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DOI 10.1007/s10958-015-2220-0 Journal of Mathematical Sciences, Vol. 204, No. 6, February, 2015

CLASSIFICATION OF INTEGRABLE CASES IN THE DYNAMICS OF A FOUR-DIMENSIONAL RIGID BODY IN A NONCONSERVATIVE FIELD IN THE PRESENCE OF A TRACKING FORCE M. V. Shamolin UDC 517; 531.01

Abstract. This pap er is a survey of integrable cases in the dynamics of a four-dimensional rigid b ody under the action of a nonconservative force field. We review b oth new results and results obtained earlier. The problems examined are describ ed by dynamical systems with so-called variable dissipation with zero mean. The problem of a search for complete sets of transcendental first integrals of systems with dissipation is quite current; a large numb er of works are devoted to it. We introduce a new class of dynamical systems that have a p eriodic coordinate. Due to the existence of a nontrivial symmetry group of such systems, we can prove that these systems p ossess variable dissipation with zero mean, which means that on the average for a p eriod with resp ect to the p eriodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either energy pumping or dissipation can occur. Based on the results obtained, we analyze dynamical systems that app ear in the dynamics of a four-dimensional rigid b ody and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions that can b e expressed through a finite combination of elementary functions.

CONTENTS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1. Integrability in Elementary Functions of Some Classes of Nonconservative Systems 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Dynamical Systems with Variable Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . 3. Systems with Symmetries and Variable Dissipation with Zero Mean . . . . . . . . . . . . 4. Systems in the Plane and on a Two-Dimensional Cylinder . . . . . . . . . . . . . . . . . . 5. Systems of the Tangent Bundle of the Two-Dimensional Sphere . . . . . . . . . . . . . . . 6. Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2. Cases of Integrability Corresp onding to the Motion of a Rigid Body in Four-Dimensional Space, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. General Problem of Motion Under a Tracing Force . . . . . . . . . . . . . . . . . . . . . . 9. Case Where the Moment of a Nonconservative Force Is Indep endent of the Angular Velocity 10. Case Where the Moment of a Nonconservative Force Dep ends on the Angular Velocity . . Chapter 3. Cases of Integrability Corresp onding to the Motion of a Rigid Body in Four-Dimensional Space, I I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. General Problem of Motion Under a Tracing Force . . . . . . . . . . . . . . . . . . . . . . 12. Case Where the Moment of a Nonconservative Force Is Indep endent of the Angular Velocity 13. Case Where the Moment of a Nonconservative Force Dep ends on the Angular Velocity . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 810 811 812 814 816 818 820 823 823 825 830 836 843 844 849 854 862

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 88, Geometry and Mechanics, 2013.

808

1072­3374/15/2046­0808 c 2015 Springer Science+Business Media New York


Intro duction This pap er is a survey of integrable cases in the dynamics of a four-dimensional rigid b ody under the action of a nonconservative force field. We review b oth new results and results obtained earlier. The problems examined are describ ed by dynamical systems with so-called variable dissipation with zero mean. We study nonconservative systems for which the usual methods of studying Hamiltonian systems are not applicable. Thus, for such systems, we must "directly" integrate the main equation of dynamics (see [45, 47­52, 58, 60, 64, 65, 70, 79, 127]. We generalize previously known cases and obtain new cases of complete integrability in transcendental functions of the equation of dynamics of a four-dimensional rigid b ody in a nonconservative force field. Of course, in the general case, the construction of a theory of integration of nonconservative systems (even of low dimension) is a quite difficult task. In a numb er of cases where the systems considered have additional symmetries, we succeed in finding first integrals expressed through finite combinations of elementary functions (see [5, 6, 12, 17, 18, 21, 28, 29, 32­35, 37, 42, 91, 92, 146, 148, 151]. We obtain a series of complete integrable nonconservative dynamical systems with nontrivial symmetries. Moreover, in almost all cases, all first integrals are expressed through finite combinations of elementary functions; these first integrals are transcendental functions of their variables. In this case, transcendence is understood in the sense of complex analysis, when the analytic continuation of a function into the complex plane has essentially singular p oints. This fact is caused by the existence of attracting and rep elling limit sets in the system (for example, attracting and rep elling focuses). We discover new integrable cases of the motion of a rigid b ody, including the classical problem of the motion of a multi-dimensional spherical p endulum in a running flow of a medium. Chapter 1 is devoted to general asp ects of the integrability of dynamical systems with variable dissipation. First, we prop ose a descriptive characteristic of such systems. The term "variable dissipation" refers to the p ossibility of alternation of sign rather than to the value of the dissipation coefficient (therefore, it is more reasonable to use the term "sign-alternating"). Later, we define systems with variable dissipation with zero (nonzero) mean based on the divergence of the vector field of the system, which characterizes the change of the phase volume in the phase space of the system considered (see [91, 95, 103, 107, 109, 110, 112, 118, 120]). We introduce a class of autonomous dynamical systems with one p eriodic phase coordinate p ossessing certain symmetries that are typical for p endulum-typ e systems. We show that this class of systems can b e naturally emb edded in the class of systems with variable dissipation with zero mean, i.e., on the average for the p eriod with resp ect to the p eriodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either energy pumping or dissipation can occur, but they balance each other in a certain sense. We present some examples of p endulum-typ e systems on lower-dimension manifolds from the dynamics of a rigid b ody in a nonconservative field. For multi-parametric third-order systems, we present sufficient conditions of existence of first integrals that are expressed through finite combinations of elementary functions. We deal with three prop erties that seem, at first glance, indep endent: (1) a class of systems with symmetries sp ecified ab ove; (2) the fact that this class consists of systems with variable dissipation with zero mean (with resp ect to the existing p eriodic variable), which allows us to consider them as "almost" conservative systems; (3) in certain (although lower-dimensional) cases, these systems have a complete set of first integrals, which, in general, are transcendental (in the sense of complex analysis). 809


In Chap. 2, we recall general asp ects of the dynamics of a free multi-dimensional rigid b ody: the notion of the tensor of angular velocity of the b ody, the joint dynamical equations of motion on the direct product Rn â so(n), and the Euler and Rivals formulas in the multi-dimensional case. We also consider the tensor of inertia of a four-dimensional (4D) rigid b ody. In this work, we study two p ossible cases in which there exist two relations b etween the principal moments of inertia: (i) there are three equal principal moments of inertia (I2 = I3 = I4 ); (ii) there are two pairs of equal principal moments of inertia (I1 = I2 and I3 = I4 ). In Chaps. 2 and 3, we systematize results on the study of equations of motion of a four-dimensional (4D) rigid b ody in a nonconservative force field for the case (i). The form of these equations is taken from the dynamics of realistic rigid b odies of lesser dimension that interact with a resisting medium by laws of jet flow when the b ody is influenced by a nonconservative tracing force. Under the action of this force, the following two cases are p ossible. In the first case, the velocity of some characteristic p oint of the b ody remains constant, which means that the system p ossesses a nonintegrable servo constraint (see Chap. 2). In the second case, the b ody is sub jected to a nonconservative tracing force such that throughout the motion the center of mass of the b ody moves rectilinearly and uniformly; this means that in the system there exists a nonconservative couple of forces (see Chap. 3). See also [19­21, 27, 44, 54, 55, 57, 59, 61­63, 66­68, 72­78, 80, 82­87, 90­94, 96­107, 111­117, 119­125, 128­138, 140­144, 149]. Moreover, in Chap. 2, b esides the four existing analytic invariant relations (a nonintegrable connection and three integrals that show that the comp onents of the tensor of angular sp eed vanish), we obtain four additional transcendental first integrals expressed in terms of finite combinations of elementary functions. In Chap. 3, we find additional transcendental first integrals b esides the four known analytic first integrals (the squared velocity of the center of mass and the three integrals that show that the comp onents of the tensor of angular sp eed vanish). The results relate to the case where all interaction of the medium with the b ody part is concentrated on a part of the surface of the b ody that has the form of a three-dimensional disk, while the action of the force is concentrated in the direction p erp endicular to this disk. These results are systematized and are preserved in invariant form. Moreover, we introduce an extra dep endence of the moment of the nonconservative force on the angular velocity. This dep endence can b e further extended to cases of motion in spaces of higher dimension. Many results of this pap er were regularly presented at scientific seminars, including the Trofimov seminar "Current problems of geometry and mechanics" (see [22]) under the sup ervision of D. V. Georgievskii and M. V. Shamolin [1, 2, 23­26].

Chapter 1 INTEGRABILITY IN ELEMENTARY FUNCTIONS OF SOME CLASSES OF NONCONSERVATIVE SYSTEMS
We study nonconservative systems for which the usual methods of studying Hamiltonian systems are not applicable. Thus, for such systems, we must "directly" integrate the main equation of dynamics. We recall known facts in a more universal form and also present some new cases of complete integrability in transcendental functions in the dynamics of a 4D-rigid b ody in a nonconservative field. The results of the present pap er develop previous studies, including an applied problem from the dynamics of a rigid b ody (see [42, 44, 46­48, 149]), for which complete lists of transcendental first integrals that can b e expressed through finite combinations of elementary functions were obtained. 810


Later, this allows us to p erform a complete analysis of all phase tra jectories and to sp ecify their rough prop erties that are preserved for systems of a more general form. The complete integrability of such systems is related to hidden symmetries. As is well known, the notion of integrability is, generally sp eaking, quite vague. We must always take into account in what sense this notion is understood (what criterion allows one to judge whether tra jectories of a dynamical system are simple in one sense or another), in what functional class the first integrals are searched, etc. (see [6, 33, 34, 44, 54, 57]). In this pap er, we consider first integrals that b elong to the functional class consisting of transcendental elementary functions. Here the term "transcendental" is meant in the sense of complex analysis, i.e., a transcendental function is a function that p ossesses essential singularities after an analytic continuation in the complex plane (see [12, 44]).

1.

Preliminaries

The construction of a theory of integration of nonconservative systems (even lower-dimensional) is a difficult problem. However, in some cases where the systems b eing studied p ossess additional symmetries, one can find first integrals in the form of finite combinations of elementary functions (see [91]). The present pap er is a development of the planar problem on the motion of a rigid b ody in a resisting medium in which the domain of the contact b etween the b ody and the medium is a planar part of the exterior surface of the b ody. The force field in this problem is constructed by accounting for the action of the medium on the b ody in the quasi-stationary jet or separated flow. It turns out that the study of such motions can b e reduced to systems with dissipation of energy ((purely) dissipative systems or systems in dissipative fields) or to systems with energy pumping (so-called systems with antidissipation or systems with accelerating forces). Note that similar problems have earlier app eared in applied aerodynamics (see [14, 15]). The problems that were considered earlier stimulated the development of qualitative tools that substantially supplement the qualitative theory of nonconservative systems with dissipation of any sign (see [91]). Nonlinear effects in problems of planar and spatial dynamics of a rigid b ody were examined by qualitative methods. We justify the need to introduce the notions of relative roughness and relative nonroughness of different orders (see [4, 28, 29, 39, 52, 60, 64, 88, 91, 149]. In the present work, the following results are obtained. (1) We develop methods of qualitative analysis of dissipative and antidissipative systems, which allows us to obtain bifurcation conditions for the app earance of stable and unstable self-oscillations and conditions of the absence of singular tra jectories. We succeed in extending the study of planar top ographical Poincar´ systems and comparison systems to higher dimensions. We obe tain sufficient Poisson-stability conditions (everywhere density near itself ) of some classes of nonclosed tra jectories of dynamical systems (see [91]); (2) in 2D- and 3D-dynamics of a rigid b ody, we obtain complete lists of first integrals of dissipative and antidissipative systems that are transcendental (in the sense of the classification of their singularities) functions, which, in some cases, can b e expressed through elementary functions. We introduce the notions of relative roughness and relative nonroughness of different orders for integrated systems (see [4, 28, 29, 39, 52, 60, 64, 88, 91, 149]; (3) we obtain multi-parameter families of top ologically nonequivalent phase p ortraits that app ear in purely dissipative systems (i.e., systems with variable dissipation with nonzero (p ositive) mean). Almost all p ortraits of such families are rough (see [91]); 811


(4) we detect new qualitative analogies b etween the motion of a free b ody in a resisting medium and the motion of a fixed b ody in a flow of a running medium.

2.

Dynamical Systems with Variable Dissipation

2.1. Descriptive characteristics of dynamical systems with variable dissipation. As the initial modeling of the action of a medium on a rigid b ody, we used exp erimental information on the prop erties of jet flow. Naturally it b ecame necessary to study the class of dynamical systems that p ossess the prop erty of (relative) roughness (relative structural stability). Therefore, it is natural to introduce these notions for such systems. Many of the systems considered are rough in the sense of Andronov and Pontryagin (see [4, 28, 29, 39, 52, 60, 64, 88, 91, 149]. After some transformations (for example, in 2D-dynamics), the dynamical part of the general system of the equations of plane-parallel motion can b e reduced to a p endulum system of second order containing a linear nonconservative (sign-alternating dissipative) force with a coefficient, which can change sign for different values of the p eriodic phase coordinate of the system. Thus, in this case, we sp eak of systems with so-called variable dissipation, where the term "variable" refers not only to the value of the dissipation coefficient but to its sign (and so the term "signalternating" is more adequate). On the average by a p eriod (with resp ect to the p eriodic coordinate), dissipation can b e p ositive ("purely" dissipative systems), negative (systems with accelerating forces), or zero (but does not vanish identically). In the last case, we sp eak of systems with variable dissipation with zero mean (these systems can b e associated with "almost" conservative systems). As was noted ab ove, we obtain imp ortant mechanical analogies in comparing the qualitative properties of a free b ody and the equilibrium of a p endulum in a flow of a medium. Such analogies have a deep sense since they allow one to transfer prop erties of a nonlinear dynamical system for a p endulum to dynamical systems for a free b ody. Both systems b elong to the class of so-called p endulum dynamical systems with variable dissipation with zero mean. Under additional conditions, the equivalence describ ed ab ove can b e extended to the case of spatial motion, which allows one to sp eak of a general character of symmetries of systems with variable dissipation with zero mean in plane-parallel and spatial motions (for planar and spatial versions of a p endulum in a flow of a medium, see also [91]). Subsequently, we present some classes of nonlinear systems of the second, third, and higher orders that are integrable in the class of transcendental (in the sense of the theory of functions of complex variables) elementary functions, for example, five-parameter dynamical systems including the ma jority of systems examined earlier in the dynamics of a low-dimensional (2D and 3D) rigid b ody interacting with a medium: = a sin + b + 1 sin5 + 2 sin4 + 3 2 sin3 + 4 3 sin2 + 5 4 sin , = c sin cos + d cos + 1 sin4 cos + 2 2 sin3 cos + 3 3 sin2 cos + + 4 4 sin cos + 5 5 cos . Purely dissipative dynamical systems (and also (purely) antidissipative systems), which, in our case, can b elong to the class of systems with variable dissipation with nonzero mean, are, as a rule, structurally stable ((absolutely) rough), whereas systems with variable dissipation with zero mean (which usually p ossess additional symmetries) are either structurally unstable (nonrough) or only relatively structurally stable (relatively rough). However, the proof of the last assertion in the general case is a difficult problem. 812


For example, the dynamical system of the form = + sin , = - sin cos (2.1)

is relatively structurally stable (relatively rough) and is top ologically equivalent to the system describing a fixed p endulum in a running flow of a medium (see [91]). One can obtain its first integral, which is a transcendental (in the sense of the theory of functions of a complex variable, as a function whose analytical continuation in the complex plane has essential singularities) function of phase variables that can b e expressed through a finite combination of elementary functions (see [91]). The phase cylinder R2 {, } of quasi-velocictes of the system considered has an interesting top ological structure of a splitting into tra jectories. Although the dynamical system considered is not conservative, in the rotational domain (and only in this domain) of its phase plane R2 {, }, it admits the preservation of invariant measure with variable density. This prop erty characterizes this system as a system with variable dissipation with zero mean (see [91]). 2.2. A definition of a system with variable dissipation with zero mean. We study systems of ordinary differential equations that have a p eriodic phase coordinate. Such systems p ossess symmetries under which their average phase volume with resp ect to the p eriodic coordinate is preserved. For example, the following p endulum system with smooth and p eriodic (of p eriod T ) with resp ect to right-hand side V(, ) of the form = - + f (), f ( + T ) = f (), = g(), g ( + T ) = g (), preserves its phase area on the phase cylinder within the p eriod T :
T T T

(2.2)

div V(, )d =
0 0

(- + f ()) + g () d =

f ()d = 0.
0

(2.3)

