necessary to take into account in what sense it is understood (it is meant that a certain criterion according to which one makes a conclusion that the structure of trajectories of the dynamical system considered is especially “attractive and simple”), in which function classes the first integrals are sought for, etc. (see also [1, 4, 14, 17, 20–22, 35, 40–42, 47, 83–85, 117, 120]). In this work, the author applies such an approach such that as first integrals, transcendental functions are elementary. Here, the transcendence is understood not in the sense of elementary functions (e.g., trigonometrical functions) but in the sense that they have essentially singular points (by the classification accepted in the theory of functions of one complex variable according to which a function has essentially singular points). In this case, it is necessary to continue them formally to the complex plane. As a rule, such systems are strongly nonconservative (see also [142, 149, 167, 203–206, 216, 218, 233, 236, 238, 239, 252, 253, 258, 260, 261, 264–266, 269–274, 276–287, 289–292, 295–297, 299–307, 309–315, 320]).

CONTENTS 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Various Aspects of Problem Consideration . . . . . . . . . . . . . . . . Variable Dissipation Dynamical Systems and Their General Properties One of the Definitions of Zero Mean Variable Dissipation System . . . Zero Mean Variable Dissipation Systems with Symmetries . . . . . . . Systems on S1 {α mod 2π} × R1 {ω} . . . . . . . . . . . . . . . . . . . Systems on S1 {α mod 2π} \ {α = 0, α = π} × R2 {z1 , z2 } . . . . . . . Systems from Dynamics of a Four-Dimensional Rigid Body Interacting Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.

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734 736 737 739 740 745 746 754 759 759

Introduction

Of course, in the general case, it is sufficiently difficult to construct a certain theory of integration of nonconservative systems (although of lower dimension). But, in a number of cases where the systems considered have additional symmetries, one succeeds in finding their first integrals expressed through finite combinations of elementary functions [114, 317, 318, 323, 326, 329, 332, 339, 341, 343–345, 349, 351, 354]. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 62, Geometry and Mechanics, 2009.

734

c 2009 Springer Science+Business Media, Inc. 1072–3374/09/1615–0734

In this work, we present examples of systems from the dynamics of a rigid body interacting with a medium (see also [30, 44, 55, 56, 143, 145–148, 170–172]). In our previous works, we considered the class of problems from the rigid body dynamics in which the characteristic time of the body motion with respect to its center of masses is commensurable with the characteristic time of the motion of the center itself. The complexity of solving such problems depends on many factors, including the character of the external force field. For example, in the case of the conservative (gravity) force field, the body motion around its center of masses can be strongly chaotic (the classical problem of the heavy rigid body motion around a fixed point) [18, 21]. In this case, it is impossible to construct a somewhat general theory of integration; a natural possibility for further development is to impose certain restriction on the rigid body geometry and also to claim that the force field considered has some groups of symmetries, although latent. The presented work appeared from the problem of the motion of a body in a resisting medium such that the contact surface with the medium is only the plane part of its exterior surface. In this case, the force field is constructed when the action of the medium on the body is executed under streamline (or separation) flow and quasi-stationarity conditions. It turns out that studying the motion of such a class of bodies reduces to that of systems either with energy dissipation (dissipative systems) or with pumping (the so-called anti-dissipation systems). Note that similar problems already appeared in applied aerodynamics in studies of the Central Aero-Hydrodynamical Institute (see also [2, 39, 69–77, 106, 122, 124, 134, 162, 192, 203, 207–211, 213, 374, 375, 390, 391, 393]). Also, the classes of plane-parallel and spatial motions of rigid bodies interacting with a medium were considered; among them (depending on the number of degrees of freedom) we can highlight the following: the motions of free bodies in a medium immovable at infinity and bodies partially clamped and situated in the over-running medium flow [58, 59, 217, 219]. One such problem, which is of the most applied significance, was studied in detail; this is the problem of body free drag in a resisting medium. Moreover, the problem of the free body motion under the presence of a tracking force and also the problem of oscillations of a clumped pendulum placed in the over-running medium flow were considered (for more detail, see also [220, 224, 226, 229, 232, 234–237, 242–247, 249–251, 254–257, 259]). The problems considered previously stimulate the development of qualitative tools, which essentially complements the qualitative theory of dissipation and nonconservative anti-dissipation systems [262, 263, 267, 268, 275, 288, 293, 294, 298, 308, 316, 319, 321, 322]. Therefore, the study of dynamical equations of motion, which arise in studying the plane and spatial dynamics of a rigid body interacting with a medium, was carried out; this study stimulates the author to possible generalization of the previous methods to general systems arising in qualitative theory of ordinary differential equations and dynamical systems, as well as to oscillation theory. Also, the qualitatively nonlinear effects in the plane and spatial dynamics of a rigid body interacting with a medium were studied; justification of the necessity of introducing the concepts of relative roughness and relative non-roughness of various degrees was carried out on the qualitative level (see also [3, 6, 8–13, 23, 27, 90, 92, 105, 115, 129, 138, 160, 168, 174, 175, 178, 180–187, 221, 225, 227–231, 240, 325, 326, 367–369]). Performing a qualitative analysis, the author previously did the following: (a) he elaborated the methods for qualitatively studying dissipative systems and anti-dissipative systems, which allowed him to obtain the condition for the bifurcation of birth of stable and unstable auto-oscillations and also the conditions for the absence of any singular trajectories. He succeeded in generalization of the Poincaré topographical systems and comparison of systems to higher dimensions. He obtained the sufficient conditions for the Poisson stability (everywhere dense around itself) of certain classes of unclosed dynamical system trajectories [5, 7, 15, 16, 19, 24–29, 32–34, 46, 50–54, 67, 78, 81, 82, 96–98, 111–113, 139–141, 157, 188–190, 194, 222, 223, 241, 280, 281, 329, 331, 332, 336–338, 364, 372, 382, 385, 386]; 735

(b) in plane and spatial rigid body dynamics, he found complete lists of first integrals of dissipative and anti-dissipative systems, which are transcendental (in the sense of classification of their singularities) functions that are expressed through elementary functions in a number of cases. He introduced new definitions of properties of relative roughness and non-roughness of various degrees, which the integrated systems have [280, 281, 333–335, 350]; (c) he obtained multiparameter families of topologically non-equivalent phase portraits arising in the problem of a free drag. Almost every portrait of such families is (absolutely) rough [327, 340, 346–348, 351, 353]; (d) he detected new qualitative analogies between the properties of free body motions in a resisting medium immovable at infinity and clamped bodies situated in the over-running medium flow [36, 73–75, 91, 93, 116, 195, 354, 357, 358, 360, 361, 363, 380]. Many results of this work were regularly reported at the workshop “Actual problems of geometry and mechanics” named after professor V. V. Trofimov [86]. 2.

Various Aspects of Problem Consideration

In this section, we consider some aspects of mathematical modelling of the medium action on a rigid body under quasi-stationarity conditions. These aspects form an initial idea of the problems of qualitative character arising in what follows. In the past, only one aspect of the problem of the body motion in a resisting medium was mainly considered. Specifically, the interests of investigators were focused on obtaining concrete trajectories, although in an approximate but explicit form. Moreover, the problem of a more exact modelling of the body interaction with a resisting medium was considered in parallel. For interesting experimental phenomena, see also the works [38, 109, 144, 154, 163, 173, 176, 177, 191, 193, 387] of the scientists of the 19th century: Hubert Airy, Magnus Blix, Bret Onniere, Otto Liliental, Marey, Mouillard, Parseval, S. E. Peal, Rayleigh, and Weyher. The matter is that a plane plate is the simplest body that allows one to study various peculiarities of the motion in a medium. The dynamical effects related to the influence of attached masses (the classical Kirchhoff problem) are demonstrated in the classical textbook [134] of G. Lamb by examining the body-plate motion in a fluid (as is known, the study was initiated by Thomson, Tate, and Kirchhoff). The Kirchhoff problem posed in the second half of the 19th century opened the second aspect of the problem. It is related with integrability problems of the nonlinear system of differential equations, which describes a given motion (the problems of existence of analytic (smooth, meromorphic, or more specific) first integrals). Up to now, because of complexity, various variants of the Kirchhoff problem were considered from the integrability viewpoint, and only in some cases, the qualitative analysis of a number of trajectories was carried out. In the works of Kirchhoff, Clebsch, Steklov, Lyapunov, Chaplygin, Kharlamov, and others, the conditions for the existence of an additional analytical first integral were presented. At present, the solution of this problem is improved: in [181], A. M. Perelomov constructs the theory of integrable cases (the construction of the L-A-pair), and in [127], V. V. Kozlov and D. A. Onishchenko show the conditions for the nonexistence of an additional first integral of the Kirchhoff equations (see also the works [39–41, 169, 196] of O. I. Bogoyavlenskii, S. P. Novikov, C. T. Sadetov, and others.). Also, let us mention the third aspect of the problem, precisely, a complete qualitative analysis of the system of differential equations, which describes a given motion (fiberings of the phase space, the qualitative location of phase trajectories, symmetries, etc.). Although the listed problems are closely related to the integrability, their solution is of independent character. This aspect naturally stimulates the development of the qualitative tools.

736

3.

Variable Dissipation Dynamical Systems and Their General Properties

3.1. General characteristic of variable dissipation dynamical systems. In general, the dynamics of a rigid body interacting with a medium is just a field in which there arise either dissipative systems or systems with the so-called anti-dissipation (energy pumping inside a system itself). Here, it becomes urgent to construct a methodology precisely for those classes of systems which arise in modelling the motion of a body whose surface of contact with a medium is a plane part (the simplest form) of its exterior surface. Since in such a modelling, the experimental information about the properties of the streamline flow around is used, there arises a necessity to study the class of dynamical systems that have the property of (relative) structural stability. Therefore, it is quite natural to introduce the definitions of relative roughness for such systems. Moreover, many of the systems considered are (absolutely) rough in the sense of Andronov–Pontryagin [10]. After certain simplifications, the general system of equation of the plane-parallel motion can reduce to pendulum second-order systems in which there exists a linear dissipative force with variable coefficient, which can alternate its sign for different values of a periodic phase variable existing in the system. In this case, we speak about systems with the so-called variable dissipation, where the term “variable” mostly refers not to the value of the dissipation coefficient but to a possible alternation of its sign (therefore, it is of reason to use the term “sign-alternating”). In the mean during the period in the periodic coordinate, the dissipation can be either positive (“purely” dissipative systems), negative (systems with dispersive forces), or equal to zero. In the latter case, we speak about zero mean variable dissipation systems (such systems can be associated with “almost” conservative systems). As was already mentioned above, there exist important mechanical analogies arising in comparing the qualitative properties of the free body stationary motion and the pendulum equilibrium in the medium flow. Such analogies have a deep support meaning, since they allow us to extend the properties of pendulum nonlinear dynamical systems to the free body dynamical systems. Both classes of systems belong to the class of the so-called pendulum dynamical systems with zero mean variable dissipation. Under additional conditions, the equivalence described above also extends to the case of spatial motion, which allows us to speak about the general character of symmetries existing in a zero mean variable dissipation system both in the plane-parallel and spatial motions (for plane and spatial variants of a pendulum in the medium flow, see also [280, 281]). 3.2. Examples from dynamics. In what follows, we highlight classes of essentially nonlinear second- and third-order systems integrable in transcendental (in the sense of theory of functions of one complex variable) elementary functions. For example, these are the following five-parameter dynamical systems including the majority of systems studied previously in the dynamics of a rigid body interacting with a medium: α˙ = a sin α + bω + γ1 sin5 α + γ2 ω sin4 α + γ3 ω 2 sin3 α + γ4 ω 3 sin2 α + γ5 ω 4 sin α, ω˙ = c sin α cos α + dω cos α + γ1 ω sin4 α cos α + γ2 ω 2 sin3 α cos α +γ3 ω 3 sin2 α cos α + γ4 ω 4 sin α cos α + γ5 ω 5 cos α. In this connection, it is reasonable to introduce the definitions of relative structural stability (relative roughness) and relative structural instability (relative non-roughness) of various degrees. The latter properties are proved for the systems arising in the dynamics of a rigid body interacting with a medium, e.g., in [280, 322, 355]. As is known, (purely) dissipative dynamical systems (as well as (purely) anti-dissipative), which can belong to zero mean variable dissipation systems in our case are, as a rule, structurally stable ((absolutely) rough), whereas zero mean variable dissipation systems (which, as a rule, have additional 737