This system is equivalent to the equation of a p endulum - f () + g () = 0, ¨ (2.4)

in which the integral of the coefficient f () of the dissipative term over the p eriod is equal to zero. We see that this system has symmetries under which it b ecomes a system with variable dissipation with zero mean in the sense of the following definition (see [91]). Definition 2.1. Consider a smooth autonomous system of order on the cylinder Rn {x} â S1 { mod 2 }, where is a p eriodic divergence of the right-hand side V(x, ) (which, in general, is a does not vanish identically) of this system is denoted by div V(x, with variable dissipation with zero (resp ectively, nonzero) mean if
T

(n + 1) in the normal form defined coordinate of p eriod T > 0. The function of all phase variables and ). This system is called a system the function (2.5)

div V(x, )d
0

vanishes (resp ectively, does not vanish) identically. In some cases (for example, when at some p oints of the circle S1 { mod 2 } singularities app ear), this integral is understood in the sense of principal value. 813


We note that it is quite difficult to give a general definition of a system with variable dissipation with zero (nonzero) mean. The definition presented ab ove is based on the notion of divergence (as is well known, the divergence of the right-hand side of a system in the normal form characterizes the change of the phase volume in the phase space of the given system). 3. Systems with Symmetries and Variable Dissipation with Zero Mean

Consider a system of the following form (the dot denotes the derivative with resp ect to time): = f (, sin , cos ), k = fk (, sin , cos ), defined on the set S1 { mod 2 }\ K â Rn { }, = (1 ,... ,n ), (3.2) where sufficiently smooth functions f (u1 ,u2 ,u3 ), = , 1,... ,n, of three variables u1 , u2 , u3 are such that f (-u1 , -u2 ,u3 ) = -f (u1 ,u2 ,u3 ), f (u1 ,u2 , -u3 ) = f (u1 ,u2 ,u3 ), fk (u1 ,u2 , -u3 ) = -fk (u1 ,u2 ,u3 ); moreover, the functions fk (u1 ,u2 ,u3 ) are defined for u3 = 0 for any k = 1,... ,n. The set K is either empty or consists of a finite numb er of p oints of the circle S1 { mod 2 }. The last two variables u2 , u3 in the functions f (u1 ,u2 ,u3 ) dep end on the same parameter , but we assume that these variables b elong to different groups for the following reason. First, they cannot b e uniquely expressed through one another on their entire domain and, second, u2 is an odd function of whereas u3 is an even function, which affects the symmetries of system (3.1). We establish a corresp ondence b etween system (3.1) and the following nonautonomous system: fk (, sin , cos ) dk = , d f (, sin , cos ) k = 1,... ,n. (3.4) (3.3) k = 1,... ,n, (3.1)

By the substitution = sin , it can b e reduced to the form fk (, , k ( )) dk = , k = 1,... ,n, d f (, , ( )) (- ) = ( ), = , 1,... ,n. (3.5)

The last system, in particular, can have an algebraic right-hand side (i.e., it can b e the ratio of two p olynomials), which simplifies the search for its first integrals in explicit form. The following theorem states that the class of systems (3.1) is a sub class of the class of dynamical systems with variable dissipation with zero mean. Note that, in general, the converse is invalid. Theorem 3.1. Systems of the form (3.1) are dynamical systems with variable dissipation with zero mean. Proof. The proof of this theorem is based on certain symmetries (3.3) of system (3.1) listed ab ove and the p eriodicity of the right-hand side of the system with resp ect to . Indeed, the divergence of the vector field of system (3.1) equals f (, sin , cos ) f (, sin , cos ) cos - sin + u2 u3 The following integral of the first two terms in (3.6) vanishes: 814
n k =1

fk (, sin , cos ) . u1

(3.6)


2

0

f (, sin , cos ) f (, sin , cos ) d sin + d cos u2 u3
2

=
0

f (, sin , cos ) d = h ( ) 0, (3.7)

since the function f (, sin , cos ) is p eriodic with resp ect to . Further, by the third equation in (3.3), for any k = 1,... ,n we have gk (, sin ) fk (, sin , cos ) = cos · , u1 u1 where the function gk (u1 ,u2 ) is sufficiently smooth for any k = 1,... ,n. Then the integral over the p eriod 2 of the right-hand side of Eq. (3.8) equals
2

(3.8)

0

gk (, sin ) d sin = hk ( ) 0 u1

(3.9)

for any k = 1,... ,n. From Eqs. (3.7) and (3.9) we obtain Theorem 3.1. The converse assertion is invalid: there exist dynamical systems on the two-dimensional cylinder that are systems with variable dissipation with zero mean, but do not p ossess the symmetries listed ab ove. In this pap er, we basically consider the case where the functions f (, , k ( )) ( = , 1,... ,n) are p olynomials of and . Example 3.1. We consider p endulum systems on the two-dimensional cylinder S1 { mod 2 } â R1 { } with parameter b > 0, which app ear in the dynamics of a rigid b ody (see [91]): = - + b sin , = sin cos , and = - + b sin cos2 + b 2 sin , = sin cos - b sin2 cos + b 3 cos . (3.10)

(3.11)

We establish a corresp ondence b etween these systems in the variables (, ) and the equations with algebraic right-hand sides d = , (3.12) d - + b and + b 2 - 2 d = (3.13) d - + b + b [ 2 - 2 ] of the form (3.5), resp ectively. These systems are dynamical systems with variable dissipation with zero mean, which can b e easily verified. Indeed, the divergences of their right-hand sides are equal to b cos and b cos 4 2 +cos2 - 3sin2 , resp ectively; they b elong to the class of systems (3.1). Moreover, each of them p ossesses a first integral, which is a transcendental (in the sense of the theory of functions of complex variables) function that can b e expressed through a finite combination of elementary functions. 815


We present another imp ortant example of a higher-order system that p ossesses the prop erties listed ab ove. Example 3.2. Consider the following system with a parameter b, which is defined in the threedimensional domain S1 { mod 2 }\ { = 0, = }â R2 {z1 ,z2 } S2 = -z2 + b sin , (3.15) cos z1 = z1 z2 . sin This system describ es the motion of a rigid b ody in a resistive medium. We establish a corresp ondence b etween this system and the following nonautonomous system with algebraic right-hand side ( = sin ): 2 - z1 / dz2 = , d -z2 + b (3.16) dz1 z1 z2 / = . d -z2 + b We see that system (3.15) is a system with variable dissipation with zero mean. To obtain the full corresp ondence with the definition, we introduce the new phase variable
z1 = ln |z1 |. 2 z2 = sin cos - z1

(3.14)

(this system is separated from a system on the tangent bundle T cos , sin

of the two-dimensional sphere S2 ):

(3.17)

The divergence of the right-hand side of system (3.15) in the Cartesian coordinates , z1 , z2 is equal to b cos . Taking into account (3.14), we have (in the sense of principal value) - 0 2 - 0

lim

b cos + lim


b cos = 0.
+

(3.18)

Moreover, this system p ossesses two first integrals (i.e., a complete set) that are transcendental functions, which can b e expressed through a finite combination of elementary functions. This b ecomes p ossible after establishing a corresp ondence b etween it and the system (nonautonomous, generally sp eaking) of equations with an algebraic (p olynomial) right-hand side (3.16). Systems (3.10), (3.11), and (3.15) b elong to the class of systems (3.1), p ossess variable dissipation with zero mean, and have a complete set of transcendental first integrals that can b e expressed through a finite combination of elementary functions. So, to find the first integrals of the systems considered, it is convenient to reduce systems of the form (3.1) to systems with p olynomial right-hand sides (3.5), which allow one to p erform integration in terms of elementary functions of the initial system. Thus, we find sufficient conditions for the integrability in elementary functions of systems with p olynomial right-hand sides and examine systems of the most general form. 4. Systems in the Plane and on a Two-Dimensional Cylinder

Earlier, the author proved a series of assertions regarding many-parameter systems of ordinary differential equations with algebraic right-hand side (see, e.g., [91]). We recall some of these assertions. 816


Prop osition 4.1. A seven-parameter family of systems of equations in the plane R2 {x, y } x = a1 x + b1 y + 1 x3 + 2 x2 y + 3 xy 2 , y = c1 x + d1 y + 1 x2 y + 2 xy 2 + 3 y 3 , (4.1)

possesses a first integral (in general, transcendental ) that can be expressed through elementary functions. Corollary 4.1. For any parameters a1 , b1 , c1 , d1 , 1 , 2 , and 3 , the system = a1 sin + b1 + 1 sin3 + 2 sin2 + 3 2 sin , = c1 sin cos + d1 cos + 1 sin2 cos + 2 2 sin cos + 3 3 cos (4.2)

on the two-dimensional cylinder {(, ) R2 : mod 2 } possesses a first integral (in general, transcendental ) that can be expressed through elementary functions. In particular, systems (3.10) and (3.11) can b e obtained from this system if a1 = b, and a1 = b, b1 = -1, c1 = 1, d1 = -b, 1 = -b, 2 = 0, 3 = b, resp ectively. The ab ove arguments can b e easily generalized. We consider the p ossibility of complete integration (in elementary functions) of systems of a more general form: the nonlinearity is characterized by an arbitrary homogeneous form of odd degree 2n - 1. In this case, we have the following assertion, which is more general than Prop osition 4.1. Prop osition 4.2. The (2n +3)-parameter family of systems of equations x = a1 x + b1 y + 1 x y = c1 x + d1 y + 1 x
2n-1 2n-2 2n-2

b1 = -1,

c1 = 1,

d1 = 1 = 2 = 3 = 0

+ 2 x

y + ··· + 2n

-2

x2 y

2n-3

+ 2n

-1

xy y

2n-2

,

y + 2 x

2n-3 2

y + ··· + 2n

-2

xy

2n-2

+ 2n

-1

2n-1

(4.3)

in the plane R2 {x, y } possesses a first integral (in general, transcendental ), which can be expressed through elementary functions. Indeed, the family of Eqs. (4.3) dep ends on 2n - 1 + 4 indep endent parameters since the total nonlinearity of an odd degree is characterized by 4n parameters sub ject to 2n + 1 conditions (the other four parameters are contained in the linear part). Corollary 4.2. For any parameters a1 , b1 , c1 , d1 , and 1 ,... ,2n = a sin + b + 1 sin
2n-1 2n-2

+ 2 sin

+ ··· + 2n

-1



-1 , 2n-2

the systems sin ,
-1

cos (4.4) 2 : mo d 2 } possesses a transcendental first integral, on the two-dimensional cylinder {(, ) R which can be expressed through elementary functions. Systems (3.10), (3.11), and (3.15) are relatively rough (see [91]), but if we violate the symmetries (3.3) introduced for systems of general form (3.1) (for example, by introducing additional terms in their right-hand sides), then the numb er of top ologically distinct phase p ortraits can substantially change. In [91], we obtained a multi-parametric family of phase p ortraits of a system with variable dissipation with nonzero mean (whose typical p ortraits are (absolutely) rough), which is a p erturbation of a dynamical system with variable dissipation with zero mean of the form (3.11). This family (as families 817

= c sin cos + d cos + 1 sin2n

-2

cos + 2 2 sin

2n-3

cos + ··· + 2n



2n-1


obtained earlier, see [91]) contains an infinite numb er of top ologically nonequivalent phase p ortraits on a two-dimensional phase cylinder. 5. Systems of the Tangent Bundle of the Two-Dimensional Sphere

On the tangent bundle T S2 of the two-dimensional sphere S2 {, }, we consider the following dynamical system: ¨ sin = 0, + b cos +sin cos - 2 cos (5.1) 1+cos2 ¨ = 0. + b cos + sin cos This system describ es a spherical p endulum in a flow of a running medium (see [91]). Moreover, the system p ossesses the conservative moment sin cos and the force moment, which linearly dep ends of the velocity with a variable coefficient: b cos . (5.3) (5.2)

Other coefficients in the equations are the connection coefficients, namely,


=-

sin , cos





=

1+ cos2 . sin cos

(5.4)

In fact, system (5.1) has order 3 since the variable is cyclic and the system contains only the variable . Prop osition 5.1. The equation =0 (5.5)

defines a family of integral planes for system (5.1). Moreover, Eq. (5.5) reduces system (5.1) to an equation that describes a cylindrical pendulum in a flow of a running medium (see [91]). Prop osition 5.2. System (5.1) is equivalent to the fol lowing system: = -z2 + b sin ,
2 z2 = sin cos - z1

cos , sin

z1 = z1 z2

cos , sin cos = z1 sin

(5.6)

on the tangent bund le T S2 {z1 ,z2 , , } of the two-dimensional sphere S2 {, }. Moreover, the first three equations of system (5.6) form a closed system of third order and coincide with system (3.15) (if we set = ). The fourth equation of system (5.6) has been separated due to the cyclicity of the variable . 818


Example 5.1. We examine a system of the form (3.15) , which can b e reduced to (3.16), and the following system, which app ears in the spatial (3D) dynamics of a rigid b ody interacting with a medium:
2 2 = -z2 + b z1 + z2 sin + b sin cos2 , 2 2 2 z2 = sin cos + bz2 z1 + z2 cos - bz2 sin2 cos - z1 2 2 z1 = bz1 z1 + z2 cos - bz1 sin2 cos + z1 z2

cos , sin

cos , sin

(5.7)

which corresp onds to the following system with algebraic right-hand side:
2 2 2 + bz2 z1 + z2 - bz2 2 - z1 / dz2 = , 2 2 d -z2 + b z1 + z2 + b (1 - 2 ) 2 2 bz1 z1 + z2 - bz1 2 + z1 z2 / dz1 = . 2 2 d -z2 + b z1 + z2 )+ b (1 - 2

(5.8)

Thus, we consider two systems: the initial system (5.7) and the corresp onding algebraic system (5.8). Similarly, we can pass to homogeneous coordinates uk , k = 1, 2, by the formulas zk = uk . By this change of variables, system (3.16) (see ab ove) can b e transformed to the form - u2 du2 1 + u2 = , d -u2 + b u1 u2 du1 + u1 = , d -u2 + b which, in turn, corresp onds to the equation 1 - bu2 + u2 - u2 du2 2 1 = . du1 2u1 u2 - bu1 Since the identity d 1 - u2 + u2 2 u1 + du1 = 0 (5.12) (5.11) (5.9)

(5.10)

is integrable, this equation can b e integrated in elementary functions and in the coordinates (, z1 ,z2 ) has a first integral of the form
2 2 z1 + z2 - z2 + 2 = const . z1

System (5.7) after reduction corresp onds to the system + bu2 3 u2 + u2 - bu2 3 - u2 du2 1 2 1 + u2 = , 3 u2 + u2 + b (1 - 2 ) d -u2 + b 1 2

bu1 3 u2 + u2 - bu1 3 + u1 u2 du1 1 2 + u1 = , d -u2 + b 3 u2 + u2 + b (1 - 2 ) 1 2 which can also b e reduced to (5.11).

(5.13)

819


6.