Fig. 1. Relatively rough dynamical system. symmetries) are either structurally unstable (non-rough) or only relatively structurally stable (relatively rough). It is difficult to prove the latter assertion in the general case. However, the introduction of the concept of relative roughness (and also that of relative non-roughness of various degrees) allows us to present classes of concrete systems from the rigid body dynamics, which have the properties mentioned above. So, in [352], we have qualitatively studied and have integrated two model variants of the planeparallel body motion in a resisting medium, which are described by zero mean variable dissipation dynamical systems. Such cases of motion presuppose the existence of a certain non-integrable constraint in the system considered (which is realized by means of a certain additional tracking force). For example, a dynamical system of the form α˙ = Ω + β sin α, ˙ = −β sin α cos α Ω is relatively structurally stable (relatively rough) and topologically equivalent to the system describing a clamped pendulum placed in the over-running medium flow [317] (its phase portrait is depicted in Fig. 1). We can present its first integral being a transcendental function (in the sense of theory of functions of one complex variable, i.e., a function having essentially singular points after its continuation to the complex domain) of phase variables and expressing through a finite combination of elementary functions. As is seen, the phase cylinder R2 {α, Ω} of quasi-velocities of the system considered has an interesting topological structure of partition into trajectories. On the cylinder, there are two domains (whose closure is just the phase cylinder) with quite different character of trajectories. The first domain called oscillatory or finitary (it is one-connected (Fig. 1)) is entirely filled with trajectories of the following types. Almost every such trajectory starts from the repelling point (2πk, 0) and ends at the attracting point ((2k +1)π, 0), k ∈ Z. The exclusions are the equilibrium points (πk, 0) and also the separatrices that either start from repelling points (2πk, 0) and enter the saddles S2k and S2k+1 or start from the saddles S2k+1 and S2k+2 and enter the attracting point ((2k + 1)π, 0). Here, π k Sk = − + πk, (−1) β . 2 The second domain called rotational (it is two-connected (Fig. 1)) is entirely filled with rotational motions like rotations on the phase plane of the mathematical pendulum. These trajectories envelope the phase cylinder and are periodic on it. 738

Although the dynamical system considered is nonconservative, in the rotational domain of its phase plane R2 {α, Ω}, it admits the preservation of an invariant measure with variable density. This property characterizes the system considered as a zero mean variable dissipation system. π The key separatrices (for example, the separatrix emanating from the point − , β and entering 2 3π , β ) are boundaries of domains in each of which the motion is of different character. So, the point 2 in the oscillatory domain containing attracting and repelling equilibrium points, almost all trajectories have attractors and repellers as limit sets. Hence there is not even absolutely continuous function being the density of an invariant measure in this domain. The matter is different in the case of the domain entirely filled with rotational motions. As was shown above, there exists a smooth function being the density of an invariant measure in the domain entirely filled with periodic trajectories not contractible to a point along the phase cylinder [317]. 3.3. First results. Some results of the dynamics of the plane-parallel motion also extend to the spatial case; in this connection, the author has posed the spatial problem in detail. In particular, he found a complete list of integrals in the problem of the spatial motion of a dynamically symmetric clamped rigid body placed in the over-running medium flow (see, e.g., [260]). This zero mean variable dissipation system is topologically equivalent to the spatial motion of a rigid body in a resisting medium where a non-integrable constraint is imposed on the body (which is realized by a certain tracking force). The spatial motion of a rigid body in a resisting medium in which the center of masses executes the rectilinear uniform motion is also a zero mean variable dissipation dynamical system. Its qualitative study allows us to present a convenient spatial comparison system for studying many nonzero mean variable dissipation systems. Also, note that in [296, 297, 317], we have obtained a family of phase portraits in the problem of spatial body free drag in a resisting medium. In these works, we develop the technique of studying a neighborhood of a singular equilibrium state, i.e., the state at which the right-hand sides of dynamical systems are defined only by continuity. For example, for small phase variables α and Ω, the right-hand 1 side of a system has a singularity of the type . This difficulty is overcame by a specific construction α of the Lyapunov function [273, 280]. As a result of all this, we obtained the family of three-dimensional phase portraits [293] analogous to the plane-parallel dynamics (see [280]). 4.

One of the Definitions of Zero Mean Variable Dissipation System

We study systems of ordinary differential equations having a periodic phase coordinate. The systems studied have those symmetries under which their phase volume preserves in the mean during a period of the periodic coordinate. So, for example, the following pendulum system with smooth right-hand side V(α, ω) periodic in α with period T : α˙ = −ω + f (α),

ω˙ = g(α),

f (α + T ) = f (α),

g(α + T ) = g(α),

preserves its phase area on the phase cylinder for the period T : T

T div V(α, ω) dα = 0

0

T ∂ ∂ (−ω + f (α)) + g(α) dα = f (α) dα = 0. ∂α ∂ω 0

The system considered is equivalent to the pendulum equation α ¨ − f (α)α˙ + g(α) = 0 in which the integral of the coefficient f (α) standing by the dissipative term α˙ is equal to zero in the mean during the period. 739

It is seen that the system considered has those symmetries under which it becomes the so-called zero mean variable dissipation in the sense of the following definition (see also [317]). Definition. Let us consider a smooth autonomous system of the (n + 1)th order and normal form given on the cylinder Rn {x} × S1 {α mod 2π}, where α is a periodic coordinate of period T > 0. Denote by div(x, α) the divergence of the right-hand side (which, in general, is a function of all phase variables and is not identically equal to zero. Such a system is called a zero (nonzero) mean variable dissipation system if the function T div(x, α) dα 0

is identically equal (not equal) to zero. Moreover, in some cases (for example, where at separate points of the circle S1 {α mod 2π}, there arise singularities), this integral is understood in the sense of principal value. It should be noted that it is sufficiently difficult to give the general definition of zero (nonzero) mean variable dissipation system. The just presented definition uses the concept of divergence (as is known, the divergence of the right-hand side of a system in the normal form characterizes the variation of the phase volume in the phase space of this system). 5.

Zero Mean Variable Dissipation Systems with Symmetries

Let us consider systems of the following form (the dot denotes the derivative in time): α˙ = fα (ω, sin α, cos α),

ω˙k = fk (ω, sin α, cos α),

k = 1, . . . , n,

(1)

defined on the set S1 {α mod 2π} \ K × Rn {ω}, ω = (ω1 , . . . , ωn ), where the functions fλ (u1 , u2 , u3 ), λ = α, 1, . . . , n, of three variables u1 , u2 , and u3 are as follows: fλ (−u1 , −u2 , u3 ) = −fλ (u1 , u2 , u3 ),

fα (u1 , u2 , −u3 ) = fα (u1 , u2 , u3 ),

fk (u1 , u2 , −u3 ) = −fk (u1 , u2 , u3 ). The set K is either empty or consists of finitely many points of the circle S1 {α mod 2π}. The latter two variables u2 and u3 in the functions fλ (u1 , u2 , u3 ) depend on the same parameter α, but they are isolated into different groups for the following reasons. First, not in their whole domain, they uniquely express through one another, and, second, the first of them is odd, whereas the second is an even function of α, which differently influence on the symmetries of system (1). To this system, we put in correspondence the non-autonomous system fk (ω, sin α, cos α) dωk = ; dα fα (ω, sin α, cos α) by substitution τ = sin α, it reduces to the form fk (ω, τ, ϕk (τ )) dωk = , dτ fα (ω, τ, ϕα (τ ))

k = 1, . . . , n,

ϕλ (−τ ) = ϕλ (τ ),

λ = α, 1, . . . , n.

In particular, the right-hand side of the latter system can be algebraic (i.e., it can be the ratio of two polynomials); sometimes, this helps us to seek for its first integrals in explicit form. The following assertion includes the class of system (1) in the class of zero mean variable dissipation systems. The inverse inclusion is not true in general. Proposition 1. Systems of the form (1) are zero mean variable dissipation dynamical systems. 740

In this work, we consider the case where the functions fλ (ω, τ, ϕk (τ )), λ = α, 1, . . . , n, are polynomials in ω and τ . We first consider a certain class of systems on the two-dimensional cylinder S1{α mod 2π}× R1 {ω}. So, for example, to the following pendulum systems (arising in the dynamics of a rigid body interacting with a medium) with parameter β > 0 [203, 205, 317] (see also Fig. 1 for ω ↔ Ω, α ↔ −α): α˙ = −ω + β sin α, ω˙ = sin α cos α,

(2)

α˙ = −ω + β sin α cos2 α + βω 2 sin α, ω˙ = sin α cos α − βω sin2 α cos α + βω 3 cos α

(3)

in the variables (ω, τ ), we put in correspondence the equations dω τ = , dτ −ω + βτ τ + βω[ω 2 − τ 2 ] dω = dτ −ω + βτ + βτ [ω 2 − τ 2 ] with algebraic right-hand sides, respectively. In this case, systems (2) and (3) are zero mean variable dissipation dynamical system, which is easily directly verified. Moreover, each of then has a first integral being a transcendental (in the sense of theory of one complex variable) function expressed through a finite combination of elementary functions [317]. For example, system (2) has the first integral of the following form (depending on the value of the constant β, three cases are possible; they correspond to the existence of foci, nodes, or degenerate nodes in the phase portrait of the system): β2 − 4 < 0 : 2β 2Ω + β sin α arctan = const; [Ω2 + βΩ sin α + sin2 α] × exp −β 2 + 4 −β 2 + 4

|2Ω + (β +

β 2 − 4 sin α)|

√

β2 − 4 > 0 : β 2 −4−β

× |2Ω + (β −

√ 2 β 2 − 4 sin α)| β −4+β = const;

β2 − 4 = 0 : β sin α |2Ω + β sin α| × exp − = const . 2Ω + β sin α System (3) has the phase portrait of three different types depending on different values of the parameter β (Figs. 2, 3, and 4, ω ↔ Ω, α ↔ −α). In the expression of its first integral, also depending on the value of the constant β, three cases are possible; they correspond to the existence of foci, nodes, and degenerate nodes in the phase portrait of the system. Let us represent the parameter β in the form of the product: β = σ 2 n20 , after that, to system (3), we put in correspondence a differential equation of the form −n20 τ + σω[ω 2 − n20 τ 2 ] dω = , dτ ω + σn20 τ + στ [ω 2 − n20 τ 2 ]

τ = − sin α. 741

Fig. 2. Relatively rough dynamical system with heteroclinic situation.

Fig. 3. Relatively non-rough dynamical system.