Some Generalizations z 2 /x + ey z 2 /x + ey c2 zy/x + c3 y 2 /x , fz i2 zy/x + i3 y 2 /x , fz

The following question arises: Can the system ax + by + cz + c1 dz = dx d1 x gx + hy + iz + i1 dy = dx d1 x + + + +

(6.1)

p ossessing a singularity of typ e 1/x, b e integrated in elementary functions? This system is a generalization of systems (3.16) and (5.8) in three-dimensional phase domains. A series of results concerning this question has already b een obtained (see [91]); here we present a brief review of these results. As ab ove, we introduce the substitutions y = ux, and reduce system (6.1) to the following form: ax + bux + cv x + dv +v = dx d1 x + gx + hux + iv x + du +u = x dx d1 x + x which is equivalent to ax + bux +(c - d1 )vx +(c1 - f )v 2 x +(c2 - e)vux + c3 u2 x dv = , dx d1 x + eux + fv x gx +(h - d1 )ux + iv x + i1 v 2 x +(i2 - f )vux +(i3 - e)u2 x du = . x dx d1 x + eux + fv x x (6.5) (6.6) c1 v 2 eux i1 v 2 eux x + c2 vux + c3 u2 x , + fv x x + i2 vux + i3 u2 x , + fv x (6.3) (6.4) z = vx (6.2)

We establish a corresp ondence b etween this system and the following nonautonomous equation with algebraic right-hand side: a + bu + cv + c1 v 2 + c2 vu + c3 u2 - v [d1 + eu + fv ] dv = . du g + hu + iv + i1 v 2 + i2 vu + i3 u2 - u[d1 + eu + fv ] Integration of this equation reduces to integration of the equation in complete differentials g + hu + iv + i1 v 2 + i2 vu + i3 u2 - d1 u - eu2 - fuv dv = a + bu + cv + c1 v 2 + c2 vu + c3 u2 - d1 v - euv - fv
2

(6.7)

du. (6.8)

Generally sp eaking, we have a 15-parameter family of equations of the form (6.8). To integrate the last identity in elementary functions as a homogeneous equation, it suffices to imp ose the following six restrictions: (6.9) g = 0, i = 0, i1 = 0, e = c2 , h = c, i2 = 2c1 - f. We introduce nine parameters 1 ,... ,9 and consider them as indep endent: 1 = a, 2 = b, 3 = c, 4 = c1 , 5 = c2 , 6 = c3 , 7 = d1 , 8 = f, 9 = i3 . (6.10)

Thus, Eq. (6.8) under conditions (6.9) and (6.10) is reduced to the form 1 + 2 u +(3 - 7 )v +(4 - 8 )v 2 + 6 u2 dv = , du (3 - 7 )u +2(4 - 8 )vu +(9 - 5 )u2 820 (6.11)


whereas system (6.5) (6.6) is reduced to the form 1 + 2 u +(3 - 7 )v +(4 - 8 )v 2 + 6 u2 dv = , dx 7 + 5 u + 8 v (3 - 7 )u +2(4 - 8 )vu +(9 - 5 )u2 du = , x dx 7 + 5 u + 8 v x after which Eq. (6.11) can b e integrated by a finite combination of elementary functions. Indeed, integrating identity (6.8), we obtain d (3 - 7 )v (4 - 8 )v +d u u
2

(6.12) (6.13)

+ d[(9 - 5 )v ]+ d

1 - d[2 ln |u|] - d[6 u] = 0, u

(6.14)

which implies the following invariant relation: 1 (3 - 7 )v (4 - 8 )v 2 + +(9 - 5 )v + - 2 ln |u|- 6 u = C1 = const, u u u and then in the coordinates (x, y , z ), the first integral in the form (4 - 8 )z 2 - 6 y 2 +(3 - 7 )zx +(9 - 5 )zy + 1 x2 y - 2 ln = const . yx x (6.15)

(6.16)

Therefore, we can conclude that the following, generally sp eaking nonconservative, system of third order dep ending on nine parameters is integrable in elementary functions: 1 x + 2 y + 3 z + 4 z 2 /x + 5 zy/x + 6 y 2 /x dz = , dx 7 x + 5 y + 8 z dy 3 y +(24 - 8 )zy/x + 9 y 2 /x = . dx 7 x + 5 y + 8 z Corollary 6.1. On the set S1 { the third-order system = 7 sin + 5 z1 + 8 z2 , (6.19) cos cos 2 + 9 z1 , z1 = 3 z1 cos +(24 - 8 )z1 z2 sin sin which depend on nine parameters and possesses, general ly speaking, a transcendental first integral, which can be expressed through elementary functions :
2 2 z1 (4 - 8 )z2 - 6 z1 +(3 - 7 )z2 sin +(9 - 5 )z2 z1 + 1 sin2 2 - 2 ln = const . z1 sin sin 2 z2 = 1 sin cos + 2 z1 cos + 3 z2 cos + 4 z2

(6.17)

mod 2 }\ { = 0, = }â R2 {z1 ,z2 },

(6.18)

cos cos 2 cos + 5 z1 z2 + 6 z1 , sin sin sin

(6.20)

In particular, system (6.19) for 1 = 1, 2 = 3 = 4 = 5 = 9 = 0, 6 = 8 = -1, and 7 = b coincides with system (3.15). To find an additional first integral of the nonautonomous system (6.1), we can use the first integral (6.16) , which is expressed through a finite combination of elementary functions. First, we transform relation (6.15) as follows: (4 - 8 )v 2 +[(9 - 5 )u +(3 - 7 )] v + f1 (u) = 0, where f1 (u) = 1 - 6 u2 - 2 u ln |u|- C1 u. 821 (6.21)


Formally, v can b e found from the relation v1,2 (u) = where f2 (u) = A1 + A2 u + A3 u2 + A4 u ln |u|, A1 = (3 - 7 )2 - 41 (4 - 8 ),
2

1 (5 - 9 )u +(7 - 3 ) ± 2(4 - 8 )

f2 (u) ,

(6.22)

A2 = 2(9 - 5 )(3 - 7 )+4C1 (4 - 8 ), A4 = 42 (4 - 8 ).

A3 = (9 - 5 ) +46 (4 - 8 ),

Then the required quadrature for the additional (in general, transcendental) first integral (for example, of system (6.12), (6.13) or (6.5), (6.6), where Eq. (6.13) is used) b ecomes dx = x [7 + 5 u + 8 v1,2 (u)]du = (3 - 7 )u +(9 - 5 )u2 +2(4 - 8 )uv1,2 (u) B1 + B2 u + B
3

, B4 u f2 (u) Bk = const, k = 1,... , 4. (6.23)

f2 (u) du

The required quadrature for the search for an additional (in general, transcendental) first integral (for system (6.12), (6.13) or (6.5), (6.6), where Eq. (6.12) is used) b ecomes dx = x [7 + 5 u(v )+ 8 v ]dv ; 1 + 2 u(v )+(3 - 7 )v +(4 - 8 )v 2 + 6 u2 (v ) (6.24)

in this case, the function u(v ) must b e obtained by solving the implicit equation (6.15) with resp ect to u (which, in the general case, is not evident). Sufficient conditions of expressability of integrals in (6.24) through finite combinations of elementary functions are stated by the following lemma. Lemma 6.1. For A4 = 0, i.e., for 2 = 0 or for 4 = 8 , (6.26) the indefinite integral in (6.24) can be expressed through a finite combinations of elementary functions. Theorem 6.1. Under the sufficient conditions of Lemma 6.1 (in this case, property (6.25) holds), system (6.19) possesses a complete set of first integrals that can be expressed through a finite combination of elementary functions. The dynamical systems considered in the present pap er are systems with variable dissipation with zero mean with resp ect to the p eriodic coordinate. In many cases, such systems p ossess a complete set of first integrals that can b e expressed through elementary functions. We have presented several cases of complete integrability in the dynamics of the spatial (3D) motion of a b ody in a nonconservative field. Moreover, we are dealing with three prop erties that, at first glance, seem to b e indep endent: (1) the class of systems (3.1) with marked symmetries sp ecified ab ove; (2) this class of systems p ossesses variable dissipation with zero mean (with resp ect to the variable ); this allows one to consider them as "almost" conservative systems; (3) in some (sufficiently low-dimensional) cases, these systems p ossess a complete set of (generally sp eaking, transcendental from the standp oint of complex analysis) first integrals. 822 (6.25)


The method of reduction of initial systems whose right-hand sides contain p olynomials of trigonometric functions to systems with p olynomial right-hand sides allows one to find (or to prove the absence) of first integrals for systems of a more general form that p erhaps do not p ossess the symmetries mentioned ab ove (see [91]).

Chapter 2 CASES OF INTEGRABILITY CORRESPONDING TO THE MOTION OF A RIGID BODY IN FOUR-DIMENSIONAL SPACE, I
In this chapter, we systematize some earlier and newer results on the study of the equations of motion of axisymmetric four-dimensional (4D) rigid b odies in nonconservative force fields. The form of these equations is taken from the dynamics of real lower-dimensional rigid b odies interacting with a resisting medium by the laws of jet flows, where a b ody is influenced by a nonconservative tracing force; under the action of this force, the velocity of some characteristic p oint of the b ody remains constant, which means that the system p ossesses a nonintegrable servo constraint (see [5, 31, 36, 46, 53, 71, 77, 81, 88, 139, 152]). Earlier (see [42, 81]), the present author proved the complete integrability of the equations of planeparallel motion of a b ody in a resisting medium under jet flow conditions when the system of dynamical equations p ossesses a first integral which is a transcendental (in the sense of the theory of functions of a complex variable) function of quasi-velocities having essential singularities. It was assumed that the interaction of the medium with the b ody is concentrated on a part of the surface of the b ody that has the form of a (one-dimensional) plate. Subsequently (see [76, 77, 95]), the planar problem was generalized to the spatial (three-dimensional) case, where the system of dynamical equations p ossesses a complete set of transcendental first integrals. In this case, it was assumed that the interaction of the medium with the b ody is concentrated on a part of the surface of the b ody that has the form of a planar (two-dimensional) disk. In this chapter, we discuss results, b oth new results and results obtained earlier, concerning the case where the interaction of the medium with the b ody is concentrated on a part of the surface of the b ody that has the form of a three-dimensional disk and the force acts in the direction p erp endicular to the disk. We systematize these results and formulate them in invariant form. We also introduce an additional dep endence of the moment of a nonconservative force on the angular velocity; this dep endence can b e generalized to the motion in higher-dimensional spaces. 7. General Discussion

7.1. Two cases of dynamical symmetry of a four-dimensional b o dy. Assume that a fourdimensional rigid b ody of mass m with smooth three-dimensional b oundary is under the influence of a nonconservative force field; this can b e interpreted as motion of the b ody in a resisting medium that fills up the four-dimensional domain of Euclidean space E4 . We assume that the b ody is dynamically symmetric. If the b ody has two indep endent principal moments of inertia, then in some coordinate system Dx1 x2 x3 x4 attached to the b ody the op erator of inertia has the form diag{I1 ,I2 ,I2 ,I2 } or the form diag{I1 ,I1 ,I3 ,I3 }. (7.2) 823 (7.1)


In the first case, the b ody is dynamically symmetric in the hyp erplane Dx2 x3 x4 while in the second case the two-dimensional planes Dx1 x2 and Dx3 x4 are planes of dynamical symmetry of the b ody. 7.2. Dynamics on so(4) and R4 . The configuration space of a free, n-dimensional rigid b ody is the direct product Rn â SO(n) (7.3) of the space Rn , which defines the coordinates of the center of mass of the b ody, and the rotation group SO(n), which defines rotations of the b ody ab out its center of mass and has dimension n(n +1) n(n - 1) = . 2 2 Therefore, the dynamical part of the equations of motion has the same dimension, whereas the dimension of the phase space is equal to n(n +1). In particular, if is the tensor of the angular velocity of a four-dimensional rigid b ody (it is a secondrank tensor, see [18­22, 25­27, 40, 41, 66]), so(4), then the part of the dynamical equations of motion corresp onding to the Lie algebra so(4) has the following form (see [7, 9, 10, 13, 66, 147­149]): + +[, + ] = M, (7.4) n+ where = diag{1 ,2 ,3 ,4 }, - I1 + I2 + I3 + I4 I1 - I2 + I3 + I4 , 2 = , 1 = 2 2 I1 + I2 - I3 + I4 I1 + I2 + I3 - I4 , 4 = , 3 = 2 2 M = MF is the natural pro jection of the moment of external forces F acting on the b ody in R4 natural coordinates of the Lie algebra so(4) and [ ] is the commutator in so(4). The skew-sym matrix corresp onding to this second-rank tensor so(4) is represented in the form 0 -6 5 -3 6 0 -4 2 , -5 4 0 -1 3 -2 1 0 (7.5)

on the metric

(7.6)

where 1 , 2 , 3 , 4 , 5 , and 6 are the comp onents of the tensor of angular velocity corresp onding to the pro jections on the coordinates of the Lie algebra so(4). Obviously, the following relations hold: i - j = Ij - Ii , i, j = 1,... , 4. (7.7)

To calculate the moment of an external force acting to the b ody, we need to construct the mapping R4 â R4 so(4) that maps a pair of vectors (DN, F) R4 â R
2N 4

(7.8) (7.9) (7.10)

into an element of the Lie algebra so(4), where DN = {0,x ,x
3N

,x

4N

},

F = {F1 ,F2 ,F3 ,F4 },

and F is an external force acting on the b ody. For this purp ose, we construct the following auxiliary matrix 0 x2N x3N x4N . (7.11) F1 F2 F3 F4 824


Then the right-hand side of system (7.4) takes the form M= x
3N

F4 - x

4N

F3 , x

4N

F2 - x

2N

F4 , -x

4N

F1 , x

2N

F3 - x

3N

F2 , x

3N

F1 , -x

2N

F1 .

(7.12)

The dynamical systems studied in the following chapters, generally sp eaking, are not conservative and in fact are dynamical systems with variable dissipation with zero mean (see [91]). We need to examine by direct methods one part of the main system of dynamical equations, namely, the Newton equation, which serves as the equation of motion of the center of mass, i.e., the part of the dynamical equations of motion corresp onding to the space R4 : mw C = F , (7.13) where wC is the acceleration of the center of mass C of the b ody and m is its mass. Moreover, due to the higher-dimensional Rivals formula (it can b e obtained by the op erator method) we have the following relations: (7.14) wC = wD +2 DC + E DC, wD = vD +vD , E = , where wD is the acceleration of the p oint D , F is the external force acting on the b ody (in our case, F = S), and E is the tensor of angular acceleration (second-rank tensor). Thus, the system of equations (7.4) and (7.13) of tenth order on the manifold R4 â so(4) is a closed system of dynamical equations of the motion of a free four-dimensional rigid b ody under the action of an external force F. This system is separated from the kinematic part of the equations of motion on the manifold (7.3) and can b e examined indep endently. 8. General Problem of Motion Under a Tracing Force

Consider a motion of a homogeneous, dynamically symmetric (case (7.1)), rigid b ody with "front end face" (a three-dimensional disk interacting with a medium that fills four-dimensional space) in the field of a resistance force S under quasi-stationarity conditions (see [16, 17, 30, 35, 36, 42, 43, 89, 108, 126, 145, 152]. Let (v, , 1 ,2 ) b e the (generalized) spherical coordinates of the velocity vector of the center of the three-dimensional disk lying on the axis of symmetry of the b ody, let 0 -6 5 -3 0 -4 2 = 6 -5 4 0 -1 3 -2 1 0 be to of I4 the tensor of angular velocity of the b ody, and let Dx1 x2 x3 x4 b e the coordinate system attached the b ody such that the axis of symmetry CD coicides with the axis Dx1 (recall that C is the center mass) and the axes Dx2 , Dx3 , and Dx4 lie in the hyp erplane of the disk, while I1 , I2 , I3 = I2 , = I2 , and m are the characteristics of inertia and mass. We adopt the following expansions in pro jections onto the axes of the coordinate system Dx1 x2 x3 x4 : DC = {-, 0, 0, 0}, (8.1) vD = v cos , v sin cos 1 , v sin sin 1 cos 2 , v sin sin 1 sin 2 .

In the case (7.1) we additionally have an expansion for the function of the influence of the medium on the four-dimensional b ody: S = {-S, 0, 0, 0}, (8.2) i.e., in this case F = S. Then the part of the dynamical equations of motion (including the analytic Chaplygin functions [16, 17]; see b elow) that describ es the motion of the center of mass and corresp onds to the space R4 , in 825


which tangent forces of the influence of the medium on the three-dimensional disk vanish, takes the form v cos - v sin - 6 v sin cos 1 + 5 v sin sin 1 cos 2 - 3 v sin sin 1 sin 2 (8.3) S 2 2 2 + 6 + 5 + 3 = - , m v sin cos 1 + v cos cos 1 - 1 v sin sin 1 + 6 v cos - 4 v sin sin 1 cos 2 +2 v sin sin 1 sin 2 - (4 5 + 2 3 ) - 6 = 0, v sin sin 1 cos 2 + v cos sin 1 cos 2 + 1 v sin cos 1 cos 2 - 2 v sin sin 1 sin 2 -5 v cos + 4 v sin cos 1 - 1 v sin sin 1 sin 2 - (-1 2 + 4 6 )+ 5 = 0, v sin sin 1 sin 2 + v cos sin 1 sin 2 + 1 v sin cos 1 sin 2 + 2 v sin sin 1 cos 2 +3 v cos - 2 v sin cos 1 + 1 v sin sin 1 cos 2 + (2 6 + 1 5 ) - 3 = 0, where S = s()v 2 , = CD , v > 0. (8.7) Further, the auxiliary matrix (7.11) for calculating the moment of the resistance force has the form 0 x2N -S 0 x
3N

(8.4)

(8.5)

(8.6)

x

4N

0

0

;

(8.8)

then the part of the dynamical equations of motion that describ es the motion of the b ody ab out the center of mass and corresp onds to the Lie algebra so(4), b ecomes (4 + 3 ) 1 +(3 - 4 )(3 5 + 2 4 ) = 0, (2 + 4 ) 2 +(2 - 4 )(3 6 - 1 4 ) = 0, (4 + 1 ) 3 +(4 - 1 )(2 6 + 1 5 ) = x
4N

(8.9) (8.10) , 1 ,2 , v v s()v 2 , (8.11) (8.12)
3N

(3 + 2 ) 4 +(2 - 3 )(5 6 + 1 2 ) = 0, (1 + 3 ) 5 +(3 - 1 )(4 6 - 1 3 ) = -x (1 + 2 ) 6 +(1 - 2 )(4 5 + 2 3 ) = x , 1 ,2 , , 1 ,2 , v s()v 2 , s()v 2 .

(8.13) (8.14)

2N

Thus, the phase space of system (8.3)­(8.6), (8.9)­(8.14) of tenth order is the direct product of the four-dimensional manifold and the Lie algebra so(4): R1 â S3 â so(4). We note that system (8.3)­(8.6), (8.9)­(8.14), due to the existing dynamical symmetry I2 = I3 = I4 , p ossesses cyclic first integrals
0 1 1 = const, 0 2 2 = const, 0 4 4 = const .