Fig. 4. Relatively rough dynamical system with homoclinic situation. Introduce the following notation C1 = 2−σn0 , C2 = σn0 , and C3 = −2−σn0 . Performing a number of changes of variables according to the formulas ω − n0 τ = u1 ; u 1 = v1 t 1 ; 742

ω + n 0 τ = v1 ; v12 = p1 ,

where v1 = 0, we obtain the Bernoulli-type equation

dp1 2σ t 1 p1 = [C3 − C1 t21 ]. 2p1 C1 t1 + C2 + n0 dt1 By the known change p−1 = q1 for p1 = 0, we reduce this equation to the form q˙1 = a1 (t1 )q1 + a2 (t1 ), where

2(C1 t1 + C2 ) 4σt1 . , a2 (t1 ) = 2 C1 t1 − C3 n0 (C1 t21 − C3 ) (Here, the dot denotes the derivative in t1 .) The general solution of linear homogeneous equation is represented in the form a1 (t1 ) =

q1hom (t1 ) = k(C1 t21 − C3 )Q(t1 ),

k = const,

where depending on the value of the constant C1 , the function Q has the form ⎧ ⎪ C1 = 0; et1 , ⎪ ⎪ ⎪ √ √ ⎪ ⎨ 2(C2 / −C1 C3 )arctan −(C1 /C3 )t1 , C1 > 0; e Q(t1 ) = √ ⎪ √ √ ⎪ ⎪ −C t + −C C2 / C1 C3 ⎪ ⎪ ⎩ √ 1 1 √ 3 , C1 < 0. −C1 t1 − −C3 To obtain the solution of the inhomogeneous equation, we assume that the quantity k is a function of t1 , which is found by the quadrature 4σ t1 dt1 . k(t1 ) = Q−1 (t1 ) 2 n0 (C1 t1 − C3 )2 Therefore, the transcendental first integral of system (3) takes the form −1

Q

(t1 )q1 (C1 t21

−1

− C3 )

4σ − n0

t1 t0

Q−1 (τ1 )

τ1 dτ1 = C 0 , − C3 )2

(C1 τ12

C0

= const. where As is seen, the final form of the first integral depends on the sign of the constant C1 ; as a result of this, three variants are possible. Let us examine each of them. First variant. C1 = 0. After an elementary calculation, we obtain an additional integral in the form

σ 2 u1 +1 = const . e−u1 /v1 v1−2 + 2 v1 Therefore, the transcendental first integral of system (3) is expressed through elementary functions for C1 = 0. Second variant. C1 > 0. The integration leads to the function C2 σ −2(C2 /√−C1 C3 )ζ √ e sin 2ζ + cos 2ζ + const, − 4n0 −C1 C3 where C1 ζ = arctan − t1 . C3 As is seen, in the case C1 > 0, the additional first integral is expressed through elementary functions. Third variant. C1 < 0. By equivalent transformations, the integral transforms into the form

1−γ ζ ζ −γ ζ −1−γ σ 2 −3 + + const, C1 C2 n0 γ − 1 γ γ+1 743

where C2 > 1, γ=√ C1 C3

√ √ −C1 t1 + −C3 √ ζ=√ . −C1 t1 − −C3

Therefore, in the case C1 < 0, the additional integral is also expressed through elementary functions. Therefore, we study the connection of the following three properties, which are independent apparently, but are sufficiently harmonically combined on systems from the rigid body dynamics: (1) the distinguished class of systems (1) with mentioned symmetries; (2) this class of systems has a zero mean variable dissipation (in the variable α), which allows us to consider them as “almost” conservative systems; (3) in some (although lower-dimensional) cases, the systems have, in general, transcendental first integral. Let us present one more important example of a higher-order system having just listed properties. To the system α˙ = −z2 + β sin α, cos α , z˙2 = sin α cos α − z12 sin α cos α , z˙1 = z1 z2 sin α

(4)

which now is considered in the three-dimensional domain S1 {α mod 2π} \ {α = 0, α = π} × R2 {z1 , z2 } (such a system can also reduce to the equivalent system on the tangent bundle of the two-dimentional sphere) and describes the rigid body spatial motion in a resisting medium [317], we put in correspondence the following system with algebraic right-hand side: z12 dz2 τ , = dτ −z2 + βτ τ−

z1 z2 dz1 τ = . dτ −z2 + βτ

(5)

In this case, it is also seen that system (4) is a zero mean variable dissipation system. To attain a full correspondence with the definition, it suffices to introduce the following new phase variable: z1∗ = ln |z1 |. Moreover, it has two first integrals (i.e., a complete list) being transcendental functions and expressed through a finite combination of elementary functions [317]; as was mentioned above, this became possible after the assignment to it a (in general, non-autonomous) system of equations with algebraic (polynomial) right-hand side (5). Therefore, the systems from the rigid body dynamics presented above not only enter the class of systems (1) and have a zero mean variable dissipation, but they have a complete list of transcendental first integrals expressed trough a finite combination of elementary functions. Moreover, the integration of systems (2) and (3) reduces to the integration of the corresponding equations with algebraic righthand sides. As was noted, to seek for the first integrals of the systems considered, it is better to reduce systems of the form (1) to systems with polynomial right-hand sides; on the form of the latter ones the possibility of integrating in elementary functions of the initial system depends. Therefore, we proceed further as follows: let us seek for sufficient conditions for integrability in elementary functions of systems of equations with polynomial right-hand sides studying systems of the most general form in this process. 744

6.

Systems on S1 {α mod 2π} × R1 {ω}

Let us consider the possibilities of complete integrating (in elementary functions) systems of the form x˙ = ax + by + f1 x3 + f2 x2 y + f3 xy 2 + f4 y 3 , y˙ = cx + dy + g1 x3 + g2 x2 y + g3 xy 2 + g4 y 3 on the plane R2 {x, y}. In this case, for definiteness, we assume that the roots of the characteristic equation of the λ-matrix a−λ b c d−λ are real. In this case, we can assume that b = 0. In the case where there are complex-conjugate roots of this equation, we can assume that ad − bc = 1. Applying the substitution y = tx, which is characteristic for homogeneous systems, we arrive at the integration of the identity [at + bt2 + f1 tx2 + f2 t2 x2 + f3 t3 x2 + f4 t4 x2 − c − dt − g1 x2 − g2 tx2 −g3 t2 x2 − g4 t3 x2 ]dx + [ax + btx + f1 x3 + f2 tx3 + f3 t2 x3 + f4 t3 x3 ]dt = 0. It is seen that the nonlinearity is characterized by eight parameters. To integrate the latter identity in elementary functions as a homogeneous equation, it suffices to impose five independent relations g1 = 0, f1 = g2 = β1 ,

f2 = g3 = β2 ,

f3 = g4 = β3 ,

f4 = 0.

Proposition 2. The seven-parametric family of equations x˙ = ax + by + β1 x3 + β2 x2 y + β3 xy 2 , y˙ = cx + dy + β1 x2 y + β2 xy 2 + β3 y 3 on the plane R2 {x, y} has a first (in general, transcendental ) integral expressed through elementary functions. A scheme of proof. After simple transformations, we arrive at the necessity of integration the Bernoulli equation dx + [a + bt]x + [β1 + β2 t + β3 t2 ]x3 = 0; [at + bt2 − c − dt] dt by the change x−3 = u, it reduces to the linear homogeneous equation

at + bt2 − c − dt du + [a + bt]u = −β1 − β2 t − β3 t2 . − 2 dt Since the primitive of the function a + bt at + bt2 − c − dt is expressed through elementary functions, it follows that the general solution of the homogeneous part of the latter equation is also expressed through elementary functions. If, without loss of generality, we assume that b = 0, then a particular solution of the latter inhomogeneous equation is expressed through a finite combination of elementary functions. Remark. The first integral can be also sought for from the equation · 2 x x x = (a − d) + b − c . y y y 745

Corollary 1. For any parameters a, b, c, d, β1 , β2 , and β3 , systems of the form α˙ = a sin α + bω + β1 sin3 α + β2 ω sin2 α + β3 ω 2 sin α, ω˙ = c sin α cos α + dω cos α + β1 ω sin2 α cos α + β2 ω 2 sin α cos α + β3 ω 3 cos α have a transcendental first integral expressed through elementary functions. In particular, system (2) is obtained from the latter system for a = β, b = −1, c = 1, and d = β1 = β2 = β3 = 0, whereas the system (3) is obtained for a = β, b = −1, c = 1, d = −β, β1 = −β, β2 = 0, and β3 = β. Let us generalize the previous arguments. Let us consider the possibility of complete integrating (in elementary functions) systems of a more general form. Precisely, let the nonlinearity be characterized by an arbitrary homogeneous form of odd degree 2n − 1. Then the following assertion more general than Proposition 2 holds. Proposition 3. The (2n + 3)-parameter family of equations x˙ = ax + by + δ1 x2n−1 + δ2 x2n−2 y + · · · + δ2n−2 x2 y 2n−3 + δ2n−1 xy 2n−2 , y˙ = cx + dy + δ1 x2n−2 y + δ2 x2n−3 y 2 + · · · + δ2n−2 xy 2n−2 + δ2n−1 y 2n−1

(6)

on the plane has a (in general, transcendental ) first integral, which is expressed through elementary functions. The family of Eqs. (6) indeed depends on 2n − 1 + 4 independent parameters, since the general nonlinearity of odd degree in this case is characterized by 4n parameters on which 2n + 1 conditions are imposed (four more parameters are in the linear part). Corollary 2. For any parameters a, b, c, d, δ1 , . . . , δ2n−1 , systems of the form α˙ = a sin α + bω + δ1 sin2n−1 α + δ2 ω sin2n−2 α + · · · + δ2n−1 ω 2n−2 sin α, ω˙ = c sin α cos α + dω cos α + δ1 ω sin2n−2 α cos α + δ2 ω 2 sin2n−3 α cos α + · · · + δ2n−1 ω 2n−1 cos α have a transcendental first integral expressed through elementary functions. Let us make the following important remark. As was mentioned above, systems (2) and (3) are relatively rough, but if we violate the symmetries in these systems (for example, if we add additional terms to their right-hand sides), which were introduced for systems of a more general form (1), then the number of obtained topologically different phase portraits can considerably change. So, for example, the following systems being a “perturbation” of a system of the form (3) become nonzero mean variable dissipation dynamical systems and have infinitely many topologically nonequivalent phase portraits (see, e.g., Figs. 5–14, the portraits of a multiparameter family of portraits without limit cycles, and Figs. 15–22, the portraits of a multiparameter family of portraits when there are simple and complicated limit cycles in the system under certain conditions) [317]. 7.

Systems on S1 {α mod 2π} \ {α = 0, α = π} × R2 {z1 , z2 }

Let us study a system of the form (4), which reduces to (5), and the following system, which also arises in the spatial dynamics of a rigid body interacting with a medium [317]: α˙ = −z2 + β(z12 + z22 ) sin α + β sin α cos2 α, cos α , sin α cos α ; z˙1 = βz1 (z12 + z22 ) cos α − βz1 sin2 α cos α + z1 z2 sin α

z˙2 = sin α cos α + βz2 (z12 + z22 ) cos α − βz2 sin2 α cos α − z12

746

Fig. 5. Family of phase portraits without limit cycles.


Fig. 7. Family of phase portraits without limit cycles. it corresponds to the following system with algebraic right-hand side: z12 dz2 τ , = dτ −z2 + βτ (z12 + z22 ) + βτ (1 − τ 2 ) z1 z2 βz1 (z12 + z22 ) − βz1 τ 2 + dz1 τ = . dτ −z2 + βτ (z12 + z22 ) + βτ (1 − τ 2 ) τ + βz2 (z12 + z22 ) − βz2 τ 2 −

(7)

747



Fig. 10. Family of phase portraits without limit cycles. In a similar way, we pass to the homogeneous coordinates uk , k = 1, 2, according to the formulas zk = uk τ . System (5) reduces to the system τ 748

τ − u21 τ du2 + u2 = , dτ −u2 τ + βτ

τ

du1 u1 u2 τ + u1 = , dτ −u2 τ + βτ



Fig. 13. Family of phase portraits without limit cycles. which, in turn, corresponds to the equation 1 − βu2 + u22 − u21 du2 = . du1 2u1 u2 − βu1 This equation is integrable in elementary functions, since the identity 1 − βu2 + u22 + du1 = 0 d u1

(8)

749


Fig. 15. Family of portraits with limit cycles.

Fig. 16. Family of phase portraits without limit cycles. is integrable, and in the coordinates (τ, z1 , z2 ), it has a first integral of the form (compare with [317]) z12 + z22 − βz2 τ + τ 2 = const . z1 τ After the reduction, system (7) corresponds to the system τ 750

τ + βu2 τ 3 (u21 + u22 ) − βu2 τ 3 − u21 τ du2 + u2 = , dτ −u2 τ + βτ 3 (u21 + u22 ) + βτ (1 − τ 2 )



Fig. 19. Family of phase portraits without limit cycles. τ

βu1 τ 3 (u21 + u22 ) − βu1 τ 3 + u1 u2 τ du1 + u1 = , dτ −u2 τ + βτ 3 (u21 + u22 ) + βτ (1 − τ 2 )

which also reduces to (8). A system of the form (4) is equivalent to the following system on the tangent bundle T∗ S2 of the two-dimensional sphere (in this case, β → −β): sin θ = 0, θ¨ + β θ˙ cos θ + sin θ cos θ − ψ˙ 2 cos θ 751




1 + cos2 θ ¨ ˙ ˙ ˙ ψ + β ψ cos θ + θψ = 0. sin θ cos θ sin α ). Its phase portrait is depicted in Fig. 23 (for α = θ and z1 = ψ˙ cos α Let us pose the following question: which are the possibilities of integration in elementary functions of the following system of a more general form including systems (5) and (7) in three-dimensional 752

Fig. 23. Relatively rough phase portrait in a three-dimensional domain. phase domains: z2 zy y2 + c + c 2 3 dz x x x , = dx dx + ey + f z z2 zy y2 gx + hy + iz + i + i + i 1 2 3 dy x x x , = dx dx + ey + f z ax + by + cz + c1

(9)

which have a singularity of the type 1/x? As before, introducing the substitutions y = ux and z = vx, we obtain that system (9) reduces to the system dv ax + bux + cvx + c1 v 2 x + c2 vux + c3 u2 x +v = , dx dx + eux + f vx gx + hux + ivx + i1 v 2 x + i2 vux + i3 u2 x du +u= ; x dx dx + eux + f vx x

to this system, we put in correspondence the following equation with algebraic right-hand side: a + bu + cv + c1 v 2 + c2 vu + c3 u2 − v[d + eu + f v] dv = . du g + hu + iv + i1 v 2 + i2 vu + i3 u2 − u[d + eu + f v] The integration of the latter equation reduces to the integration of the following equation in total differentials: [g + hu + iv + i1 v 2 + i2 vu + i3 u2 − du − eu2 − f uv]dv = = [a + bu + cv + c1 v 2 + c2 vu + c3 u2 − dv − euv − f v 2 ]du.

(10)

In general, we have a 15-parameter family of equations of the form (10). To integrate the latter identity in elementary functions as a homogeneous equation, it suffices to impose seven relations g = 0,

i3 = e,

i1 = 0,

i = 0,

c2 = e,

c = h,

2c1 = i2 + f.

(11)

Let us introduce eight parameters β1 , . . . , β8 and consider them as independent: g = 0, f = β5 ,

h = β1 , a = β6 ,

i1 = 0, b = β7 ,

i = 0,

i2 = β2 ,

c = β1 ,

i3 = β3 , d = β4 , e = β3 , β2 + β5 , c2 = β3 , c3 = β8 . c1 = 2 753

Therefore, under the group of conditions (11), Eq. (10) reduces to the form dv = du

v2 + β8 u2 2 ; (β1 − β4 )u + (β2 − β5 )vu

β6 + β7 u + (β1 − β4 )v + (β2 − β5 )

(12)

after that, Eq. (12) is integrated in elementary functions. Indeed, integrating identity (10), we obtain the relation

(β2 − β5 )v 2 β6 (β1 − β4 )v + d[] +d − d[β7 ln |u|] − d[β8 u] = 0; d u 2u u in the coordinates (x, y, z), it allows us to obtain a first integral in the form (β2 − β5 )

z2 − β8 y 2 + (β1 − β4 )zx + β6 x2 y 2 − β7 ln = const . yx x

(13)

Therefore, we can make a conclusion on the integrability in elementary functions of the following third-order system, which is nonconservative in general and depends on eight parameters: dz = dx

z2 zy y2 + β3 + β8 2x x x , β4 x + β3 y + β5 z zy y2 β + β y + β 1 2 3 dy x x . = dx β4 x + β3 y + β5 z

β6 x + β7 y + β1 z + (β2 − β5 )

Corollary 3. The third-order system α˙ = β4 sin α + β3 z1 + β5 z2 , β2 + β5 2 cos α cos α cos α z2 + β3 z1 z2 + β8 z12 , z˙2 = β6 sin α cos α + β7 z1 cos α + β1 z2 cos α + 2 sin α sin α sin α cos α cos α + β3 z12 z˙1 = β1 z1 cos α + β2 z1 z2 sin α sin α on the set S1 {α mod 2π} \ {α = 0, α = π} × R2 {z1 , z2 }

(14)

depending on eight parameters has in general a transcendental first integral expressed trough elementary functions. In particular, for β1 = β3 = β7 = 0, β2 = β6 = 1, β5 = β8 = −1, and β4 = β, system (14) reduces to system (4). To find an additional first integral of the non-autonomous system (9), we use the just found first integral (13) expressed through a finite combination of elementary functions. 8.

Systems from Dynamics of a Four-Dimensional Rigid Body Interacting with a Medium

8.1. Enlarging dimension in rigid body dynamics. In [203–206], the authors showed the complete integrability of the plane problem (i.e., a body is situated in the Euclidean plane E2 ) of the rigid body motion in a resisting medium under the streamline flow around conditions in the case where the system of dynamical equations has one first integral being a transcendental function (in the sense of theory of functions of one complex variable, having essentially singular points) of quasi-velocities. There it was assumed that the whole interaction of the medium with the body is concentrated on the part of the body surface having the form of a (one-dimensional ) plate. 754

Later on [317], the plane problem was generalized to the spatial (three-dimensional ) case (i.e., the motion is executed in the Euclidean space E3 ), and, moreover, the system of dynamical equations has a complete tuple of transcendental first integrals. It was assumed here that the whole interaction of the medium with the body is concentrated on the part of the body surface having the form of a plane (two-dimensional ) disk. The structure of dynamical equations of motion is often preserved under the extension of dynamical properties to cases of higher dimension. For example, at present (see also [39–41, 169]), the theory of four-dimensional (and, moreover, n-dimensional) rigid body motion. The author of the recent works (the school of S. P. Novikov and O. I. Bogoyavlenskii) succeeded in showing the Hamiltonian property for the equations of motion of a many-dimensional rigid body around a fixed point. The present section is devoted to studying the motion of a four-dimensional rigid body interacting with a medium according to the laws of “streamline flow around” and presents the results of studying this problem for the first time. It is assumed that the whole interaction of a (four-dimensional) rigid body with a medium is concentrated on the part of the (smooth three-dimensional ) body surface that has the form of a (threedimensional ) ball. In this case, the tensor of the body motion angular velocity is six-dimensional, whereas the velocity of the center of masses is four-dimensional. 8.2. Problem statement and equation on Lie algebra so(4). Let a four-dimensional rigid body move in a resisting medium filling a four-dimensional domain of the Euclidean space E4 , and let the whole interaction of the medium with the body be concentrated on the part of (smooth threedimensional) body surface that has the form of the three-dimensional disk D3 . The distance from the point N of application of the resistance force to the center D of the disk is a function of one parameter, the angle of attack α, which is made by the velocity v of the point D and the middle perpendicular to the disk dropped from the body center of masses C in the four-dimensional space (compare with [39–41]). The resistance force is orthogonal to the disk D3 in the four-dimensional space, and its value has the form S = s1 (α)v 2 , where s1 is the nonnegative resistance coefficient. To the body, let us relate the coordinate system Dx1 x2 x3 x4 whose axis Dx1 coincides with the axis CD and the axes Dx2 , Dx3 , and Dx4 lie in the disk hyperplane. If in the coordinate system Dx1 x2 x3 x4 , the operator of inertia has the diagonal form diag{I1 , I2 , I3 , I4 }, where Ω is the rigid body angular velocity tensor, Ω ∈ so(4), then the part of the equations of the four-dimensional body motion corresponding to the algebra so(4) has the following form [79, 80, 379–381]: ˙ + ΛΩ ˙ + [Ω, ΩΛ + ΛΩ] = M, ΩΛ (15) where Λ = diag{λ1 , λ2 , λ3 , λ4 }, 1 λ1 = (−I1 + I2 + I3 + I4 ), 2 1 λ2 = (I1 − I2 + I3 + I4 ), 2 1 λ3 = (I1 + I2 − I3 + I4 ), 2 1 λ4 = (I1 + I2 + I3 − I4 ), 2 755

M is the moment of the exterior forces acting on the body in R4 being projected on the natural coordinates in the algebra so(4), and [. . . ] is the commutator in so(4). It is convenient to represent the matrix Ω ∈ so(4) in the natural coordinates: ⎞ ⎛ 0 −ω6 ω5 −ω3 ⎜ ω6 0 −ω4 ω2 ⎟ ⎟, ⎜ (16) ⎝−ω5 ω4 0 −ω1 ⎠ ω3 −ω2 ω1 0 where ω1 , ω2 , ω3 , ω4 , ω5 , and ω6 are the components of angular velocity tensor in projections on the natural coordinates in the algebra so(4). It is convenient to represent the resistance coefficient s1 in the form s1 (α) = s(α) sign cos α. If (0, x2N , x3N , x4N ) are the coordinates of the point N in the system Dx1 x2 x3 x4 and {−S, 0, 0, 0} are the coordinates of the resistance force in the same system, then in calculating the resistance force moment, it is necessary to construct the mapping R4 × R4 −→ so(4),

(17)

which transforms a pair of vectors in R4 into a certain element from the algebra so(4). In projections on the coordinates in the group so(4), the resistance force moment has the form (0, 0, x4N S, 0, −x3N S, x2N S) ∈ R6 ∼ = M ∈ so(4).

(18)

Here, it is necessary to take into account that if (v, α, β1 , β2 ) are spherical coordinates of the center D of the three-dimensional disk velocity in R4 , then x2N = R(α) cos β1 ,

x3N = R(α) sin β1 cos β2 ,

x4N = R(α) sin β1 sin β2 .

Taking all this into account, we can obtain the following equations of motion in the considered nonconservative resistance force field: (λ4 + λ3 )ω˙1 + (λ3 − λ4 )(ω3 ω5 + ω2 ω4 ) = 0,

(19)

(λ2 + λ4 )ω˙2 + (λ2 − λ4 )(ω3 ω6 − ω1 ω4 ) = 0,

(20)

(λ4 + λ1 )ω˙3 + (λ4 − λ1 )(ω2 ω6 + ω1 ω5 ) = x4N S,

(21)

(λ3 + λ2 )ω˙4 + (λ2 − λ3 )(ω5 ω6 + ω1 ω2 ) = 0,

(22)

(λ1 + λ3 )ω˙5 + (λ3 − λ1 )(ω4 ω6 − ω1 ω3 ) = −x3N S,

(23)

(λ1 + λ2 )ω˙6 + (λ1 − λ2 )(ω4 ω5 + ω2 ω3 ) = x2N S.