(8.15)

(8.16)

(8.17)

We will henceforth consider the dynamics of the system on zero levels:
0 0 0 1 = 2 = 4 = 0.

(8.18)

826


If we consider a more general problem on the motion of a b ody under a tracing force T that lies on the straight line CD = Dx1 and assume that the relation v const T - s()v 2 , (8.19) is satisfied throughout the motion (see [91]), then instead of F1 system (8.3)­(8.6), (8.9)­(8.14) contains = DC. (8.20) Choosing the value T of the tracing force appropriately, we can assume Eq. (8.19) throughout the motion. Indeed, expressing T on the basis of system (8.3)­(8.6) , (8.9)­(8.14), we obtain for cos = 0 the relation m sin 2 2 2 v , 1 ,2 , , (8.21) T = Tv (, 1 ,2 , ) = m 3 + 5 + 6 + s()v 2 1 - 2I2 cos v where
v

, 1 ,2 ,

v

=x

4N

, 1 ,2 ,

v

sin 1 sin 2 + x

3N

, 1 ,2 , +x
2N

v

sin 1 cos 2 , 1 ,2 , v cos 1 ; (8.22)

here we used conditions (8.17)­(8.19) . This procedure can b e interpreted in two ways. First, we have transformed the system using the tracing force (control) that enables us to consider the class (8.19) of motions of interest. Second, we can treat this as an order-reduction procedure. Indeed, system (8.3)­(8.6), (8.9)­(8.14) generates the following indep endent system of sixth order: v cos cos 1 - 1 v sin sin 1 + 6 v cos - 6 = 0, v cos sin 1 cos 2 + 1 v sin cos 1 cos 2 - 2 v sin sin 1 sin 2 - 5 v cos + 5 = 0, v cos sin 1 sin 2 + 1 v sin cos 1 sin 2 + 2 v sin sin 1 cos 2 + 3 v cos - 3 = 0, 2I2 3 = x
4N

(8.23) (8.24) (8.25) (8.26) (8.27) (8.28)

2I2 5 = -x 2I2 6 = x

3N

2N

s()v 2 , v s()v 2 , , 1 ,2 , v s()v 2 , , 1 ,2 , v , 1 ,2 ,

which, in addition to the p ermanent parameters sp ecified ab ove, contains the parameter v . System (8.23)­(8.28) is equivalent to the system v cos + v cos [6 cos 1 - 5 sin 1 cos 2 + 3 sin 1 sin 2 ] + [-6 cos 1 + 5 sin 1 cos 2 - 3 sin 1 sin 2 ] = 0, 1 v sin - v cos [5 cos 1 cos 2 + 6 sin 1 - 3 cos 1 sin 2 ] + [5 cos 1 cos 2 6 sin 1 - 3 cos 1 sin 2 ] = 0, 2 v sin sin 1 + v cos [3 cos 2 + 5 sin 2 ]+ [-3 cos 2 - 5 sin 2 ] = 0, 3 = v2 x 2I2
4N

(8.29)

(8.30)

(8.31) (8.32) 827

, 1 ,2 ,

v

s(),


5 = -

v2 x 2I2

3N

, 1 ,2 ,

v

s(),

(8.33)

6 =

v2 x 2I2

2N

, 1 ,2 ,

v

s().

(8.34)

We introduce new quasi-velocities. For this purp ose, we transform 3 , 5 , and 6 by means of two rotations: z1 3 z2 = T1 (-1 ) T3 (-2 ) 5 , (8.35) z3 6 where 1 0 0 T1 (1 ) = 0 cos 1 - sin 1 , 0 sin 1 cos 1 z1 = 3 cos 2 + 5 sin 2 , z2 = -3 cos 1 sin 2 + 5 cos 1 cos 2 + 6 sin 1 , z3 = 3 sin 1 sin 2 - 5 sin 1 cos 2 + 6 cos 1 . As we see from (8.29) ­(8.34) , we cannot solve the system with resp ect to , 1 , and 2 on the manifold (8.38) O1 = (, 1 ,2 ,3 ,5 ,6 ) R6 : = k, 1 = l, k , l Z . 2 Therefore, on the manifold (8.38) the uniqueness theorem is formally violated. Moreover, for even k and any l, an indeterminate form app ears due to the degeneration of the spherical coordinates (v, , 1 ,2 ). For odd k, the uniqueness theorem is obviously violated since the first equation (8.29) is degenerate. This implies that system (8.29)­(8.34) outside (and only outside) the manifold (8.38) is equivalent to the system = -z3 + v s() 2I2 cos
v

cos 2 - sin 2 0 T3 (2 ) = sin 2 cos 2 0 . 0 0 1



(8.36)

Therefore, the following relations hold: (8.37)

, 1 ,2 ,

v

, v s() cos - z2 sin 2I2 sin v

(8.39)

z3 =

v2 s()v , 1 ,2 , 2I2 v v s() z1 v , 1 ,2 + 2I2 sin

2 2 - z1 + z2

v

, 1 ,2 ,

, v

(8.40)

,

z2 = -

v2 cos 2 cos cos 1 s()v , 1 ,2 , + z2 z3 + z1 + 2I2 v sin sin sin 1 v s() cos 1 v s() z3 v , 1 ,2 , -- z1 v , 1 ,2 , + 2I2 sin v 2I2 sin sin 1 v v2 s()v , 1 ,2 , 2I2 v v s() z3 v , 1 ,2 - 2I2 sin + z1 z3 , v cos cos cos 1 - z1 z2 - sin sin sin 1 v s() cos 1 + z2 v , 1 ,2 , 2I2 sin sin 1 v

(8.41) ,

z1 =

(8.42) ,

828


cos v s() 1 = z2 + sin 2I2 sin 2 = -z1 where
v

v

, 1 ,2 ,

v
v

, v

(8.43)

cos v s() + sin sin 1 2I2 sin sin 1

, 1 ,2 ,

,

(8.44)

, 1 ,2 ,

v

=x

4N

, 1 ,2 ,

v

cos 1 sin 2 + x sin 1 , cos 2 - x

3N

, 1 ,2 ,

v

cos 1 cos 2 (8.45)

-x v

2N

, 1 ,2 , v , 1 ,2 , v



v

, 1 ,2 ,

=x

4N

3N

, 1 ,2 ,

v

sin 2 ,

(8.46)

and the function v (, 1 ,2 , /v ) can b e represented in the form (8.22). Here and in the discussion that follows, the dep endence on the group of variables (, 1 ,2 , /v ) is meant as a comp osite dep endence on (, 1 ,2 ,z1 /v , z2 /v , z3 /v ) due to (8.37) . The uniqueness theorem for system (8.29)­(8.34) on the manifold (8.38) for odd k is violated in the following sense: for odd k , a nonsingular phase tra jectory of system (8.29)­(8.34) passes through almost all p oints of the manifold (8.38), intersecting the manifold (8.38) at right angle, and there exists a phase tra jectory that at any moment of time completely coincides with the p oint sp ecified. However, physically these tra jectories are different since they corresp ond to different values of the tracing force. We prove this assertion. As was shown ab ove, to maintain a constraint of the form (8.19), we must take a value of T for cos = 0 according to (8.21) . Let s() v , 1 ,2 , = L 1 ,2 , . (8.47) lim v v /2 cos Note that |L| < + if and only if
/2

lim





v

, 1 ,2 ,

v

s()

< + .

(8.48)

For = /2, the required value of the tracing force is defined by the equation T = Tv 2 2 ,1 ,2 , = m 3 + 5 + 2
2 6

-

m Lv 2 . 2I2

(8.49)

where 3 , 5 , and 6 are arbitrary. On the other hand, maintaining the rotation ab out some p oint W by the tracing force, we must choose this force according to the relation T = Tv mv 2 ,1 ,2 , = , 2 R0 (8.50)

where R0 is the distance b etween C and W . Relations (8.21) and (8.50) define, in general, different values of the tracing force T for almost all p oints of the manifold (8.38), which proves our assertion. 829


9.

Case Where the Moment of a Nonconservative Force Is Indep endent of the Angular Velo city

9.1. Reduced system. As in the choice of Chaplygin analytic functions (see [16, 17]), we take the dynamical functions s, x2N , x3N , and x4N in the following form: s() = B cos , x x x
2N

3N

4N

v , 1 ,2 , v , 1 ,2 , v , 1 ,2 ,

=x =x =x

2N 0

(, 1 ,2 ) = A sin cos 1 , (, 1 ,2 ) = A sin sin 1 cos 2 , (, 1 ,2 ) = A sin sin 1 sin 2 , (9.1)

3N 0

4N 0

where A, B > 0 and v = 0. We see that in the system considered, the moment of nonconservative forces in indep endent of the angular velocity (but dep ends on the angles , 1 , and 2 ). Moreover, the functions v (, 1 ,2 , /v ), v (, 1 ,2 , /v ), v (, 1 ,2 , /v ) in system (8.39)­(8.44) assume the following form:
v

, 1 ,2 ,

v

= A sin ,



v

, 1 ,2 ,

v



v

, 1 ,2 ,

v

0.

(9.2)

Then, due to the nonintegrable constraint (8.19) outside the manifold (8.38), the dynamical part of the equations of motion (system (8.39)­(8.44)) has the form of the following analytic system: = -z3 + z3 = AB v sin , 2I2 (9.3) (9.4) (9.5) (9.6) (9.7) (9.8) s: (9.9)

AB v 2 2 2 cos sin cos - z1 + z2 , 2I2 sin cos 2 cos cos 1 + z1 , z2 = z2 z3 sin sin sin 1 cos cos cos 1 - z1 z2 , z1 = z1 z3 sin sin sin 1 cos , 1 = z2 sin cos 2 = -z1 . sin sin 1 Further, introducing dimensionless variables and parameters and a new differentiation as follow AB , b = n0 , = n0 v , zk n0 vzk , k = 1, 2, 3, n2 = 0 2I2 we reduce system (9.3)­(9.8) to the form = -z3 + b sin , z3 = z2 = z1 = 1 = 830
2 2 cos , sin cos - z1 + z2 sin cos 2 cos cos 1 z2 z3 + z1 , sin sin sin 1 cos cos cos 1 - z1 z2 z1 z3 , sin sin sin 1 cos , z2 sin

(9.10) (9.11) (9.12) (9.13) (9.14)


2 = -z1

cos . sin sin 1

(9.15)

We see that the sixth-order system (9.10)­(9.15) (which can b e considered as a system on the tangent bundle T S3 of the three-dimensional sphere S3 , see b elow) contains the indep endent fifthorder system (9.10)­(9.14) on its own five-dimensional manifold. For the complete integration of system (9.10) ­(9.15), in general, we need five indep endent first integrals. However, after the change of variables z1 z2 z , z z=
2 2 z1 + z2 ,

z = z2 /z1 ,

(9.16)

system (9.10)­(9.15) splits as follows: = -z3 + b sin , z3 = z= z = 1 = 2 = cos sin cos - z 2 , sin cos zz3 , sin cos cos 1 2 (±)z 1+ z , sin sin 1 zz cos (±) , 2 1+ z sin z cos () . 2 sin sin 1 1+ z (9.17) (9.18) (9.19) (9.20) (9.21) (9.22)

We see that the sixth-order system splits into indep endent subsystems of lower order: system (9.17)­ (9.19) has order three and system (9.20), (9.21) (after a change of the indep endent variable) order two. Thus, for the complete integration of system (9.17)­(9.22) it suffices to sp ecify two indep endent first integrals of system (9.17)­(9.19) , one first integral of system (9.20), (9.21) , and an additional first integral that "attaches" Eq. (9.22). Note that system (9.17)­(9.19) can b e considered on the tangent bundle T S2 of the two-dimensional sphere S2 . 9.2. Complete list of invariant relations. System (9.17)­(9.19) has the form of a system that app ears in the dynamics of a three-dimensional (3D) rigid b ody in a field of nonconservative forces. First, we establish a corresp ondence b etween the third-order system (9.17) ­(9.19) and the nonautonomous second-order system sin cos - z 2 cos / sin dz3 = , d -z3 + b sin zz3 cos / sin dz = . d -z3 + b sin Applying the substitution = sin , we rewrite system (9.23) in algebraic form - z 2 / dz3 = , d -z3 + b zz3 / dz = . d -z3 + b Introducing homogeneous variables by the formulas z = u1 , z3 = u2 , (9.25) 831

(9.23)

(9.24)


we reduce system (9.24) to the following form: 1 - u2 du2 1 + u2 = , d -u2 + b u1 u2 du1 + u1 = , d -u2 + b which is equivalent to the system du2 1 - u2 + u2 - bu2 1 2 = , d -u2 + b du1 2u1 u2 - bu1 = . d -u2 + b

(9.26)

(9.27)

We establish a corresp ondence b etween the second-order system (9.27) and the nonautonomous first-order equation 1 - u2 + u2 - bu2 du2 1 2 = , (9.28) du1 2u1 u2 - bu1 which can b e easily reduced to exact-differential form: d u2 + u2 - bu2 +1 2 1 u1 = 0. (9.29)

Thus, Eq. (9.28) has the following first integral: u2 + u2 - bu2 +1 2 1 = C1 = const, u1 which expresses in terms of the previous variables has the form
2 z3 + z 2 - bz3 sin +sin2 = C1 = const . z sin

(9.30)

(9.31)

Remark 9.1. Consider system (9.17)­(9.19) with variable dissipation with zero mean (see [91]) which b ecomes conservative for b = 0: = -z3 , cos , z3 = sin cos - z 2 (9.32) sin cos . z = zz3 sin It p ossesses two analytic first integrals of the form
2 z3 + z 2 +sin2 = C1 = const,

(9.33) (9.34)

z sin = C = const .

2

Obviously, the ratio of the two first integrals (9.33) and (9.34) is also a first integral of system (9.32) . However, for b = 0, neither of the functions
2 z3 + z 2 - bz3 sin +sin2

(9.35)

and (9.34) is a first integral of system (9.17) ­(9.19) but their ratio is a first integral for any b. Further, we find the explicit form of the additional first integral of the third-order system (9.17)­ (9.19). For this purp ose, we transform the invariant relation (9.30) for u1 = 0 as follows: u2 - 832 b 2
2

+ u1 -

C1 2

2

=

2 b2 + C1 - 1. 4

(9.36)


We see that the parameters of this invariant relation satisfy the condition
2 b2 + C1 - 4 0,

(9.37)

and the phase space of system (9.17)­(9.19) is stratified into a family of surfaces defined by Eq. (9.36) . Thus, by relation (9.30), the first equation of system (9.27) has the form where U1 (C1 ,u2 ) = 1 2 C1 ±
2 C1 - 4 u2 - bu2 +1 2

2 1 - bu2 + u2 - C1 U1 (C1 ,u2 ) du2 2 = , d -u2 + b ;

(9.38)

(9.39)

the integration constant C1 is defined by condition (9.37) . Therefore, the quadrature for the search for the additional first integral of system (9.17)­(9.19) b ecomes (b - u2 )du2 d = . (9.40) 2 2 1 - bu2 + u2 - C1 C1 ± C1 - 4 u2 - bu2 +1 /2 2 2 Obviously, the left-hand side (up to an additive constant) equals ln | sin |. If b 2 = w1 , b2 = b2 + C1 - 4, 1 2 then the right-hand side of Eq. (9.40) has the form u2 - - 1 4 d b2 - 4w 1 b2 - 4w 1
2 1 2 1

(9.41)

(9.42)

±C

1

b2 - 4w 1

2 1

-b

dw1
2 (b2 - 4w1 ) ± C 1 1

b2 - 4w 1 1 = - ln 2

2 1 2 b2 - 4w1 b 1 ± 1 ± I1 , (9.43) C1 2

where I1 = dw3
2 b - w3 (w3 ± C1 ) 2 1

,

w3 =

2 b2 - 4w1 . 1

(9.44)

In the calculation of the integral (9.44), the following three cases are p ossible. I. b > 2: 1 ln I1 = - 2 b2 - 4
2 b2 - 4+ b2 - w3 C1 1 ± + w3 ± C1 b2 - 4 2 b 2 - 4 - b2 - w3 C1 1 1 ln + const; (9.45) + 2-4 w3 ± C 1 2b b2 - 4

II. b < 2:

1 ±C1 w3 + b2 1 + const; arcsin I1 = b1 (w3 ± C1 ) 4 - b2 I1 =
2 b2 - w3 1 +const . C1 (w3 ± C1 )

(9.46)

III. b = 2:

(9.47)

833


Returning to the variable w1 = z3 b -, sin 2 (9.48)

we obtain the final expression for I1 : I. b > 2: 1 b2 - 4 ± 2w1 C1 I1 = - ln ± 2 - 4w 2 ± C 2 b2 - 4 b2 - 4 b1 1 1 1 + ln 2 b2 - 4 II. b < 2: I1 = III. b = 2: I1 = C
1



C1 + const; (9.49) b2 - 4 b - 4w ± C1
2 1 2 1

b2 - 4 2w1

2 ±C1 b2 - 4w1 + b2 1 1 1 arcsin + const; 2 - 4w 2 ± C 4 - b2 b1 b1 1 1

(9.50)

2w1
2 b2 - 4w1 ± C 1 1

+const .