(24)

8.3. Dynamics on R4 . By analogy with the three-dimensional case, we can deduce formulas analogous to the Euler and Rivals formulas: the velocities and accelerations of any two points A and B of a four-dimensional rigid body are related by the following relations in any coordinate system: vB = vA + ΩAB,

wB = wA + Ω2 AB + EAB,

(25)

where Ω ∈ so(4) and E = Ω˙ ∈ so(4). The matrix E is called the matrix (corresponding to the tensor ) of acceleration of a four-dimensional rigid body. Using formulas (25), we can obtain the equations of the four-dimensional rigid body center-of-masses motion in R4 . 756

8.4. Motion in a resisting medium under action of a non-integrable constraint (compare with [317]). Let us consider a class of body motions such that the following condition holds all the time: v = |v| = const .

(26)

Moreover, assume that a certain (tracking) traction force acts on the body that ensures the fulfilment of condition (26) and is a reaction of this non-integrable constraint (compare with the two- and threedimensional cases [333]). By a certain choice of the tracking force value along the line CD, the fulfilment of condition (26) can be attained (see also [379–381]). 8.5. Case of a dynamically symmetric rigid body. Analogously to the three-dimensional case, let the following relations hold: I2 = I3 = I4 .

(27)

In such a case, there exist the following three cyclic first integrals of Eqs. (19)–(24): ω1 = ω10 ,

ω2 = ω20 ,

ω4 = ω40 .

For simplicity, let us consider the motions on their zero levels: ω10 = ω20 = ω40 = 0.

(28)

To describe the body motion, we use the pair of dynamical functions (R(α), s(α)) the information about which is of qualitative character. By analogy with the cases of sufficiently lower dimensions, without loss of generality [58, 59], we can assume that R(α) = A sin α,

A > 0;

s(α) = B cos α,

B > 0.

As a result of all this, the equations corresponding to so(4) take the following form (here, n20 =

(29) AB ): 2I2

ω˙3 = n20 v 2 sin α cos α sin β1 sin β2 ,

(30)

ω˙5 = −n20 v 2 sin α cos α sin β1 cos β2 ,

(31)

ω˙6 =

n20 v 2 sin α cos α cos β1 .

(32)

If we introduce the natural change of angular velocities by the formulas z1 = ω3 cos β2 + ω5 sin β2 ,

(33)

z2 = −ω3 sin β2 cos β1 + ω5 cos β2 cos β1 + ω6 sin β1 ,

(34)

z3 = ω3 sin β2 sin β1 − ω5 cos β2 sin β1 + ω6 cos β1 ,

(35)

then the complete system of dynamical equations of motion on the direct product so(4) × R4 (after taking four Eqs. (26) and (28) into account) takes the symmetric form α˙ = −z3 + σn20 v sin α, cos α , sin α cos α cos α cos β1 + z12 , z˙2 = z2 z3 sin α sin α sin β1 cos α cos α cos β1 − z1 z2 , z˙1 = z1 z3 sin α sin α sin β1 cos α , β˙1 = z2 sin α cos α β˙2 = −z1 . sin α sin β1

z˙3 = n20 v 2 sin α cos α − (z12 + z22 )

(36)

(37)

757

The sixth-order system (36), (37) has the independent fifth-order subsystem (36). To integrate this system completely, it is in general necessary to know five independent first integrals. However, after the change of variables z2 (38) z1 , z2 −→ z = z12 + z22 , z∗ = , z1 the system (36), (37) reduces to the form α˙ = −z3 + σn20 v sin α, cos α , z˙3 = n20 v 2 sin α cos α − z 2 sin α cos α , z˙ = zz3 sin α cos α cos β1 z˙∗ = 1 + z∗2 z , sin α sin β1 zz∗ cos α , β˙1 = 1 + z∗2 sin α cos α . β˙2 = −Z1 (z, z∗ ) sin α sin β1

(39)

(40) (41)

It is seen that the fifth-order system (36) falls into independent subsystems of lower order: system (39) is of the third order, and system (40) is of the second order (of course, after the change of the independent variable). Therefore, for the complete integrability of the system (39)–(41), it suffices to find two independent first integrals of system (39): one for system (40) and an additional first integral “coupling” Eq. (41). System (39) arises in the three-dimensional rigid body dynamics [317]. It has the following two transcendental integrals: z 2 + z32 − σn20 vz3 sin α + n20 v 2 sin2 α = C1 = const, z sin α z3 z , , sin α = C2 = const . G sin α sin α System (40) has a first integral of the form 1 + z∗2 = C3 = const, sin β1

(42) (43)

(44)

and the corresponding additional first integral “coupling” Eq. (41) has the form cos β1 = sin{C3 (β2 + C4 )}, ± 2 C3 − 1

C4 = const .

(45)

8.6. A new direction. Previously only those four-dimensional body motions were considered for which M ≡ 0 (or the force field is conservative (see, e.g., [31, 45, 65, 66, 95, 378, 379, 381, 384])). This work opens a new direction developed by the author in studying the equations of the rigid body motion on the direct product so(4) × R4 (M is not identically equal to zero) in a nonconservative force field (see also [87–89, 258, 274, 283, 299, 317, 359]). Sometimes the methodology for integrating the dynamical systems considered can be also extended to the space so(n) × Rn of an arbitrary dynamically symmetric n-dimensional rigid body. 758

9.

Conclusion

Therefore, the dynamical systems considered in this work refer to zero mean variable dissipation systems with respect to the existing periodic coordinate. Moreover, such systems often have a complete list of first integrals expressed through elementary functions. The method for reducing the initial systems with right-hand sides containing polynomials in trigonometrical functions to systems with polynomial right-hand sides allows one to seek for (or to prove the absence of) first integrals for system of a more general form but not only for the systems having the above symmetries (see also [23, 37, 43, 48, 49, 57, 60–64, 68, 94, 99–104, 107, 108, 118–133, 135–137, 150–153, 155–162, 164–166, 197–202, 212, 214, 215, 240, 317, 357, 362, 366, 370–373, 376, 377, 383, 388, 389]). Acknowledgment. This work was supported by the Russian Foundation for Basic Research, Grant No. 08-01-00231-A. REFERENCES 1. S. A. Agafonov, D. V. Georgievskii, and M. V. Shamolin, “Some actual problems of geometry and mechanics”, In: Abstract of Sessions of Workshop Actual Problems of Geometry and Mechanics, Contemporary Mathematics. Fundamental Directions [in Russian], Vol. 23 (2007), p. 34. 2. G. A. Al’ev, “Spatial problem of submegence of a disk in an incompressible fluid,” Izv. Akad. Nauk SSSR, Mekh. Zh. Gaz. 1, 17–20 (1988). 3. V. V. Amel’kin, N. A. Lukashevich, and A. P. Sadovskii, Nonlinear Oscillations in Second-Order Systems [in Russian], BGU, Minsk (1982). 4. A. A. Andronov, Collection of Works [in Russian], Izd. Akad. Nauk SSSR, Moscow (1956). 5. A. A. Andronov and E. A. Leontovich, “To theory of variations of qualitative structure of plane partition into trajectories,” Dokl. Akad. Nauk SSSR, 21, No. 9 (1938). 6. A. A. Andronov and E. A, Leontovich, “Birth of limit cycles from a nonrough focus or center and from a nonrough limit cycle,” Math. Sb. 40, No. 2 (1956). 7. A. A. Andronov and E. A. Leontovich,“On birth of limit cycles from a separatrix loop and from separatrix of saddle-node equilibrium state,” Mat. Sb., 48, No. 3 (1959). 8. A. A. Andronov and E. A. Leontovich, “Dynamical systems of the first degree of non-roughness on the plane,” Mat. Sb., 68, No. 3 (1965). 9. A. A. Andronov and E. A. Lentovich, “Sufficient conditions for non-roughness of the first degree of a dynamical system on the plane,” Differents. Uravn., 6, No. 12 (1970). 10. A. A. Andronov and L. S. Pontryagin, “Rough systems,” Dokl. Akad. Nauk SSSR, 14, No. 5, 247– 250 (1937). 11. A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Oscillation Theory [in Russian], Nauka, Moscow (1981). 12. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Qualitative Theory of SecondOrder Dynamical Systems [in Russian], Nauka, Moscow (1966). 13. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Bifurcation Theory of Dynamical Systems on the Plane [in Russian], Nauka, Moscow (1967). 14. P. Appel, Theoretical Mechanics, Vols. I, II [Russian translation], Fizmatgiz, Moscow (1960). 15. S. Kh. Aranson, “Dynamical systems on two-dimensional manifolds,” In: Proceedings of the 5th International Conference on Nonlinear Oscillations, Vol. 2 [in Russian], Institute of Mathematics, Academy of Sciences of UkrSSR (1970). 16. S. Kh. Aranson and V. Z. Grines, “Topological classification of flows on two-dimensional manifolds,” Usp. Mat. Nauk, 41, No. 1 (1986). 17. V. I. Arnol’d, “Hamiltionian property of Euler equations of rigid body dynamics in ideal fluid,” Usp. Mat. Nauk, 24, No. 3, 225–226 (1969). 759

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228. M. V. Shamolin, “On relative roughness of dynamical systems in problem of body motion in a resisting medium,” In: Abstracts of Reports of Chebyshev Readings, Vestn. VGU, Ser. 1, Mat., Mekh., 6, 17 (1995). 229. M. V. Shamolin, “Definition of relative roughness and two-parameter family of phase portraits in rigid body dynamics,” Usp. Mat. Nauk, 51, No. 1, 175–176 (1996). 230. M. V. Shamolin, “Periodic and Poisson stable trajectories in problem of body motion in a resisting medium,” Izv. Ross. Akad. Nauk, Mekhanika Tverdogo Tela, 2, 55–63 (1996). 231. M. V. Shamolin, “Spatial Poincaré topographical systems and comparison systems,” In: Abstracts of Reports of Mathematical Conference ‘Erugin Readings,’ Brest, May 14–16, 1996 [in Russian], Brest (1996), p. 107. 232. M. V. Shamolin, “Introduction to spatial dynamics of rigid body motion in resisting medium.” In: Materials of International Conference and Chebyshev Readings Devoted to the 175th Anniversary of P. L. Chebyshev, Moscow, May 14–19, 1996, Vol. 2 [in Russian], MGU, Moscow (1996), pp. 371–373. 233. M. V. Shamolin, “A list of integrals of dynamical equations in spatial problem of body motion in a resisting medium,” In: Modelling and Study of Stability of Systems, Scientific Conference, May 20–24, 1996. Absracts of Reports (Study of Systems) [in Russian], Kiev (1996), p. 142. 234. M. V. Shamolin, “Variety of types of phase portraits in dynamics of a rigid body interacting with a resisting medium,” Dokl. Ross. Akad. Nauk, 349, No. 2, 193–197 (1996). 235. M. V. Shamolin, “Qualitative methods in dynamics of a rigid body interacting with a medium,” In: II Siberian Congress in Applied and Industrial Mathematics, Novosibirsk, June 25–30, 1996. Abstracts of Reports. Pt. III [in Russian], Novosibirsk (1996), p. 267. 236. M. V. Shamolin, “On a certain integrable case in dynamics of spatial body motion in a resisting medium,” In: II Symposium in Classical and Celestial Mechanics. Abstracts of Reports, Velikie Luki, August 23–28, 1996 [in Russian], Moscow–Velikie Luki (1996), pp. 91–92. 237. M. V. Shamolin, “Introduction to problem of body drag in a resisting medium and a new twoparameter family of phase portraits, ” Vestn. MGU, Ser. 1, Mat., Mekh., 4, 57–69 (1996). 238. M. V. Shamolin, “On an integrable case in spatial dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekhanika Tverdogo Tela, 2, 65–68 (1997). 239. M. V. Shamolin, “Jacobi integrability of problem of a spatial pendulum placed in over-running medium flow,” In: Modelling and Study of Systems. Scientific Conference, May, 19–23, 1997. Abstracts of Reports [in Russian], Kiev (1997), p. 143. 240. M. V. Shamolin, “Partial stabilization of body rotational motions in a medium under a free drag,” In: Abstracts of Reports of All-Russian Conference with International Participation ‘Problems of Celestial Mechanics,’ St. Petersburg, June 3–6, 1997, Institute of Theoretical Astronomy [in Russian], Institute of theoretical Astronomy, Russian Academy of Sciences, St. Petersburg (1997), pp. 183–184. 241. M. V. Shamolin, “Spatial Poincaré topographical systems and comparison systems,” Usp. Mat. Nauk, 52, No. 3, 177–178 (1997). 242. M. V. Shamolin, “Mathematical modelling of dynamics of a spatial pendulum flowing around by a medium,” In: Proceedings of VII International Symposium ‘Methods of Discrete Singularities in Problems of Mathematical Physics’, June 26–29, 1997, Feodociya [in Russian], Kherson State Technical University, Kherson (1997), pp. 153–154. 243. M. V. Shamolin, “Spatial dynamics of a rigid body interacting with a medium,” In: Workshop in Mechanics of Systems and Problems of Motion Control and Navigation, Izv. Ross. Akad. Nauk, Mekhanika Tverdogo Tela, 4, 174 (1997). 244. M. V. Shamolin, “Qualitative methods in dynamics of a rigid body interacting with a medium,” In: YSTM’96: ‘Young People and Science, the Third Millenium,’ Proceedings of International Conference (Ser. Professional) [in Russian], Vol. 2, NTA “APFN,” Moscow (1997), pp. I–4.

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245. M. V. Shamolin, “Qualitative and numerical methods in some problems of spatial dynamics of a rigid body interacting with a medium,” In: Abstracts of Reports of 5th International ConferenceWorkshop ‘Engineering-Physical Problems of New Technics,’ Moscow, May 19–22, 1998 [in Russian], Moscow State Technical University, Moscow (1998), pp. 154–155. 246. M. V. Shamolin, “Some problems of spatial dynamics of a rigid body interacting with a medium under quasi-stationarity conditions,” In: Abstracts of Reports of All-Russian Scientific-Technical Conference of Young Scientists ‘Modern Problems of Aero-Cosmous Science,’ Zhukovskii, May 27–29, 1998 [in Russian], Central Aero-Hydrodynamical Institute, Moscow (1998), pp. 89–90. 247. M. V. Shamolin, “Absolute and relative structural stability in spatial dynamics of a rigid body interacting with a medium,” In: Proceedings of International Conference ‘Mathematics in Industry’, ICIM–98, Taganrog, June 29– July 03, 1998 [in Russian], Taganrog State Pedagogical Institute, Taganrog (1998), pp. 332–333. 248. M. V. Shamolin, “On integrability in transcendental functions,” Usp. Mat. Nauk, 53, No. 3, 209–210 (1998). 249. M. V. Shamolin, “Families of three-dimensional phase portraits in spatial dynamics of a rigid body interacting with a medium,” In: III International Symposium in Classical and Celestial Mechanics, August 23–27, 1998, Velikie Luki. Abstracts of Reports [in Russian], Computational Center of Russian Academy of Sciences, Moscow–Velikie Luki (1998), pp. 165–167. 250. M. V. Shamolin, “Methods of nonlinear analysis in dynamics of a rigid body interacting with a medium,” In: CD–Proceedings of the Congress ‘Nonlinear Analysis and Its Applications’, Moscow, Russia, Sept. 1–5, 1998 [in Russian], Moscow (1999), pp. 497–508. 251. M. V. Shamolin, “Family of portraits with limit cycles in plane dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekhanika Tverdogo Tela, 6, 29–37 (1998). 252. M. V. Shamolin, “Certain classes of partial solutions in dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekhanika Tverdogo Tela, 2, 178–189 (1999). 253. M. V. Shamolin, “New Jacobi integrable cases in dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 364, No. 5, 627–629 (1999). 254. M. V. Shamolin, “On roughness of dissipative systems and relative roughness and non-roughness of variable dissipation systems,” Usp. Mat. Nauk, 54, No. 5, 181–182 (1999). 255. M. V. Shamolin, “A new family of phase portraits in spatial dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 371, No. 4, 480–483 (2000). 256. M. V. Shamolin, “On roughness of disspative systems and relative roughness of variable dissipation systems,” In: Abstracts of Reports of Workshop in Vector and Tensor Analysis Named after P. K. Rashevskii, Vestn. MGU, Ser. 1, Mat., Mekh., 2, 63 (2000). 257. M. V. Shamolin, “On limit sets of differential equations near singular points,” Usp. Mat. Nauk, 55, No. 3, 187–188 (2000). 258. M. V. Shamolin, “Jacobi integrability in problem of four-dimensional rigid body motion in a resisting medium,” Dokl. Ross. Akad. Nauk, 375, No. 3, 343–346. (2000). 259. M. V. Shamolin, “On stability of motion of a body twisted around its longitudinal axis in a resisting medium,” Izv. Ross. Akad. Nauk, Mekhanika Tverdogo Tela 1, 189–193 (2001). 260. M. V. Shamolin, “Complete integrability of equations for motion of a spatial pendulum in overrunning medium flow,” Vestn. MGU, Ser. 1, Mat., Mekh., 5, 22–28 (2001). 261. M. V. Shamolin, “Problem of four-dimensional body motion in a resisting medium and a certain case of integrability,” In: Book of Abstracts of the Third International Conference “Differential Equations and Applications,” St. Petersburg, Russia, June 12–17, 2000 [in Russian], St. Petersburg State University, St. Petersburg (2000), p. 198. 262. M. V. Shamolin, “On limit sets of differential equations near singular points,”, Usp. Mat. Nauk, 55, No. 3, 187–188 (2000). 263. M. V. Shamolin, “Many-dimensional topographical Poincaré systems and transcendental integrability,” In: IV Siberian Congress in Applied and Industrial Mathematics, Novosibirsk, June 770

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26–July 01, 2000. Abstracts of Reports, Pt. I. [in Russian], Novosibirsk, Institute of Mathematics (2000), pp. 25–26. M. V. Shamolin, “Jacobi integrability of problem of four-dimensional body motion in a resisting medium,” In: Abstracts of Reports of International Conference in Differential Equations and Dynamical Systems, Suzdal’, August 21–26, 2000 [in Russian], Vladimir, Vladimir State University (2000), pp. 196–197. M. V. Shamolin, “Comparison of certain integrability cases from two-, three-, and fourdimensional dynamics of a rigid body interacting with a medium,” In: Abstracts of Reports of V Crimeanian International Mathematical School ‘Lyapunov Function Method and Its Application,’ (LFM–2000), Crimea, Alushta, September 5–13, 2000 [in Russian], Simpheropol’ (2000), p. 169. M. V. Shamolin, “On a certain case of Jacobi integrability in dynamics of a four-dimensional rigid body interacting with a medium,” In: Abstracts of Reports of International Conference in Differential and Integral Equations, Odessa, September 12–14, 2000 [in Russian], AstroPrint, Odessa (2000), pp. 294–295. M. V. Shamolin, “On stability of motion of a rigid body twisted around its longitudinal axis in a resisting medium,” Izv. Ross. Akad. Nauk, Mekhanika Tverdogo Tela, 1, 189–193 (2001). M. V. Shamolin, “Variety of types of phase portraits in dynamics of a rigid body interacting with a medium,” In: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Fund. Prikl. Mat., 7, No. 1, 302–303 (2001). M. V. Shamolin, “Integrability of a problem of four-dimensional rigid body in a resisting medium,” In: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Fund. Prikl. Mat., 7, No. 1, 309 (2001). M. V. Shamolin, “New integrable cases in dynamics of a four-dimensional rigid body interacting with a medium,” In: Abstracts of Reports of Scientific Conference, May 22–25, 2001 [in Russian], Kiev (2001), p. 344. M. V. Shamolin, “Integrability cases of equations for spatial dynamics of a rigid body,” Prikl. Mekh., 37, No. 6, 74–82 (2001). M. V. Shamolin, “New Jacobi integrable cases in dynamics of two-, three-, and four-dimensional rigid body interacting with a medium,” In: Absracts of Reports of VIII All-Russian Congress in Theoretical and Applied Mechanics, Perm’, August 23–29, 2001 [in Russian], Ural Department of Russian Academy of Sciencesm Ekaterinburg (2001), pp. 599–600. M. V. Shamolin, “On integrability of certain classes of nonconservative systems,” Usp. Mat. Nauk, 57, No. 1, 169–170 (2002). M. V. Shamolin, “New integrable cases in dynamics of a two-, three-, and four-dimensional rigid body interacting with a medium,” In: Abstracts of Reports of International Conference in Differential Equations and Dynamical Systems, Suzdal’, July 1–6, 2002 [in Russian], Vladimir State University, Vladimir (2002), pp. 142–144. M. V. Shamolin, “On a certain spatial problem of rigid body motion in a resisting medium,” In: Abstracts of Reports of International Scientific Conference ‘Third Polyakhov Readings,’ St. Petersburg, February 4–6, 2003 [in Russian], NIIKh St. Petersburg Univ, (2003), pp. 170–171. M. V. Shamolin, “Integrability in transcendental functions in rigid body dynamics,” In: Abstracts of Reports of Scientific Conference ‘Lomonosov Readings,’ Sec. Mechanics, April 17–27, 2003, Moscow , M. V. Lomonosov Moscow State University [in Russian], MGU, Moscow (2003), p. 130. M. V. Shamolin, “On integrability of nonconservative dynamical systems in transcendental functions,” In: Modelling and Study of Stability of Systems, Scientific Conference, May 27–30, 2003, Abstracts of Reports [in Russian], Kiev (2003), p. 277. M. V. Shamolin, “Geometric representation of motion in a certain problem of body interaction with a medium,” Prikl. Mekh., 40, No. 4, 137–144 (2004).