(9.51)

Thus, we have found an additional first integral for the third-order system (9.17)­(9.19) and we have the complete set of first integrals that are transcendental functions of their phase variables. Remark 9.2. We must substitute the left-hand side of the first integral (9.30) into the expression of this first integral instead of C1 . Then the additional first integral obtained has the following structure (similar to the transcendental first integral in planar dynamics): z3 z , = C2 = const . (9.52) ln | sin | + G2 sin , sin sin Thus, for the integration of the sixth-order system (9.17) ­(9.22), we have found two indep endent first integrals. As was mentioned ab ove, to integrate it completely, it suffices to find one first integral for (p otentially separated) system (9.20), (9.21) and an additional first integral that "attaches" Eq. (9.22) . To find a first integral for (p otentially separated) system (9.20), (9.21), we establish a corresp ondence b etween it and the following nonautonomous first-order equation:
2 1+ z cos 1 dz = . d1 z sin 1 After integration, this leads to the invariant relation 2 1+ z = C3 = const, sin 1 which in the variables z1 and z2 has the form 2 2 z1 + z2 = C3 = const . z1 sin 1

(9.53)

(9.54)

(9.55)

Further, to find an additional first integral that "attaches" Eq. (9.22), we establish a corresp ondence b etween Eqs. (9.22) and (9.20) and the following nonautonomous equation: dz 2 = - 1+ z cos 1 . d2 Since, by (9.54) ,
2 2 C3 cos 1 = ± C3 - 1 - z ,

(9.56)

(9.57)

834


dz 1 2 2 2 1+ z = C3 - 1 - z . d2 C3 Integrating the last relation, we arrive at the following quadrature: C3 dz , C4 = const . (2 + C4 ) = 2 2) 2 (1 + z C3 - 1 - z Integrating this relation we obtain tan(2 + C4 ) = C3 z
2 C - 1 - z 2 3

we have

(9.58)

(9.59)

,

C4 = const .

(9.60)

In the variables z1 and z2 the last invariant relation has the form C3 z2 tan(2 + C4 ) = , C4 = const . 2 2 2 C3 - 1 z1 - z2 C3 z
2 C - 1 - z 2 3

(9.61)

Finally, we have the following form of the additional first integral that "attaches" Eq. (9.22) : arctan or arctan ± 2 = C4 , ± 2 = C4 , C4 = const (9.62)

C3 z2
2 2 2 C3 - 1 z1 - z2

C4 = const .

(9.63)

Thus, in the case considered, the system of dynamical equations (8.3)­(8.6) and (8.9)­(8.14) under condition (9.1) has eight invariant relations: the nonintegrable analytic constraint of the form (8.19) , the cyclic first integrals of the form (8.17), (8.18), the first integral of the form (9.31), the first integral expressed by relations (9.45)­(9.52), which is a transcendental function of the phase variables (in the sense of complex analysis) expressed through a finite combination of elementary functions, and, finally, the transcendental first integrals of the form (9.54) (or (9.55)) and (9.62) (or (9.63)). Theorem 9.1. System (8.3)­(8.6) , (8.9)­(8.14) under conditions (8.19) , (9.1), (8.18) possesses eight invariant relations (complete set), four of which are transcendental functions from the point of view of complex analysis. Moreover, al l the relations are expressed through finite combinations of elementary functions. Top ological analogies. Consider the following fifth-order system: sin ¨ = 0, + b cos +sin cos - 1 2 + 2 2 sin2 1 cos 2 1+cos - 2 2 sin 1 cos 1 = 0, (9.64) 1 + b 1 cos + 1 ¨ cos sin 2 1+cos +21 2 cos 1 = 0, b > 0, 2 + b 2 cos + 2 ¨ cos sin cos 1 which describ es a fixed four-dimensional p endulum in a flow of a running medium for which the moment of forces is indep endent of the angular velocity, i.e., a mechanical system in a nonconservative field (see [14, 15, 150]). In general, the order of such a system is equal to 6, but the phase variable 2 is a cyclic variable, which leads to the stratification of the phase space and reduces the order of the system. The phase space of this system is the tangent bundle T S3 , 1 , 2 , ,1 ,2 (9.65) 835 9.3.


of the three-dimensional sphere S3 {, 1 ,2 }. The equation that transforms system (9.64) into the system on the tangent bundle of the two-dimensional sphere 2 0, and the equations of great circles 1 0, 2 0 (9.67) (9.66)

define families of integral manifolds. It is easy to verify that system (9.64) is equivalent to a dynamical system with variable dissipation with zero mean on the tangent bundle (9.65) of the three-dimensional sphere. Moreover, the following theorem holds. Theorem 9.2. System (8.3)­(8.6) , (8.9)­(8.14) under conditions (8.19), (9.1), and (8.18) is equivalent to the dynamical system (9.64) . Proof. Indeed it suffices to set = , 1 = 1 , 2 = 2 , and b = -b . For more general top ological analogies, see [91].

10.

Case Where the Moment of a Nonconservative Force Dep ends on the Angular Velo city

10.1. Intro duction of the dep endence on the angular velo city. This chapter is devoted to the dynamics of a four-dimensional rigid b ody in four-dimensional space. Since the present section is devoted to the study of motion in the case where the moment of forces dep ends on the tensor of angular velocity, we introduce this dep endence in a more general situation. This also allows us to introduce this dep endence for multi-dimensional b odies. Let x = (x1N ,x2N ,x3N ,x4N ) b e the coordinates of the p oint N of application of a nonconservative force (influence of the medium) acting on the three-dimensional disk and let Q = (Q1 ,Q2 ,Q3 ,Q4 ) b e the comp onents of the force S of the influence of the medium indep endent of the tensor of the angular velocity. We consider only the linear dep endence of the functions (x1N ,x2N ,x3N ,x4N ) on the tensor of angular velocity since this introduction is not itself a priori obvious (see [14, 15]). We adopt the following dep endence: x = Q + R, where R = (R1 ,R2 ,R3 ,R4 ) is a vector-valued function containing angular velocity. The dep endence of the function R on the com velocity is gyroscopic: 0 -6 5 -3 R1 R2 1 6 0 -4 2 R= =- -5 4 R3 0 -1 v R4 3 -2 1 0 (10.1) the comp onents of the tensor of p onents of the tensor of angular h1 h2 h3 h4 , (10.2)

where (h1 ,h2 ,h3 ,h4 ) are some p ositive parameters (cf. [14, 15, 91]). Since x1N 0, we have x 836
2N

= Q2 - h1

6 , v

x

3N

= Q3 + h1

5 , v

x

4N

= Q4 - h1

3 . v

(10.3)


10.2.

Reduced system. As in the choice of the Chaplygin analytic functions (see [16, 17]) Q2 = A sin cos 1 , Q3 = A sin sin 1 cos 2 , Q4 = A sin sin 1 sin 2 , A > 0, (10.4)

we take the dynamical functions s, x v , 1 ,2 , v , 1 ,2 , v , 1 ,2 ,

2N

,x

3N

, and x

4N

in the following form: 6 , v

s() = B cos , B > 0, x x x
2N

= A sin cos 1 - h

h = h1 > 0, v = 0, h = h1 > 0, v = 0, h = h1 > 0, v = 0. tain the the sys(10.5)

3N

4N

5 , v 3 = A sin sin 1 sin 2 - h , v = A sin sin 1 cos 2 + h

This shows that in the problem considered, there is an additional damping (but accelerating in cer domains of the phase space) moment of a nonconservative force (i.e., there is a dep endence of moment on the comp onents of the tensor of angular velocity). Moreover, h2 = h3 = h4 due to dynamical symmetry of the b ody. In this case, the functions v (, 1 ,2 , /v ), v (, 1 ,2 , /v ), and v (, 1 ,2 , /v ) in the tem (8.39)­(8.44) have the following form:
v

v

v

v , 1 ,2 , v , 1 ,2 , v , 1 ,2 ,

= A sin - h z2 , v h = - z1 . v =

h z3 , v (10.6)

Then, due to the nonintegrable constraint (8.19), outside the manifold (8.38) the dynamical part of the equations of motion (system (8.39) ­(8.44)) takes the form of the analytic system = - 1+ z3 = z2 = z1 = 1 = 2 = B h 2I2 AB v sin , 2I2 B h 2 2 cos - 1+ z1 + z2 2I2 sin B h cos 2 cos z3 + 1+ z1 sin 2I2 sin B h cos cos - 1+ z3 z1 z2 sin 2I2 sin cos , sin cos z1 . sin sin 1 z3 + AB , 2I2 (10.7) Bhv - z3 cos , 2I2 cos 1 Bhv - z2 cos , sin 1 2I2 cos 1 Bhv - z1 cos , sin 1 2I2 (10.8) (10.9) (10.10) (10.11) (10.12)

AB v 2 sin cos 2I2 B h 1+ z2 2I2 B h 1+ z1 2I2 B h 1+ z2 2I2 B h - 1+ 2I2

Introducing dimensionless variables and parameters and a new differentiation as follows: zk n0 vzk , k = 1, 2, 3, n2 = 0 b = n0 , H1 = Bh , 2I2 n0 = n0 v , (10.13) 837


we reduce system (10.7)­(10.12) to the form = - (1 + bH1 ) z3 + b sin , z3 = sin cos - (1 + z2 = (1 + bH1 ) z2 z3
2 2 cos - H1 z3 cos , bH1 ) z1 + z2 sin 2 cos cos 1 +(1+ bH1 ) z1 - H1 z2 cos , sin sin 1 cos cos 1 - (1 + bH1 ) z1 z2 - H1 z1 cos , sin sin 1

(10.14) (10.15) (10.16) (10.17) (10.18) (10.19)

cos sin cos z1 = (1 + bH1 ) z1 z3 sin cos 1 = (1 + bH1 ) z2 , sin cos 2 = - (1 + bH1 ) z1 . sin sin 1

We see that the sixth-order system (10.14)­(10.19) (which can b e considered on the tangent bundle T S3 of the three-dimensional sphere S3 ), contains an indep endent fifth-order system (10.14) ­(10.18) on its own five-dimensional manifold. For complete integration of system (10.14)­(10.19) , we need, in general, five indep endent first integrals. However, after the change of variables z1 z2 z , z z=
2 2 z1 + z2 ,

z = z2 /z1 ,

(10.20)

system (10.14)­(10.19) splits as follows: = -(1 + bH1 )z3 + b sin , z3 = sin cos - (1 + bH1 )z z = (1 + bH1 )zz3
2

(10.21) cos - H1 z3 cos , sin (10.22) (10.23) (10.24) (10.25) (10.26) sysdent two and two-

cos - H1 z cos , sin cos cos 1 2 , z = (±)(1 + bH1 )z 1+ z sin sin 1 zz cos , 1 = (±)(1 + bH1 ) 2 1+ z sin z cos . 2 = ()(1 + bH1 ) 2 sin sin 1 1+ z

We see that the sixth-order system splits into indep endent subsystems of lower orders: tem (10.21)­(10.23) of order 3 and system (10.24), (10.25) (certainly, after a choice of the indep en variables) of order 2. Thus, to integrate system (10.21)­(10.26) completely, it suffices to find indep endent first integrals of system (10.21)­(10.23), one first integral of system (10.24), (10.25), an additional first integral that "attaches" Eq. (10.26) . Note that system (10.21)­(10.23) can b e considered on the tangent bundle T S2 of the dimensional sphere S2 .

10.3. Complete list of invariant relation. System (10.21)­(10.23) has the form of a system of equations that app ears in the dynamics of a three-dimensional (3D) rigid b ody in a nonconservative field. 838


First, we establish a corresp ondence b etween the third-order system (10.21) ­(10.23) and the nonautonomous second-order system sin cos - (1 + bH1 )z 2 cos / sin - H1 z3 cos dz3 = , d -(1 + bH1 )z3 + b sin dz (1 + bH1 )zz3 cos / sin - H1 z cos = . d -(1 + bH1 )z3 + b sin Using the substitution = sin , we rewrite system (10.27) in the algebraic form: - (1 + bH1 )z 2 / - H1 z3 dz3 = , d -(1 + bH1 )z3 + b (1 + bH1 )zz3 / - H1 z dz = . d -(1 + bH1 )z3 + b Further, introducing homogeneous variables by the formulas z = u1 , we reduce system (10.28) to the following form: 1 - (1 + bH1 )u2 - H1 u2 du2 1 + u2 = , d -(1 + bH1 )u2 + b (1 + bH1 )u1 u2 - H1 u1 du1 + u1 = , d -(1 + bH1 )u2 + b which is equivalent to (1 + bH1 ) u2 - u2 - (b + H1 )u2 +1 du2 2 1 = , d -(1 + bH1 )u2 + b du1 2(1 + bH1 )u1 u2 - (b + H1 )u1 = . d -(1 + bH1 )u2 + b z3 = u2 , (10.29)

(10.27)

(10.28)

(10.30)

(10.31)

We establish a corresp ondence b etween the second-order system (10.31) and the nonautonomous first-order equation 1 - (1 + bH1 ) u2 - u2 - (b + H1 )u2 du2 1 2 = , du1 2(1 + bH1 )u1 u2 - (b + H1 )u1 which can b e easily reduced to exact-differential form: d (1 + bH1 ) u2 + u2 - (b + H1 )u2 +1 2 1 u1 = 0. (10.33) (10.32)

Thus, Eq. (10.32) has the following first integral: (1 + bH1 ) u2 + u2 - (b + H1 )u2 +1 2 1 = C1 = const, u1 which in the original variables has the form
2 (1 + bH1 ) z3 + z 2

(10.34)

- (b + H1 )z3 sin +sin2 = C1 = const . z sin

(10.35) 839


Remark 10.1. Consider system (10.21)­(10.23) with variable dissipation with zero mean (see [91]), which b ecomes conservative for b = H1 : = - 1+ b2 z3 + b sin , z3 = sin cos - 1+ b2 z
2

cos - bz cos . z = 1+ b2 zz3 sin It p ossesses the following two analytic first integrals: 1+ b
2 2 z3 + z 2

cos - bz3 cos , sin

(10.36)

- 2bz3 sin +sin2 = C1 = const,

(10.37) (10.38)

z sin = C = const .