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279. M. V. Shamolin, “Integrability of nonconservative systems in elementary functions,” In: X Mathematical International Conference Named after Academician M. Kravchuk, September 3– 15, 2004, Kiev [in Russian], Kiev (2004), p. 279. 280. M. V. Shamolin, Methods for Analysis of Classes of Nonconservative Systems in Dynamics of a Rigid Body Interacting with a Medium [in Russian], Doctorial Dissertation, MGU, Moscow (2004), p. 329. 281. M. V. Shamolin, Some Problems of Differential and Topological Diagnostics [in Russian], Ekzamen, Moscow (2004). 282. M. V. Shamolin, “On rigid body motion in a resisting medium taking account of rotational derivatives of areodynamic force moment in angular velocity,” In: Modelling and Studying of Systems, Scientific Conference, May 23–25, 2005. Abstracts of Reports [in Russian], Kiev (2005), p. 351. 283. M. V. Shamolin, “Cases of complete integrability in dynamics of a four-dimensional rigid body interacting with a medium,” In: Abstracts of Reports of International Conference ‘Functional Spaces, Approximation Theory, and Nonlinear Analysis’ Devoted to the 100th Anniversary of A. M. Nikol’skii, Moscow, May 23–29, 2005 [in Russian], V. A. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow (2005), p. 244. 284. M. V. Shamolin, “On a certain integrable case in dynamics on so(4) × R4 ,” In: Abstracts of Reports of All-Russian Conference ‘Differential Equations and Their Applications,’ (SamDif– 2005), Samara, June 27–Jily 2, 2005 [in Russian], Univers-Grupp, Samara (2005), pp. 97–98. 285. M. V. Shamolin, “A case of complete integrability in spatial dynamics of a rigid body interacting with a medium taking account of rotational derivatives of force moment in angular velocity,” Dokl. Ross. Akad. Nauk, 403, No. 4, 482–485 (2005). 286. M. V. Shamolin, “Comparison of Jacobi integrable cases of plane and spatial body motions in a medium under streamline flow around,” Prikl. Mat. Mekh., 69, No. 6, 1003–1010 (2005). 287. M. V. Shamolin, “On body motion in a resisting medium taking account of rotational derivatives of aerodynamic force moment in angular velocity,” In: Abstracts of Reports of Scientific Conference ‘Lomonosov Readings–2005,’ Sec. Mechanics, April, 2005, Moscow, M. V. Lomonosov Moscow State University [in Russian], MGU, Moscow (2005), p. 182. 288. M. V. Shamolin, “Variable dissipation dynamical systems in dynamics of a rigid body interacting with a medium,” In: Differential Equations and Computer Algebra Tools, Materials of International Conference, Brest, October 5–8, 2005, Pt. 1. [in Russian], BGPU, Minsk (2005), pp. 231–233. 289. M. V. Shamolin, “On a certain integrable case of equations of dynamics in so(4) × R4 ,” Usp. Mat. Nauk, 60, No. 6, 233–234 (2005). 290. M. V. Shamolin,“Integrability in transcendental functions in rigid body dynamics,” In: Mathematical Conference ‘Modern Problems of Applied Mathematics and Mathematical Modelling, Voronezh, December 12–17, 2005 [in Russian], Voronezh State Academy, Voronezh (2005), p. 240. 291. M. V. Shamolin, “Variable dissipation systems in dynamics of a rigid body interacting with a medium,” In: Fourth Polyakhov Readings, Abstracts of Reports of International Scientific Conference on Mechanics, St. Petersburg, February 7–10, 2006 [in Russian], VVM, St. Petersburg (2006), p. 86. 292. M. V. Shamolin, “Model problem of body motion in a resisting medium taking account of dependence of resistance force on angular velocity,” In: Scientifuc Report of Institute of Mechanics, Moscow State University [in Russian], No. 4818, Institute of Mechanics, Moscow State University, Moscow (2006), p. 44. 293. M. V. Shamolin, “To problem on rigid body spatial drag in a resisting medium,” Izv. Ross. Akad. Nauk, Mekhanika Tverdogo Tela, 3, 45–57 (2006).

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294. M. V. Shamolin, “On trajectories of characteristic points of a rigid body moving in a medium,” In: International Conference ‘Fifth Okunev Readings,’ St. Petersburg, June 26–30, 2006. Absracts of Reports [in Russian], Baltic State Technical University, St. Petersburg (2006), p. 34. 295. M. V. Shamolin, “On a case of complete integrability in four-dimensional rigid body dynamics,” In: Abstracts of Reports of International Conference in Differential Equations and Dynamical Systems, Vladimir, July 10–15, 2006 [in Russian], Vladimir State University, Vladimir (2006), pp. 226–228. 296. M. V. Shamolin, “To spatial problem of rigid body interaction with a resisting medium,”In: Absracts of Reports of IX All-Russian Congress in Theoretical and Applied Mechanics, Nizhnii Novgorod, August 22–28, 2006. Vol. I [in Russian], N. I. Lobachevskii Nizhnii Novgorod State Univesity, Nizhnii Novgorod (2006), p. 120. 297. M. V. Shamolin, “Spatial problem on rigid body motion in a resisting medium,” In: VIII Crimeanian International Mathematical School ‘Lyapunov Function Method and Its Applications,’ Abstracts of Reports, Alushta, September 10–17, 2006, Tavriya National University [in Russian], DiAiPi, Simpheropol’ (2006), p. 184. 298. M. V. Shamolin, “Variety of types of phase portraits in dynamics of a rigid body interacting with a medium,” In: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], Vol. 23 (2007), p. 17. 299. M. V. Shamolin, “Integrability of problem of four-dimensional rigid body motion in a resisting medium,” In: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], Vol. 23 (2007), p. 21. 300. M. V. Shamolin, “On account of rotational deivatives of a aerodynamic force moment on body motion in a resisting medium,” In: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], Vol. 23 (2007), p. 26. 301. M. V. Shamolin, “New integrable cases in dynamics of a four-dimensional rigid body interacting with a medium,” In: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], Vol. 23 (2007), p. 27. 302. M. V. Shamolin, “On integrability in transcendental functions,” In: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], Vol. 23 (2007), p. 34. 303. M. V. Shamolin, “On integrability of motion of four-dimensional body-pendulum situated in over-running medium flow,” In: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian] Vol. 23, (2007), p. 37. 304. M. V. Shamolin, “Integrability in elementary functions of variable dissipation systems,” In: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], Vol. 23 (2007), p. 38. 305. M. V. Shamolin, “Integrability in transcendental elementary functions,” In: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], Vol. 23 (2007), p. 40. 306. M. V. Shamolin, “On rigid body motion in a resisting medium taking account of rotational derivatives of aerodynamic force moment in angular velocity,” In: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], Vol. 23 (2007), p. 44. 307. M. V. Shamolin, “Influence of rotational derivatives of medium interaction force moment in angular velocity of a rigid body on its motion,” In: Abstracts of Sessions of Workshop ‘Actual

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Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], Vol. 23 (2007), p. 44. M. V. Shamolin, “On work of All-Russian Conference ‘Differential equations and Their Applications,’ Samara, June 27–July 29, 2005,” In: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], Vol. 23 (2007), p. 45. M. V. Shamolin, “On integrability in elementary functions of certain classes of nonconservative dynamical systems,” In: Modelling and Study of Stability of Systems, Scientific Conference, May 22–25, 2007. Abstracts of Reports [in Russian], Kiev (2007), p. 249. M. V. Shamolin, “Case of complete integrability in dynamics of a four-dimensional rigid body in nonconcervative force field,” In: ‘Nonlinear Dynamical Analysis-2007,’ Abstracts of Reports of International Congress, St. Petersburg, June 4–8, 2007 [in Russian], St. Petersburg State University, St. Petersburg (2007), p. 178. M. V. Shamolin, “Cases of complete integrability in elementary functions of certain classes of nonconservative dynamical systems,” In: Abstracts of Reports of International Conference ‘Classical Problems of Rigid Body Dynamics,’ June 9–13, 2007 [in Russian], Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk (2007), pp. 81– 82. M. V. Shamolin, “Complete integrability of equations of motion for a spatial pendulum in medium flow taking account of rotational derivatives of moment of its action force,” Izv. Ross Akad. Nauk, Mekhanika Tverdogo Tela, 3, 187–192 (2007). M. V. Shamolin, “Cases of complete integrability in dynamics of a four-dimensional rigid body in a nonconservative force field,” In: Abstracts of Reports of International Conference ‘Analysis and Singularities,’ Devoted to 70th Anniversary of V. I. Arnol’d, August 20–24, 2007, Moscow [in Russian], MIAN, Moscow (2007), pp. 110–112. M. V. Shamolin, “A case of complete integrability in dynamics on a tangent bundle of twodimensional sphere,” Usp. Mat. Nauk, 62, No. 5, 169–170 (2007). M. V. Shamolin, “Cases of complete integrability in dynamics of a rigid body interacting with a medium,” In: Abstracts of Reports of All-Russiann Conference ‘Modern Problems of Contionuous Medium Mechanics’ Devoted to Memory of L. I. Sedov in Connection with His 100th Anniversary, Moscow, November, 12–14, 2007 [in Russian], MIAN, Moscow (2007), pp. 166–167. M. V. Shamolin, “On stability of a certain regime of rigid body motion in a resisting medium,” In: Abstracts of Reports of Scientific Conference ‘Lomonosov Readings-2007,’ Sec. Mechanics, Moscow, Moscow State University, April, 2007 [in Russian], MGU, Moscow (2007), p. 153. M. V. Shamolin, Methods for Analysis of Variable Dissipation Dynamical Systems in Rigid Body Dynamics [in Russian], Ekzamen, Moscow (2007). M. V. Shamolin, Some Problems of Differential and Topological Diagnostics [in Russian], 2nd Corrected and Added Edition, Eksamen, Moscow (2007). M. V. Shamolin, “Three-parameter family of phase portraits in dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 418, No. 1. 46–51 (2008). M. V. Shamolin and S. V. Tsyptsyn, “Analytical and numerical study of trajectories of body motion in a resisting medium,” In: Scientific Report of Institute of Mechanics, Moscow State University [in Russian], No. 4289, Institute of Mechanics, Moscow State University, Moscow (1993). M. V. Shamolin and D. V. Shebarshov, “Projections of Lagrangian tori of a biharmonic oscillator on position space and dynamics of a rigid body interacting with a medium,” In: Modelling and Study of Stability of Systems, Scientific Conference May 19–23, 1997. Abstracts of Reports [in Russian], Kiev (1997), p. 142.