2

Obviously, the ratio of the two first integrals (10.37) and (10.38) is also a first integral of system (10.36). However, for b = H1 , none of the functions
2 (1 + bH1 ) z3 + z 2

- (b + H1 )z3 sin +sin2

(10.39)

and (10.38) is a first integral of system (10.21) ­(10.23) , but their ratio is a first integral of system (10.21)­(10.23) for any b and H1 . We find the explicit form of the additional first integral of the third-order system (10.21)­(10.23) . First, we transform the invariant relation (10.34) for u1 = 0 as follows: u2 - b + H1 2(1 + bH1 )
2

+ u1 -

C1 2(1 + bH1 )

2

=

2 (b - H1 )2 + C1 - 4 . 4(1 + bH1 )2

(10.40)

We see that the parameters of this invariant relation must satisfy the condition
2 (b - H1 )2 + C1 - 4 0,

(10.41)

and the phase space of system (10.21)­(10.23) is stratified into the family of surfaces defined by Eq. (10.40). Thus, due to relation (10.34) , the first equation of system (10.31) has the form where U1 (C1 ,u2 ) = U2 (C1 ,u2 ) = 2(1 + bH1 )u2 - 2(b + H1 )u2 +2 - C1 U1 (C1 ,u2 ) du2 2 = , d b - (1 + bH1 )u2 1 {C1 ± U2 (C1 ,u2 )}, 2(1 + bH1 )
2 C1 - 4(1 + bH1 ) 1 - (b + H1 )u2 +(1+ bH1 )u2 , 2

(10.42)

and the integration constant C1 is defined by condition (10.41) . Therefore, the quadrature for the search for an additional first integral of system (10.21) ­(10.23) b ecomes (b - (1 + bH1 )u2 )du2 d = . (10.43) 2(1 - (b + H1 )u2 +(1+ bH1 )u2 ) - C1 {C1 ± U2 (C1 ,u2 )}/(2(1 + bH1 )) 2 Obviously, the left-hand side (up to an additive constant) is equal to ln | sin |. b + H1 = w1 , 2(1 + bH1 ) then the right-hand side of Eq. (10.43) b ecomes u2 - 840 If
2 b2 = (b - H1 )2 + C1 - 4, 1

(10.44) (10.45)


-

1 4

d b2 - 4(1 + bH1 )w 1 b2 - 4(1 + bH1 )w 1
2 1

2 1 2 1

±C

1

b2 - 4(1 + bH1 )w 1

- (b - H1 )(1 + bH1 )

dw1 b2 - 4(1 + bH1 )w 1 1 = - ln 2
2 1

±C

1

b2 - 4(1 + bH1 )w 1

2 1

2 b - H1 b2 - 4(1 + bH1 )w1 1 I1 , (10.46) ±1 ± C1 2

where
2 , w3 = b2 - 4(1 + bH1 )w1 . 1 2 b - w3 (w3 ± C1 ) In the calculation of the integral (10.47), the following three cases are p ossible:

I1 =

dw3

2 1

(10.47)

I. |b - H1 | > 2: I1 = - 1 2 (b - H1 )2 - 4 + 1 2 (b - H
1

ln

(b - H1 )2 - 4+ w3 ± C1 ln

b2 - w 1

2 3

±
2 3

C

1

(b - H1 )2 - 4 C
1

)2

-4

(b - H1 )2 - 4 - w3 ± C1 1

b2 - w 1

(b - H1 )2 - 4

+ const; (10.48)

II. |b - H1 | < 2: I1 = III. |b - H1 | = 2: I1 = Returning to the variable w1 = we have the following final form of I1 : I. |b - H1 | > 2: I1 = - 1 2 (b - H 2
1

4 - (b - H1 )2

arcsin

±C1 w3 + b2 1 + const; b1 (w3 ± C1 )

(10.49)

2 b2 - w3 1 +const . C1 (w3 ± C1 )

(10.50)

z2 b + H1 - , sin 2(1 + bH1 )

(10.51)

)2

-4

ln

(b - H1 )2 - 4 ± 2(1 + bH1 )w1 b - 4(1 + bH
2 1 2 1 1

)2

w ±C
2 1

2 1

1

±

C

1

(b - H1 )2 - 4 C
1

+

1 (b - H
1

)2

-4

ln

(b - H1 )2 - 4 2(1 + bH1 )w1 b - 4(1 + bH ±C b
1 1

)2

w ±C

1

(b - H1 )2 - 4

+ const; (10.52)

II. |b - H1 | < 2: I1 = III. |b - H1 | = 2: I1 = 2(1 + bH1 )w1 C
1 2 b - 4(1 + bH1 )2 w1 ± C 2 1 1

1 4 - (b - H1 )2

arcsin

1

2 b2 - 4(1 + bH1 )2 w1 + b2 1 1 2 b2 - 4(1 + bH1 )2 w1 ± C 1 1

+ const;

(10.53)

+const .

(10.54)

Thus, we have found an additional first integral for the third-order system (10.21) ­(10.23) and we have the complete set of first integrals that are transcendental functions of their phase variables. 841


Remark 10.2. Formally, in the expression of the first integral found, we must substitute for C1 the left-hand side of the first integral (10.34). Then the obtained additional first integral has the following structure (similar to the transcendental first integral from planar dynamics): z3 z , = C2 = const . (10.55) ln | sin | + G2 sin , sin sin Thus, to integrate the sixth-order system (10.21)­(10.26), we have already found two indep endent first integrals. As was mentioned ab ove, to integrate it completely, it suffices to find one first integral for the (p otentially separated) system (10.24) , (10.25) and an additional first integral that "attaches" Eq. (10.26). To find a first integral of the (p otentially separated) system (10.24), (10.25), we establish a corresp ondence b etween it and the following nonautonomous first-order equation:
2 1+ z cos 1 dz = . d1 z sin 1

(10.56)

After integration we obtain the required invariant relation
2 1+ z = C3 = const, sin 1

(10.57)

which in terms of the variables z1 and z2 has the form
2 2 z1 + z2 = C3 = const . z1 sin 1

(10.58)

Further, to obtain an additional first integral that "attaches" Eq. (10.26) , we establish a corresp ondence b etween Eqs. (10.26) and (10.24) and the following nonautonomous equation: dz 2 = - 1+ z cos 1 . d2 Since C3 cos 1 = ± by (10.57), we have 1 dz 2 2 2 = 1+ z C3 - 1 - z . d2 C3 Integrating this relation, we arrive at the following quadrature: (2 + C4 ) = Integration leads to the relation tan(2 + C4 ) = C3 z
2 2 C3 - 1 - z 2 2 C3 - 1 - z

(10.59)

(10.60)

(10.61)

C3 dz (1 +
2 z

)

2 2 C3 - 1 - z

,

C4 = const .

(10.62)

,

C4 = const .

(10.63)

Expressed in terms of the variables z1 and z2 this invariant relation has the form tan(2 + C4 ) = C3 z2
2 2 2 C3 - 1 z1 - z2

,

C4 = const .

(10.64)

Finally, we have the following additional first integral that "attaches" Eq. (10.26) : arctan 842 C3 z
2 C - 1 - z 2 3

± 2 = C4 ,

C4 = const

(10.65)


or arctan
2 3

C3 z2
2 2 C - 1 z1 - z2

± 2 = C4 ,

C4 = const .

(10.66)

Thus, in the case considered, the system of dynamical equations (8.3)­(8.6), (8.9)­(8.14) under condition (10.5) has eight invariant relations: the analytic nonintegrable constraint of the form (8.19), the cyclic first integrals of the form (8.17) and (8.18) , the first integral of the form (10.35) , the first integral expressed by relations (10.48)­(10.55) , which is a transcendental function of the phase variables (in the sense of complex analysis) expressed through a finite combination of functions, and the transcendental first integrals of the form (10.57) (or (10.58)) and (10.65) (or (10.66)). Theorem 10.1. System (8.3)­(8.6), (8.9)­(8.14) under conditions (8.19), (10.5), and (8.18) possesses eight invariant relations (complete set ); four of them are transcendental functions from the point of view of complex analysis. Al l the relations are expressed through finite combinations of elementary functions. Top ological analogies. Consider the following fifth-order system: sin ¨ = 0, +(b - H1 ) cos +sin cos - 1 2 + 2 2 sin2 1 cos 2 1+cos - 2 2 sin 1 cos 1 = 0, 1 +(b - H1 )1 cos + 1 ¨ (10.67) cos sin 2 1+cos +21 2 cos 1 = 0, 2 +(b - H1 )2 cos + 2 ¨ cos sin cos 1 where b > 0 and H1 > 0. This system describ es a fixed four-dimensional p endulum in a flow of a running medium for which the moment of forces dep ends on the angular velocity, i.e., a mechanical system in a nonconservative field (see [14, 15, 150]). Generally sp eaking, the order of this system must b e equal to 6, but the phase variable 2 is a cyclic variable, which leads to the stratification of the phase space and reduces the order of the system. The phase space of this system is the tangent bundle TS
3

10.4.

, 1 , 2 , ,1 ,2

(10.68)

of the three-dimensional sphere S3 {, 1 ,2 }. The equation that transforms system (9.64) into a system on the tangent bundle of the two-dimensional sphere 2 0 and the equations of great circles 1 0, 2 0 (10.70) define families of integral manifolds. It is easy to verify that system (10.67) is equivalent to a dynamical system with variable dissipation with zero mean on the tangent bundle (10.68) of the three-dimensional sphere. Moreover, the following theorem holds. Theorem 10.2. System (8.3)­(8.6) , (8.9)­(8.14) under conditions (8.19), (10.5), and (8.18) is equivalent to the dynamical system (10.67) . Proof. Indeed, it suffices to set = , 1 = 1 , 2 = 2 , b = -b , and H1 = -H1 . On more general top ological analogies, see [91]. (10.69)

843


Chapter 3 CASES OF INTEGRABILITY CORRESPONDING TO THE MOTION OF A RIGID BODY IN FOUR-DIMENSIONAL SPACE, I I
In this chapter, we systematize results, b oth new results and results obtained earlier, concerning the study of the equations of motion of an axisymmetric four-dimensional (4D) rigid b ody in a field of nonconservative forces. These equations are taken from the dynamics of realistic rigid b odies of lesser dimension that interact with a resisting medium by laws of jet flow when the b ody is sub jected to a nonconservative tracing force such that throughout the motion the center of mass of the b ody moves rectilinearly and uniformly; this means that in the system there exists a nonconservative couple of forces (see [5, 31, 36, 46, 53, 71, 77, 81, 88, 139, 152]). Earlier, in [42, 81] the author proved the complete integrability of the equations of plane-parallel motion of a b ody in a resisting medium under the conditions of jet flow in the case where the system of dynamical equations p ossesses a first integral which is a transcendental (in the sense of the theory of functions of a complex variable) function of quasi-velocities. It was assumed that the interaction of the b ody with the medium is concentrated on a part of the surface of the b ody that has the form of a (one-dimensional) plate. Subsequently (see [76, 77, 95]), the planar problem was generalized to the spatial (three-dimensional) case, where the system of dynamical equations p ossesses a complete set of transcendental first integrals. In this case, it was assumed that the interaction of the medium with the b ody is concentrated on a part of the surface of the b ody that has the form of a planar (two-dimensional) disk. In this chapter, we discuss results, b oth new results and results obtained earlier, concerning the case where the interaction of the medium with the b ody is concentrated on a part of the surface of the b ody that has the form of a three-dimensional disk and the force acts in the direction p erp endicular to the disk. We systematize these results and formulate them in invariant form. We also introduce an additional dep endence of the moment of a nonconservative force on the angular velocity; this dep endence can b e generalized to motion in higher-dimensional spaces. 11. General Problem of Motion Under a Tracing Force

Consider the motion of a homogeneous, dynamically symmetric (case (7.1)), rigid b ody with "front end face" (a three-dimensional disk interacting with a medium that fills four-dimensional space) in the field of a resistance force S under quasi-stationarity conditions (see [16, 17, 30, 35, 36, 42, 43, 89, 108, 126, 145, 152]. Let (v, , 1 ,2 ) b e the (generalized) spherical coordinates of the velocity vector of the center D of the three-dimensional disk lying on the axis of symmetry of the b ody, let 0 -6 5 -3 0 -4 2 = 6 -5 4 0 -1 3 -2 1 0 b e the tensor of angular velocity of the b ody, and let Dx1 x2 x3 x4 b e the coordinate system attached to the b ody such that the axis of symmetry CD coincides with the axis Dx1 (recall that C is the center of mass), the axes Dx2 , Dx3 , and Dx4 lie in the hyp erplane of the disk, while I1 , I2 , I3 = I2 , I4 = I2 , and m are the characteristics of inertia and mass. 844


We adopt the following expansions in pro jections onto the axes of the coordinate system Dx1 x2 x3 x4 : DC = {-, 0, 0, 0}, vD = v cos , v sin cos 1 , v sin sin 1 cos 2 , v sin sin 1 sin 2 . (11.1)

In the case (7.1) we additionally have an expansion for the function of the influence of the medium on the four-dimensional b ody: S = {-S, 0, 0, 0} (11.2) i.e., in this case F = S. Then the set of dynamical equations of motion of the b ody (including the Chaplygin analytic functions, [16, 17], see b elow) that describ es the motion of the center of mass and corresp onds to the space R4 , in which the tangent forces of the influence of the medium on the three-dimensional disk vanish, takes the form v cos - v sin - 6 v sin cos 1 + 5 v sin sin 1 cos 2 - 3 v sin sin 1 sin 2
2 2 + 6 + 5 + 2 3

=-

S , m

(11.3)

v sin cos 1 + v cos cos 1 - 1 v sin sin 1 + 6 v cos - 4 v sin sin 1 cos 2 +2 v sin sin 1 sin 2 - (4 5 + 2 3 ) - 6 = 0, v sin sin 1 cos 2 + v cos sin 1 cos 2 + 1 v sin cos 1 cos 2 - 2 v sin sin 1 sin 2 -5 v cos + 4 v sin cos 1 - 1 v sin sin 1 sin 2 - (-1 2 + 4 6 )+ 5 = 0, v sin sin 1 sin 2 + v cos sin 1 sin 2 + 1 v sin cos 1 sin 2 + 2 v sin sin 1 cos 2 +3 v cos - 2 v sin cos 1 + 1 v sin sin 1 cos 2 + (2 6 + 1 5 ) - 3 = 0, where S = s()v 2 , = CD , v > 0.

(11.4)

(11.5)

(11.6)

(11.7)

Further, the auxiliary matrix (7.11) for the calculation of the moment of the resistance force takes the form 0 x2N x3N x4N ; (11.8) -S 0 0 0 then the set of dynamical equations that describ es the motion of the b ody ab out the center of mass and corresp onds to the Lie algebra so(4) takes the form (4 + 3 ) 1 +(3 - 4 )(3 5 + 2 4 ) = 0, (2 + 4 ) 2 +(2 - 4 )(3 6 - 1 4 ) = 0, (4 + 1 ) 3 +(4 - 1 )(2 6 + 1 5 ) = x
4N

(11.9) (11.10) , 1 ,2 , v v s()v 2 , (11.11) (11.12)
3N

(3 + 2 ) 4 +(2 - 3 )(5 6 + 1 2 ) = 0, (1 + 3 ) 5 +(3 - 1 )(4 6 - 1 3 ) = -x (1 + 2 ) 6 +(1 - 2 )(4 5 + 2 3 ) = x , 1 ,2 , , 1 ,2 , v s()v 2 , s()v 2 .

(11.13) (11.14)

2N

845


Thus, the phase space of the tenth-order system (11.3)­(11.6) , (11.9) ­(11.14) is the direct product of the four-dimensional manifold and the Lie algebra so(4): R1 â S3 â so(4). Note that system (11.3)­(11.6) , (11.9)­(11.14) , due to the existing dynamical symmetry I2 = I3 = I4 , p ossesses the cyclic first integrals
0 1 1 = const, 0 2 2 = const, 0 4 4 = const .

(11.15)

(11.16)

(11.17)

Henceforth, we will consider the dynamics of the system on zero levels:
0 0 0 1 = 2 = 4 = 0.

(11.18)

If we consider a more general problem on the motion of a b ody under a tracing force T lying on the straight line CD = Dx1 that assumes that throughout the motion the condition VC const (11.19)

(here VC is the velocity of the center of mass, see also [91]) is satisfied, then system (11.3)­(11.6), (11.9)­(11.14) contains zero instead of Fx , since a nonconservative couple of forces acts on the b ody: T - s()v 2 0, = DC. (11.20)

For this purp ose, obviously, we must select the value of the tracing force T in the form T = Tv (, ) = s()v 2 , T - S. (11.21)

The case (11.21) of the choice of the value T of the tracing force is a particular case of the separation of an indep endent fifth-order subsystem after a certain transformation of the sixth-order system (11.3)­ (11.6), (11.9)­(11.14) . Indeed, let the following condition for T hold:
4

T = Tv (, 1 ,2 , ) =
i,j =0, i j



i,j

, 1 ,2 ,

v

i j = T1 , 1 ,2 ,

v

v2 ,

0 = v.

(11.22)

We introduce new quasi-velocities into the system. For this purp ose, we transform 3 , 5 , and 6 by a comp osition of two rotations: 3 z1 z2 = T1 (-1 ) T3 (-2 ) 5 , (11.23) z3 6 where 1 0 0 T1 (1 ) = 0 cos 1 - sin 1 , 0 sin 1 cos 1 z1 = 3 cos 2 + 5 sin 2 , z2 = -3 cos 1 sin 2 + 5 cos 1 cos 2 + 6 sin 1 , z3 = 3 sin 1 sin 2 - 5 sin 1 cos 2 + 6 cos 1 . 846 (11.25) cos 2 - sin 2 0 T3 (2 ) = sin 2 cos 2 0 . 0 0 1

(11.24)

Thus, the following relations hold:


System (11.3)­(11.6), (11.9)­(11.14) in the cases (11.16)­(11.18) and (11.22) can b e rewritten in the form
2 2 2 v + z1 + z2 + z3 cos -

v2 s()sin · 2I2
v

v

, 1 ,2 ,
2

v

=

T1 , 1 ,2 ,

v 2 - s()v

(11.26)

m v2 s()cos · 2I2
v

cos , v

2 2 2 v + z3 v - z1 + z2 + z3 sin -

, 1 ,2 ,
v

s()v 2 - T1 , 1 ,2 , = m v s() · 1 sin - z2 cos - 2I2
v

v

2

(11.27)

sin ,

, 1 ,2 ,

v

= 0, v

(11.28)

v s() · 2 sin sin 1 + z1 cos - 2I2 3 = v2 x 2I2
4N

v

, 1 ,2 ,

= 0,

(11.29)

, 1 ,2 ,

v v

s(),

(11.30)

5 = -

v2 x 2I2

3N

, 1 ,2 ,

s(),

(11.31)

6 =

v2 x 2I2

2N

, 1 ,2 ,

v

s().

(11.32)

Introducing new dimensionless phase variables and a new differentiation by the formulas zk = n1 vZk , k = 1, 2, 3, = n1 v , n1 > 0, n1 = const, (11.33)

we reduce system (11.26)­(11.32) to the following form: v = v (, 1 ,2 ,Z ), = -Z3 + n
1 2 2 2 Z1 + Z2 + Z3 sin +

(11.34) s()cos · v (, 1 ,2 ,n1 Z ) 2I2 n1

T1 (, 1 ,2 ,n1 Z ) - s() sin , - mn1 Z3 =

(11.35)

s() 2 2 cos 2 · v (, 1 ,2 ,n1 Z ) - Z1 + Z2 sin 2I2 n1 s() s() · v (, 1 ,2 ,n1 Z )+ · v (, 1 ,2 ,n1 Z ) Z2 Z1 - 2I2 n1 sin 2I2 n1 sin - Z3 · (, 1 ,2 ,Z ) ,

(11.36)

847


Z2 = -

s() cos 2 cos cos 1 2 · v (, 1 ,2 ,n1 Z )+ Z2 Z3 sin + Z1 sin sin 2I2 n1 1 s() s() · v (, 1 ,2 ,n1 Z ) - · v (, 1 ,2 ,n1 Z ) + Z3 Z1 2I2 n1 sin 2I2 n1 sin - Z2 · (, 1 ,2 ,Z ) , s() cos cos cos 1 - Z1 Z2 · v (, 1 ,2 ,n1 Z )+ Z1 Z3 sin sin sin 1 2I2 n2 1 s() - · v (, 1 ,2 ,n1 Z ) · Z3 sin 1 - Z2 cos 1 2I2 n1 sin sin 1 - Z1 · (, 1 ,2 ,Z ) , cos s() + · v (, 1 ,2 ,n1 Z ) , sin 2I2 n1 sin cos s() + · v (, 1 ,2 ,n1 Z ) , sin sin 1 2I2 n1 sin sin 1
1 2 2 2 Z1 + Z2 + Z3 cos +

(11.37)

Z1 =

(11.38)

1 = Z2

(11.39)

2 = -Z1 where

(11.40)

(, 1 ,2 ,Z ) = -n + v

T1 (, 1 ,2 ,n1 Z ) - s() cos , mn1 , 1 ,2 , v

s()sin · v (, 1 ,2 ,n1 Z ) 2I2 n1

(11.41)



v

, 1 ,2 ,

=x

4N

sin 1 sin 2 + x cos 1 ,

3N

, 1 ,2 ,

v

sin 1 cos 2 (11.42)

+x v

2N

, 1 ,2 , v , 1 ,2 , v



v

, 1 ,2 ,

=x

4N

cos 1 sin 2 + x sin 1 , cos 2 - x

3N

, 1 ,2 ,

v

cos 1 cos 2 (11.43)

-x v

2N

, 1 ,2 , v , 1 ,2 , v



v

, 1 ,2 ,

=x

4N

3N

, 1 ,2 ,

v

sin 2 .

(11.44)

We see that the seventh-order system (11.34) ­(11.40) contains an indep endent sixth-order subsystem (11.35) ­(11.40) , which can b e separately examined in its own six-dimensional phase space. In particular, this method of separation of an indep endent sixth-order subsystem can also b e applied under condition (11.21) . Here and in what follows, the dep endence on the group of variables (, 1 ,2 , /v ) is meant as a comp osite dep endence on (, 1 ,2 ,z1 /v , z2 /v , z3 /v ) (and further of (, 1 ,2 ,n1 Z1 ,n1 Z2 ,n1 Z3 )) due to (11.25) and (11.33).

848


12.

Case Where the Moment of a Nonconservative Force Is Indep endent of the Angular Velo city

12.1. Reduced system. As in the choice of Chaplygin analytic functions (see [16, 17]), we select the dynamical functions s, x2N , x3N , and x4N in the following form: s() = B cos , x x x
2N

3N

4N

v , 1 ,2 , v , 1 ,2 , v , 1 ,2 ,

=x =x =x

2N 0

(, 1 ,2 ) = A sin cos 1 , (, 1 ) = A sin sin 1 cos 2 , (, 1 ,2 ) = A sin sin 1 sin 2 , (12.1)

3N 0

4N 0

where A, B > 0 and v = 0. We see that in the system considered, the moment of nonconservative forces is indep endent of the angular velocity and dep ends only on the angles , 1 , and 2 . The functions v (, 1 ,2 , /v ), v (, 1 ,2 , /v ), and v (, 1 ,2 , /v ) in system (11.34)­(11.40) have the following form:
v

, 1 ,2 ,

v

= A sin ,



v

, 1 ,2 ,

v



v

, 1 ,2 ,

v

0.

(12.2)

Then, due to conditions (11.19) and (12.1), the transformed dynamical part of the equations of motion (system (11.34) ­(11.40) ) b ecomes the analytic system v = v (, 1 ,2 ,Z ), = -Z3 + b Z3 = Z2 = Z1 = 1 = 2 = where
2 2 2 (, 1 ,2 ,Z ) = -b Z1 + Z2 + Z3 cos + b sin2 cos

(12.3) +
2 Z3 2

sin + b sin cos , 2 2 cos 2 2 2 + bZ3 Z1 + Z2 + Z3 cos - bZ3 sin2 cos , sin cos - Z1 + Z2 sin cos 2 cos cos 1 2 2 2 + Z1 Z2 Z3 + bZ2 Z1 + Z2 + Z3 cos - bZ2 sin2 cos , sin sin sin 1 cos cos cos 1 2 2 2 Z1 Z3 - Z1 Z2 + bZ1 Z1 + Z2 + Z3 cos - bZ1 sin2 cos , sin sin sin 1 cos Z2 , sin cos -Z1 , sin sin 1

2 Z1

+

2 Z2

(12.4) (12.5) (12.6) (12.7) (12.8) (12.9)

and the dimensionless parameter b and the constant n1 are chosen as follows: b = n0 , n2 = 0 AB , 2I2 n1 = n0 . (12.10)

Thus, system (12.3)­(12.9) can b e considered on its own seven-dimensional phase manifold W1 = R1 {v }â T S3 Z1 , Z2 , Z3 , 0 < < , 0 < 1 < , 0 2 < 2 , + (12.11)

i.e., on the direct product of the numb er half-line and the tangent bundle of the three-dimensional sphere S3 0 < < , 0 < 1 < , 0 2 < 2 . We see that the seven-dimensional system (12.3) ­(12.9) contains the indep endent sixth-order system (12.4)­(12.9) on its own six-dimensional manifold. 849


For the complete integration of system (12.3) ­(12.9) we need, in general, six indep endent first integrals. However, after the change of variables Z1 Z2 Z , Z Z=
2 2 Z1 + Z2 ,

Z = Z2 /Z1 ,

(12.12)

system (12.4)­(12.9) splits as follows:
2 = -Z3 + b Z 2 + Z3 sin + b sin cos2 , cos 2 Z3 = sin cos - Z 2 + bZ3 Z 2 + Z3 cos - bZ3 sin2 cos , sin cos 2 Z = ZZ3 + bZ Z 2 + Z3 cos - bZ sin2 cos , sin cos cos 1 2 Z = (±)Z 1+ Z , sin sin 1 ZZ cos , 1 = (±) 2 1+ Z sin

(12.13) (12.14) (12.15) (12.16) (12.17) (12.18) sysdent sufs ys two-

2 = ()

cos . 1+ Z sin sin 1
2

Z

We see that the sixth-order system also splits into indep endent subsystems of lower orders: tem (12.13)­(12.15) of order 3 and system (12.16), (12.17) (after the change of the indep en variable) of order 2. Thus, for the complete integrability of system (12.3), (12.13)­(12.18) it fices to sp ecify two indep endent first integrals of system (12.13) ­(12.15) , one first integral of tem (12.16) , (12.17) , and two additional first integrals that "attach" Eqs. (12.18) and (12.3) . Note that system (12.13)­(12.15) can b e considered on the tangent bundle T S2 of the dimensional sphere S2 .

12.2. Complete list of first integrals. System (12.13)­(12.15) has the form of a system that app ears in the dynamics of a three-dimensional (3D) rigid b ody in a nonconservative field. Note that, by (11.19), the value of the velocity of the center of mass is a first integral of system (11.26)­(11.32) (under condition (11.21)); namely, the function of phase variables 0 (v, , 1 ,2 ,z1 ,z2 ,z3 ) = v 2 +
2 2 2 2 z1 + z2 + z3 - 2z3 v sin = V 2 C

(12.19)

is constant on phase tra jectories of the system (here z1 , z2 , and z3 are chosen due to (11.25) ). Due to a nondegenerate change of the indep endent variable (for v = 0), system (12.3), (12.13)­ (12.18) also p ossesses an analytic integral, namely, the function of phase variables 1 (v, , 1 ,2 ,Z,Z ,Z3 ) = v
2 2 1+ b2 Z 2 + Z3 - 2bZ3 sin = V 2 C

(12.20)

is constant on phase tra jectories of the system. Equality (12.20) allows one to find the dep endence of the velocity of the characteristic p oint of the rigid b ody (the center D of the disk) on the other phase variables without solving system (12.3), (12.13)­(12.18); namely, for VC = 0 we have the relation v2 = Since the phase space W2 = R1 {v }â T S3 Z, Z , Z3 , 0 < < , 0 < 1 < , 0 2 < 2 + 850 (12.22) 1+ b2
2 VC . 2 Z 2 + Z3 - 2bZ3 sin

(12.21)


of system (12.3) , (12.13) ­(12.18) has dimension 7 and contains asymptotic limit sets, Eq. (12.20) defines a unique analytic (even continuous) first integral of system (12.3), (12.13)­(12.18) in the whole phase space (see [3, 4, 8, 11, 38, 39, 56, 69, 91]). We examine the existence of other (additional) first integrals of system (12.3), (12.13) ­(12.18) . Its phase space is stratified into surfaces (v, , 1 ,2 ,Z,Z ,Z3 ) W2 : VC = const ; (12.23)

the dynamics on these surfaces is determined by the first integrals of system (12.3), (12.13)­(12.18) . First, we establish a corresp ondence b etween the indep endent third-order subsystem (12.13) ­(12.15) and the nonautonomous second-order system
2 sin cos + bZ3 Z 2 + Z3 cos - bZ3 sin2 cos - Z 2 cos / sin dZ3 = , 2 d -Z3 + b Z 2 + Z3 sin + b sin cos2 2 bZ Z 2 + Z3 cos - bZ sin2 cos + ZZ3 cos / sin dZ = . 2 d -Z3 + b Z 2 + Z3 sin + b sin cos2

(12.24)

Applying the substitution = sin , we rewrite system (12.24) in algebraic form:
2 + bZ3 Z 2 + Z3 - bZ3 2 - Z 2 / dZ3 = , 2 d -Z3 + b (1 - 2 )+ b Z 2 + Z3 2 bZ Z 2 + Z3 - bZ 2 + ZZ3 / dZ = . 2 d -Z3 + b (1 - 2 )+ b Z 2 + Z3

(12.25)

Further, introducing the homogeneous variables by the formulas Z = u1 , we reduce system (12.25)) to the following form: 1 - bu2 2 + bu2 u2 + u2 2 - u2 du2 1 2 1 + u2 = , d -u2 + b 2 u2 + u2 + b (1 - 2 ) 1 2 Z3 = u2 , (12.26)

bu1 u2 + u2 2 - bu1 2 + u1 u2 du1 1 2 + u1 = , 2 u2 + u2 + b (1 - 2 ) d -u2 + b 1 2 which is equivalent to 1 - bu2 + u2 - u2 du2 2 1 = , d -u2 + b 2 u2 + u2 + b (1 - 2 ) 1 2 du1 2u1 u2 - bu1 = . 2 u2 + u2 + b (1 - 2 ) d -u2 + b 1 2

(12.27)

(12.28)

We establish a corresp ondence b etween the second-order system (12.28) and the nonautonomous first-order equation 1 - bu2 + u2 - u2 du2 2 1 = , (12.29) du1 2u1 u2 - bu1 which is easily transformed to exact-differential form: d Thus, Eq. (12.29) has the first integral u2 + u2 - bu2 +1 2 1 = C1 = const, u1 (12.31) 851 u2 + u2 - bu2 +1 2 1 u1 = 0. (12.30)


which in terms of the previous variables has the form
2 Z3 + Z 2 - bZ3 sin +sin2 = C1 = const . Z sin

(12.32)

Remark 12.1. Consider system (12.13)­(12.15) with variable dissipation with zero mean (see [91]), which b ecomes conservative for b = 0: = -Z3 , Z3 = sin cos - Z
2

cos . Z = ZZ3 sin This system p ossesses two analytic first integrals of the form
2

cos , sin

(12.33)

2 Z3 + Z 2 +sin2 = C1 = const,

(12.34) (12.35)

Z sin = C = const .

Obviously, the ratio of the two first integrals (12.34) and (12.35) is also a first integral of system (12.33). However, for b = 0, none of the functions
2 Z3 + Z 2 - bZ3 sin +sin2

(12.36)

and (12.35) is a first integral of system (12.13)­(12.15) , but their ratio is a first integral system (12.13)­ (12.15) for any b. Further, we find an additional first integral of the third-order system (12.13)­(12.15). First, we transform the invariant relation (12.31) for u1 = 0 as follows: u2 - b 2
2

+ u1 -

C1 2

2

=

2 b2 + C1 - 1. 4

(12.37)

We see that the parameters of this invariant relation must satisfy the condition
2 b2 + C1 - 4 0,

(12.38)

and the phase space of system (12.13)­(12.15) is stratified into a family of surfaces defined by (12.37). Thus, due to relation (12.31) , the first equation of system (12.28) takes the form where 1 2 C1 ± C1 - 4 u2 - bu2 +1 (12.40) 2 2 and the integration constant C1 is defined by condition (12.38), or the form of the Bernoulli equation: U1 (C1 ,u2 ) =
2 (b - u2 ) - b 3 1 - U1 (C1 ,u2 ) - u2 d 2 . = 2 du2 1 - bu2 + u2 - U1 (C1 ,u2 ) 2 2 1 - bu2 + u2 - U1 (C1 ,u2 ) du2 2 = , 2 d -u2 + b (1 - 2 )+ b 2 U1 (C1 ,u2 )+ u2 2

(12.39)

(12.41)

Using (12.40), we can transform Eq. (12.41) into the form of a nonhomogeneous linear equation:
2 2(u2 - b)p +2b 1 - U1 (C1 ,u2 ) - u2 dp 2 , = 2 du2 1 - bu2 + u2 - U1 (C1 ,u2 ) 2

p=

1 . 2

(12.42)

This means that we can find another transcendental first integral in explicit form (i.e., in the form of a finite combination of quadratures). Moreover, the general solution of Eq. (12.42) dep ends on 852


an arbitrary constant C2 . We omit complete calculations but note that the general solution of the homogeneous linear equation obtained from (12.42) in the particular case b = C1 = 2 has the form p = p0 (u2 ) = C 1 - (u2 - 1)2 ± 1 exp 1 1± 1 - (u2 - 1)2 1 - (u2 - 1)2 , C = const . (12.43)

Remark 12.2. Formally, in the expression of the first integral thus found, we must substitute for C1 the left-hand side of the first integral (12.31). Then the obtained additional first integral has the following structure (similar to the transcendental first integral from planar dynamics): z3 z , = C2 = const . (12.44) ln | sin | + G2 sin , sin sin Thus, for integration of the sixth-order system (12.13)­(12.18) we already have two indep endent first integrals. For the complete integration, it suffices to find one first integral for the (p otentially separated) system (12.16) , (12.17) and an additional first integral that "attaches" Eq. (12.18) . To find a first integral of the (p otentially separated) system (12.16), (12.17), we establish a corresp ondence b etween it and the following nonautonomous first-order equation:
2 1+ Z cos 1 dZ = . d1 Z sin 1 After integration we obtain the required invariant relation 2 1+ Z = C3 = const; sin 1 in terms of the variables Z1 and Z2 it has the form 2 2 Z1 + Z2 = C3 = const . Z1 sin 1

(12.45)

(12.46)

(12.47)

Further, to find an additional first integral that "attaches" Eq. (12.18), we establish a corresp ondence b etween Eqs. (12.18) and (12.16) and the following nonautonomous equation: dZ 2 = - 1+ Z cos 1 . d2 Since, due to (12.46),
2 2 C3 cos 1 = ± C3 - 1 - Z ,

(12.48)

(12.49) (12.50)

1 dZ 2 2 2 = C3 - 1 - Z . 1+ Z d2 C3 Integrating this relation, we obtain the following quadrature: C3 dZ , C4 = const . (2 + C4 ) = 2 2) 2 (1 + Z C3 - 1 - Z Another integration leads to the relation tan(2 + C4 ) = C3 Z
2 C - 1 - Z 2 3

we have

(12.51)

,

C4 = const .