322. M. V. Shamolin, “Structural stable vector fields in rigid body dynamics, In: Proc. of 8th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2005), Lodz, Poland, Dec. 12–15, 2005; Tech. Univ. Lodz., 1, 429–436 (2005). 323. M. V. Shamolin, “Global qualitative analysis of the nonlinear systems on the problem of a body motion in a resisting medium,” In: Fourth Colloquium on the Qualitative Theory of Differential Equations, Bolyai Institute, August 18–21, 1993, Szeged, Hungary (1993), p. 54. 324. M. V. Shamolin, “Relative structural stability on the problem of a body motion in a resisting medium,” In: ICM’94, Abstract of Short Communications, Zurich, 3–11 August, 1994, Zurich, Switzerland (1994), p. 207. 325. M. V. Shamolin, “Structural optimization of the controlled rigid motion in a resisting medium,” In: WCSMO–1, Extended Abstracts. Posters, Goslar, May 28–June 2, 1995, Goslar, Germany (1995), pp. 18–19. 326. M. V. Shamolin, “Qualitative methods to the dynamic model of an interaction of a rigid body with a resisting medium and new two-parametric families of the phase portraits,” In: DynDays’95 (Sixteenth Annual Informal Workshop), Program and Abstracts, Lyon, June 28–July 1, 1995, Lyon, France (1995), p. 185. 327. M. V. Shamolin, “New two-parameter families of the phase patterns on the problem of a body motion in a resisting medium,” In: ICIAM’95, Book of Abstracts, Hamburg, 3–7 July, 1995, Hamburg, Germany (1995), p. 436. 328. M. V. Shamolin, “Poisson-stable and dense orbits in rigid body dynamics,” In: 3rd Experimental Chaos Conference, Advance Program, Edinburg, Scotland, August 21–23, 1995, Edinburg, Scotland (1995), p. 114. 329. M. V. Shamolin, “Qualitative methods in interacting with the medium rigid body dynamics,” In: Abstracts of GAMM Wissenschaftliche Jahrestangung’96, 27–31 May, 1996, Prague, Czech Rep, Karls-Universitat Prag., Prague, (1996), pp. 129–130. 330. M. V. Shamolin, “Relative structural stability and relative structural instability of different degrees in topological dynamics,” In: Abstracts of International Topological Conference Dedicated to P. S. Alexandroff ’s 100th Birthday ‘Topology and Applications,’ Moscow, May 27–31, 1996 [in Russian], Phasys, Moscow (1996), pp. 207–208. 331. M. V. Shamolin, “Topographical Poincare systems in many dimensional spaces,” In: Fifth Colloquium on the Qualitative Theory of Differential Equations, Bolyai Institute, Regional Committee of the Hungarian Academy of Sciences, July 29–August 2, 1996, Szeged, Hungary (1996), p. 45. 332. M. V. Shamolin, “Qualitative methods in interacting with the medium rigid body dynamics,” In: Abstracts of XIXth ICTAM, Kyoto, Japan, August 25–31, 1996, Kyoto, Japan (1996), p. 285. 333. M. V. Shamolin, “Three-dimensional structural optimization of controlled rigid motion in a resisting medium,” In: Proceedings of WCSMO–2, Zakopane, Poland, May 26–30, 1997, Zakopane, Poland (1997), pp. 387–392. 334. M. V. Shamolin, “Classical problem of a three-dimensional motion of a pendulum in a jet flow,” In: 3rd EUROMECH Solid Mechanics Conference, Book of Abstracts, Stockholm, Sweden, August 18–22, 1997, Royal Inst. of Technology, Stockholm, Sweden (1997), p. 204. 335. M. V. Shamolin, “Families of three-dimensional phase portraits in dynamics of a rigid body,” In: EQUADIFF 9, Abstracts, Enlarged Abstracts, Brno, Czech Rep., August 25–29, 1997, Masaryk Univ., Brno, Czech Rep. (1997), p. 76. 336. M. V. Shamolin, “Many-dimensional topographical Poincare systems in rigid body dynamics,” In: Abstracts of GAMM Wissenschaftliche Jahrestangung’98, 6–9 April, 1998, Bremen, Germany, Universitat Bremen, Bremen (1998), p. 128. 337. M. V. Shamolin, Shebarshov D. V. Lagrange tori and equation of Hamilton–Jacobi,” In: Book of Abstracts of Conference PDE Prague’98 (Praha, August 10–16, 1998; Partial Differential Equations: Theory and Numerical Solutions), Charles University, Praha, Czech Rep. (1998), p. 88. 775

338. M. V. Shamolin, “New two-parameter families of the phase portraits in three-dimensional rigid body dynamics,” In: Abstracts of International Conference Dedicated to L. S. Pontryagin’s 90th Birthday ‘Differential Equations,’ Moscow, 31.08.–6.09, 1998 [in Russian], MGU, Moscow (1998), pp. 97–99. 339. M. V. Shamolin, “Lyapunov functions method and many-dimensional topographical systems of Poincare in rigid body dynamics,” In: Abstracts of Reports of IV Crimeanian International Mathematical School ‘Lyapunov Function Method and Its Application,’ (LFM–1998), Crimea, Alushta, September 5–12, 1998 [in Russian], Simpheropol’ (1998), p. 80. 340. M. V. Shamolin, “Some classical problems in a three-dimensional dynamics of a rigid body interacting with a medium,” In: Proc. of ICTACEM’98, Kharagpur, India, Dec. 1–5, 1998, Aerospace Engineering Dep., Indian Inst. of Technology, Kharagpur, India (1998), p. 11 (CDRome, Printed at: Printek Point, Technology Market, KGP–2). 341. M. V. Shamolin, “Integrability in terms of transcendental functions in rigid body dynamics,” In: Book of Abstr. of GAMM Annual Meeting, April 12–16, 1999, Universite de Metz, Metz, France (1999), p. 144. 342. M. V. Shamolin, “Mathematical modelling of interaction of a rigid body with a medium and new cases of integrability,” In: CD-Proc. of ECCOMAS 2000, Barcelona, Spane, September 11–14, 2000, Barcelona (2000), p. 11. 343. M. V. Shamolin, “Methods of analysis of dynamics of a rigid body interacting with a medium,” In: Book of Abstr. of Annual Scient. Conf. GAMM 2000 at the Univ. of Gottingen, April 2–7, 2000, Univ. of Gottingen, Gottingen (2000), p. 144. 344. M. V. Shamolin, “Integrability and nonintegrability in terms of transcendental functions,” In: CD–Abs. of 3rd ECM (Poster sessions), Barcelona, Spain, June 10–14, 2000 (poster No. 36, without pages), Barcelona (2000). 345. M. V. Shamolin, “About interaction of a rigid body with a resisting medium under an assumption of a jet flow,” In: Book of Abstr. II (General sessions) of 4th EUROMECH Solid Mech. Conf., Metz, France (June 26–30, 2000), Universite de Metz, Metz, France (2000), p. 703. 346. M. V. Shamolin, “New families of many-dimensional phase portraits in dynamics of a rigid body interacting with a medium,” In: CD–Proc. of 16th IMACS World Cong. 2000, Lausanne, Switzerland, August 21–25, EPFL (2000), p. 3. 347. M. V. Shamolin, “Comparison of some cases of integrability in dynamics of a rigid body interacting with a medium,” In: Book of Abstr. of Annual Scient. Conf. GAMM 2001, ETH Zurich, February 12–15, 2001, ETH Zurich (2001), p. 132. 348. M. V. Shamolin, “Pattern recognition in the model of the interaction of a rigid body with a resisting medium,” In: Col. of Abstr. of First SIAM–EMS Conf. ‘Applied Mathematics in Our Changing World,’ Berlin, Germany, Sept. 2–6, 2001, Springer, Birkhauser (2001), p. 66. 349. M. V. Shamolin, “Some questions of the qualitative theory of ordinary differential equations and dynamics of a rigid body interacting with a medium,” J. Math. Sci., 110, No. 2, 2526–2555 (2002). 350. M. V. Shamolin, “Dynamical systems with the variable dissipation in 3D-dynamics of a rigid body interacting with a medium,” In: Book of Abstr. of 4th ENOC, Moscow, Russia, August 19–23, 2002 [in Russian], Inst. Probl. Mech. Russ. Acad. Sci., Moscow (2002), p. 109. 351. M. V. Shamolin, “Methods of analysis of dynamics of a 2D-, 3D-, or 4D-rigid body with a medium,” In: Abst. Short Commun. Post. Sess. of ICM’2002, Beijing, 2002, August 20–28, Higher Education Press, Beijing, China (2002), p. 268. 352. M. V. Shamolin, “New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium,” J. Math. Sci., 114, No. 1, 919–975 (2003). 353. M. V. Shamolin, “Integrability and nonintegrability in terms of transcendental functions,” In: Book of Abstr. of Annual Scient. Conf. GAMM 2003, Abano TermePadua, Italy, March 24–28, 2003, Univ. of Padua, Italy (2003), p. 77. 776

354. M. V. Shamolin, “Global structural stability in dynamics of a rigid body interacting with a medium,” In: 5th ICIAM, Sydney, Australia, July 7–11, 2003, Univ. of Technology, Sydney (2003), p. 306. 355. M. V. Shamolin, “Classes of variable dissipation systems with nonzero mean in the dynamics of a rigid body,” J. Math. Sci., 122, No. 1, 2841–2915 (2004). 356. M. V. Shamolin, “Some cases of integrability in dynamics of a rigid body interacting with a resisting medium,” In: Abstracts of Reports of International Conference in Differential Equations and Dynamical Systems, Suzdal’, July 05–10, 2004 [in Russian], Vladimir, Vladimir State University (2004), pp. 296–298. 357. M. V. Shamolin, “Mathematical model of interaction of a rigid body with a resisting medium in a jet flow,” In: Abstr. Part 1, 76 Annual Sci. Conf. (GAMM), Luxembourg, March 28–April 1, 2005, Univ. du Luxembourg, Luxembourg (2005) pp. 94–95. 358. M. V. Shamolin, “Some cases of integrability in 3D dynamics of a rigid body interacting with a medium,” In: Book of Abstr. IMA Int. Conf. ‘Recent Advances in Nonlinear Mechanics,’ Aberdeen, Scotland, August 30–September 1, 2005, IMA, Aberdeen (2005), p. 112. 359. M. V. Shamolin, “Almost conservative systems in dynamics of a rigid body,” In: Book of Abstr., 77th Annual Meeting of GAMM, March 27–31, 2006, Technische Univ. Berlin, Technische Univ., Berlin (2006), p. 74. 360. M. V. Shamolin, “4D-rigid body and some cases of integrability,” In: Abstracts of ICIAM07, Zurich, Switzerland, June 16–20, 2007, ETH Zurich (2007), p. 311. 361. M. V. Shamolin, “The cases of complete integrability in dynamics of a rigid body interacting with a medium,” In: Book of Abstr. of Int. Conf. on the Occasion of the 150th Birthday of A. M. Lyapunov (June 24–30, 2007, Kharkiv, Ukraine) [in Russian], Verkin Inst. Low Temper. Physics Engineer. NASU, Kharkiv (2007), pp. 147–148. 362. M. V. Shamolin, “On the problem of a symmetric body motion in a resisting medium,” In: Book of Abst. of EMAC–2007 (July 1–4, 2007, Hobart, Australia), Univ. Tasmania, Hobart, Australia (2007), p. 25. 363. M. V. Shamolin, “The cases of integrability in 2D-, 3D-, and 4D-rigid body dynamics,” In: Abstr. of Short Commun. and Post. of Int. Conf. ‘Dynamical Methods and Mathematical Modelling,’ Valladolid, Spane (Sept. 18–22, 2007), ETSII, Valladolid (2007), p. 31. 364. M. V. Shamolin, “The cases of integrability in terms of transcendental functions in dynamics of a rigid body interacting with a medium,” In: Proc. of 9th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2007), Lodz, Poland, Dec. 17–20, 2007, Vol. 1, Tech. Univ. Lodz (2007), pp. 415–422. 365. O. P. Shorygin and N. A. Shul’man, “Entrance of a disk to water with angle of arttack,” Uch. Zap. TsAGI, 8, No. 1, 12–21 (1978). 366. J. L. Singh, Classical Dynamics [Russian translation], Fizmatgiz, Moscow (1963). 367. S. Smale, “Rough systems are not dense,” In: A Collection of Translations. Mathematics [in Russian], 11, No. 4, 107–112 (1967). 368. S. Smale, “Differentiable dynamical systems,” Usp. Mat. Nauk, 25, No. 1, 113–185 (1970). 369. V. M. Starzhinskii, Applied Methods of Nonlinear Oscillations [in Russian], Nauka, Moscow (1977). 370. V. A. Steklov, On Rigid Body Motion in a Fluid [in Russian], Khar’kov (1893). 371. V. V. Stepanov, A Course of Differential Equations [in Russian], Fizmatgiz, Moscow (1959). 372. E. I. Suvorova and M. V. Shamolin, “Poincaré topographical systems and comparison systems of higher orders,” In: Mathematical Conference ‘Modern Methods of Function Theory and Related Problems,’ Voronezh, January 26–February 2, 2003 [in Russian], Voronezh State University, Voronezh (2003), pp. 251–252. 373. G. K. Suslov, Theoretical Mechanics [in Russian], Gostekhizdat, Moscow (1946).

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