(12.52)

In the variables Z1 and Z2 , this invariant relation has the form C3 Z2 , C4 = const . tan(2 + C4 ) = 2 2 2 C3 - 1 Z1 - Z2

(12.53)

853


Finally, we have the following form of the additional first integral that "attaches" Eq. (12.18) : arctan or arctan C3 Z2
2 2 2 C3 - 1 Z1 - Z2

C3 Z
2 C - 1 - Z 2 3

± 2 = C4 , ± 2 = C4 ,

C4 = const,

(12.54)

C4 = const .

(12.55)

Thus, in the case considered the system of dynamical equations (11.3)­(11.6) , (11.9)­(11.14) under condition (12.1) has eight invariant relations: the analytic nonintegrable constraint of the form (11.19) corresp onding to the analytic first integral (12.19) , the cyclic first integrals of the form (11.17) and (11.18) , the first integral of the form (12.32) . Moreover, there exists the first integral that can b e found from Eq. (12.42) ; it is a transcendental function of phase variables (in the sense of complex analysis). Finally, we have the transcendental first integrals of the form (12.46) (or (12.47)) and (12.54) (or (12.55)). Theorem 12.1. System (11.3)­(11.6) , (11.9)­(11.14) under conditions (11.19) , (12.1), (11.18) , and (11.17) possesses eight invariant relations (complete set ), four of which are transcendental functions (from the point of view of complex analysis ). Moreover, seven of these eight relations are expressed through finite combinations of elementary function. 12.3. ogy. Top ological analogies. We show that there exists another mechanical and top ological anal-

Theorem 12.2. The first integral (12.32) of system (11.3) ­(11.6), (11.9)­(11.14) under conditions (11.19) , (12.1), (11.18) , and (11.17) is constant on phase trajectories of system (9.10)­(9.15) . Proof. Indeed, the first integral (12.32) can b e obtained by a change of coordinates by means of (12.31), whereas the first integral (9.31) can b e obtained by a change of coordinates by means of (9.30). But relations (12.31) and (9.30) coincide. The theorem is proved. Thus, we have the following top ological and mechanical analogies in the sense explained ab ove: (1) motion of a free rigid b ody in a nonconservative field with a tracing force (under a nonintegrable constraint); (2) motion of a fixed physical p endulum in a flow of a running medium (a nonconservative field); (3) rotation of a rigid b ody ab out the center of mass, which, in turn, moves rectilinearly and uniformly in a nonconservative field. For more general top ological analogies, see also [91]. 13. Case Where the Moment of a Nonconservative Force Dep ends on the Angular Velo city

13.1. Intro duction of the dep endence on the angular velo city and the reduced system. In this chapter, we continue to study the dynamics of a four-dimensional rigid b ody in four-dimensional space. The present section, like the analogous section of Chap. 2, is devoted to the study of motion in the case where the moment of forces dep ends on the tensor of angular velocity. Thus, we introduce this dep endence as in the previous chapter. This also allows us to introduce this dep endence for multi-dimensional b odies. Let x = (x1N ,x2N ,x3N ,x4N ) b e the coordinates of the application p oint N of the nonconservative force (influence of the medium) on the three-dimensional disk and let Q = (Q1 ,Q2 ,Q3 ,Q4 ) b e the comp onents of the force S of the influence of the medium indep endent of the tensor of angular velocity. 854


We consider only the linear dep endence of the function (x1N ,x2N ,x3N ,x4N ) on the tensor of angular velocity since this introduction itself is not obvious (see [14, 15]). We adopt the following dep endence: x = Q + R, (13.1) where R = (R1 ,R2 ,R3 ,R4 ) is a vector-valued function containing the comp onents of the tensor of angular velocity. The dep endence of the functions R on the comp onents of the tensor of angular velocity is gyroscopic: h1 0 -6 5 -3 R1 R2 1 6 0 -4 2 h2 , R= =- (13.2) R3 0 -1 h3 v -5 4 R4 3 -2 1 0 h4 where (h1 ,h2 ,h3 ,h4 ) are some p ositive parameters (cf. [91]). Since x1N 0, we have 6 5 3 x2N = Q2 - h1 , x3N = Q3 + h1 , x4N = Q4 - h1 . v v v As in to the choice of the Chaplygin analytic functions (see [16, 17]), Q2 = A sin cos 1 , Q3 = A sin sin 1 cos 2 , Q4 = A sin sin 1 sin 2 , where A > 0, and we select the dynamical functions s, x s() = B cos , B > 0, x x x
2N 2N

(13.3)

(13.4) , and x
4N

,x

3N

in the following form:

3N

4N

v , 1 ,2 , v , 1 ,2 , v , 1 ,2 ,

= A sin cos 1 - h

6 , v

h = h1 > 0, v = 0, h = h1 > 0, v = 0, h = h1 > 0, v = 0. (13.5)

5 , v 3 = A sin sin 1 sin 2 - h , v = A sin sin 1 cos 2 + h

This shows that in this problem, there is an additional damping (but accelerating in certain domains of the phase space) moment of a nonconservative force (i.e., there is a dep endence of the moment on the comp onents of the tensor of angular velocity). By the dynamical symmetry of the b ody, h2 = h3 = h4 . The functions v (, 1 ,2 , /v ), v (, 1 ,2 , /v ), and v (, 1 ,2 , /v ) in system (11.35)­ (11.40) have the following form:
v

v

v

v , 1 ,2 , v , 1 ,2 , v , 1 ,2 ,

= A sin - h z2 , v h = - z1 . v =

h z3 , v (13.6)

By conditions (11.19) and (13.5), the transformed dynamical part of the equations of motion (system (11.34) ­(11.40) ) b ecomes the following analytic system: v = v (, 1 ,2 ,Z ),
2 2 2 = -Z3 + b Z1 + Z2 + Z3 sin + b sin cos2 - bH1 Z3 cos2 ,

(13.7) (13.8) 855


2 2 Z3 = sin cos - (1 + bH1 ) Z1 + Z2 2 + bH1 Z3 sin cos - H1 Z3 cos ,

cos 2 2 2 + bZ3 Z1 + Z2 + Z3 cos - bZ3 sin2 cos sin

(13.9)

Z2 = (1 + bH1 ) Z2 Z3

cos 2 cos cos 1 2 2 2 +(1+ bH1 ) Z1 + bZ2 Z1 + Z2 + Z3 cos sin sin sin 1 - bZ2 sin2 cos + bH1 Z2 Z3 sin cos - H1 Z2 cos ,

(13.10)

Z1 = (1 + bH1 ) Z1 Z3

cos cos cos 1 2 2 2 - (1 + bH1 ) Z1 Z2 + bZ1 Z1 + Z2 + Z3 cos sin sin sin 1 - bZ1 sin2 cos + bH1 Z1 Z3 sin cos - H1 Z1 cos , cos , sin cos , sin sin 1

(13.11)

1 = (1 + bH1 ) Z2

(13.12) (13.13)

2 = - (1 + bH1 ) Z1 where

2 2 2 (, 1 ,2 ,Z ) = -b Z1 + Z2 + Z3 cos + b sin2 cos - bH1 Z3 sin cos ;

as ab ove, the dimensionless parameters b and H1 and the constant n1 are chosen as follows: b = n0 , n2 = 0 AB , 2I2 H1 = Bh , 2I2 n0 n1 = n0 . (13.14)

Thus, system (13.7)­(13.13) can b e considered on its seven-dimensional phase manifold W1 = R1 {v }â T S3 Z1 , Z2 , Z3 , 0 < < , 0 < 1 < , 0 2 < 2 , + (13.15)

i.e., on the direct product of the numb er half-line and the tangent bundle of the three-dimensional sphere S3 {0 < < , 0 < 1 < , 0 2 < 2 }. We see that the seventh-order system (13.7)­(13.13) contains the indep endent sixth-order system (13.8)­(13.13) on its own six-dimensional manifold. For complete integration of system (13.7)­(13.13) , in general, we need six indep endent first integrals. However, after the change of variables Z1 Z2 Z , Z Z=
2 2 Z1 + Z2 ,

Z = Z2 /Z1 ,

(13.16)

system (13.8)­(13.13) splits as follows:
2 = -Z3 + b Z 2 + Z3 sin + b sin cos2 - bH1 Z3 cos2 ,

(13.17)

Z3 = sin cos - (1 + bH1 ) Z

cos 2 + bZ3 Z 2 + Z3 cos - bZ3 sin2 cos sin 2 + bH1 Z3 sin cos - H1 Z3 cos ,
2

(13.18)

Z = (1 + bH1 ) ZZ3

cos 2 + bZ Z 2 + Z3 cos - bZ sin2 cos + bH1 ZZ3 sin cos - H1 Z cos , sin (13.19) 1+ Z
2

Z = (±)(1 + bH1 ) Z 856

cos cos 1 , sin sin 1

(13.20)


1 = (±)(1 + bH1 ) 2 = ()(1 + bH1 )

cos , 1+ Z sin
2

ZZ

(13.21)

cos . 1+ Z sin sin 1
2

Z

(13.22) sysdent su f s ys two-

We see that the sixth-order system splits into indep endent subsystems of lower orders: tem (13.17)­(13.19) of order 3 and system (13.20), (13.21) (after the change of the indep en variable) of order 2. This, for the complete integrability of system (13.7), (13.17)­(13.22), it fices to sp ecify two indep endent first integrals of system (13.17) ­(13.19) , one first integral of tem (13.20) , (13.21) , and two additional first integrals that "attach" Eqs. (13.22) and (13.7) . Note that system (13.17)­(13.19) can b e considered on the tangent bundle T S2 of the dimensional sphere S2 .

13.2. Complete list of first integrals. System (13.17)­(13.19) has the form of a system of equations that app ears in the dynamics of a three-dimensional (3D) rigid b ody in a nonconservative field. Note that, by (11.19), the value of the velocity of the center of mass is a first integral of system (11.26)­(11.32) (under condition (11.21)); namely, the function of phase variables 0 (v, , 1 ,2 ,z1 ,z2 ,z3 ) = v 2 +
2 2 2 2 z1 + z2 + z3 - 2z3 v sin = V 2 C

(13.23)

is constant on phase tra jectories of this system (the values of z1 , z2 , and z3 are taken by virtue of (11.25) ). Due to the nondegenerate change of the indep endent variable (for v = 0), system (13.7), (13.17)­ (13.22) also p ossesses an analytic integral, namely, the function of phase variables 1 (v, , 1 ,2 ,Z,Z ,Z3 ) = v
2 2 1+ b2 Z 2 + Z3 - 2bZ3 sin = V 2 C

(13.24)

is constant on phase tra jectories of this system. Equality (13.24) allows one to find the dep endence of the velocity of the characteristic p oint of the rigid b ody (the cemter D of the disk) on the other phase variables without solving system (13.7), (13.17)­(13.22); namely, for VC = 0 we have v2 = Since the phase space W2 = R1 {v }â T S3 Z, Z , Z3 , 0 < < , 0 < 1 < , 0 2 < 2 + (13.26) 1+ b2
2 VC . 2 Z 2 + Z3 - 2bZ3 sin

(13.25)

of system (13.7), (13.17)­(13.22) has dimension 7 and contains asymptotic limit sets, we see that Eq. (13.24) determines the unique analytic (even continuous) first integral of system (13.7), (13.17)­ (13.22) on the whole phase space (cf. [38, 91]). We examine the existence of other (additional) first integrals of system (13.7), (13.17) ­(13.22) . Its phase space is stratified into surfaces (v, , 1 ,2 ,Z,Z ,Z3 ) W2 : VC = const ; (13.27)

the dynamics on these surfaces is determined by the first integrals of system (13.7), (13.17)­(13.22) . 857


First, we establish a corresp ondence b etween the indep endent third-order subsystem (13.17) ­(13.19) and the nonautonomous second-order system R2 (, Z, Z3 ) dZ3 = , 2 + Z 2 sin + b sin cos2 - bH Z cos2 d -Z3 + b Z 13 3 R1 (, Z, Z3 ) dZ = , 2 + Z 2 sin + b sin cos2 - bH Z cos2 d -Z3 + b Z 13 3

2 R2 (, Z, Z3 ) = sin cos + bZ3 Z 2 + Z3 cos - bZ3 sin2 cos cos 2 + bH1 Z3 sin cos - H1 Z3 cos , - (1 + bH1 )Z 2 sin 2 R1 (, Z, Z3 ) = bZ Z 2 + Z3 cos - bZ sin2 cos cos +(1+ bH1 )ZZ3 + bH1 ZZ3 sin cos - H1 Z cos . sin Using the substitution = sin , we rewrite system (13.28) in algebraic form 2 2 + bZ3 Z 2 + Z3 - bZ3 2 - (1 + bH1 )Z 2 / + bH1 Z3 - H1 Z3 dZ3 = , 2 d -Z3 + b (1 - 2 )+ b Z 2 + Z3 - bH1 Z3 (1 - 2 ) 2 bZ Z 2 + Z3 - bZ1 2 +(1+ bH1 )ZZ3 / + bH1 ZZ3 - H1 Z dZ = . 2 d -Z3 + b (1 - 2 )+ b Z 2 + Z3 - bH1 Z3 (1 - 2 )

(13.28)

(13.29)

Further, introducing homogeneous variables by the formulas Z = u1 , Z3 = u2 , (13.30)

we transform system (13.29) into the following form: 1 - bu2 2 + bu2 u2 + u2 2 - (1 + bH1 )u2 - H1 u2 + bH1 u2 2 du2 1 2 1 2 + u2 = , d -u2 + b 2 u2 + u2 + b (1 - 2 ) - bH1 u2 (1 - 2 ) 1 2

bu1 u2 + u2 2 - bu1 2 +(1+ bH1 )u1 u2 - H1 u1 + bH1 u1 u2 du1 1 2 + u1 = , d -u2 + b 2 u2 + u2 + b (1 - 2 ) - bH1 u2 (1 - 2 ) 1 2 which is equivalent to 1 - (b + H1 )u2 +(1+ bH1 )u2 - (1 + bH1 )u2 du2 2 1 = , d -u2 + b 2 u2 + u2 + b (1 - 2 ) - bH1 u2 (1 - 2 ) 1 2

(13.31)

du1 2(1 + bH1 )u1 u2 - (b + H1 )u1 = . 2 u2 + u2 + b (1 - 2 ) - bH u (1 - 2 ) d -u2 + b 12 1 2

(13.32)

We establish a corresp ondence b etween the second-order system (13.32) and the first-order nonautonomous equation 1 - (b + H1 )u2 +(1+ bH1 )u2 - (1 + bH1 )u2 du2 2 1 = , (13.33) du1 2(1 + bH1 )u1 u2 - (b + H1 )u1 which can b e easily transformed into exact-differential form: d (1 + bH1 )u2 +(1+ bH1 )u2 - (b + H1 )u2 +1 2 1 u1 = 0. (13.34)

Therefore, Eq. (13.33) has the first integral (1 + bH1 )u2 +(1+ bH1 )u2 - (b + H1 )u2 +1 2 1 = C1 = const, u1 858 (13.35)


which in terms of the previous variables has the form
2 (1 + bH1 )Z3 +(1+ bH1 )Z 2 - (b + H1 )Z3 sin +sin2 = C1 = const . Z sin

(13.36)

Remark 13.1. Consider system (13.17)­(13.19) with variable dissipation with zero mean (see [91]), which b ecomes conservative for b = H1 :
2 = -Z3 + b Z 2 + Z3 sin + b sin cos2 - b2 Z3 cos2 , cos 2 Z3 = sin cos - 1+ b2 Z 2 + bZ3 Z 2 + Z3 cos - bZ3 sin2 cos sin 2 + b2 Z3 sin cos - bZ3 cos , cos 2 + bZ Z 2 + Z3 cos - bZ sin2 cos + b2 ZZ3 sin cos - bZ cos . Z = 1+ b2 ZZ2 sin (13.37) It p ossesses two analytic first integrals

1+ b2

2 Z3 + Z

2

- 2bZ3 sin +sin2 = C1 = const,

(13.38) (13.39)

Z sin = C = const .

2

Obviously, the ratio of the two first integrals (13.38) and (13.39) is also a first integral of system (13.37). However, for b = H1 , neither of the functions
2 (1 + bH1 ) Z3 + Z 2

- (b + H1 )Z3 sin +sin2

(13.40)

and (13.39) is a first integral of system (13.17) ­(13.19) , but their ratio is a first integral of system (13.17)­(13.19) for all b and H1 . We find the explicit form of an additional first integral of the third-order system (13.17)­(13.19) . For this purp ose, we transform the invariant relation (13.35) for u1 = 0 as follows: u2 - b + H1 2(1 + bH1 )
2

+ u1 -

C1 2(1 + bH1 )

2

=

2 (b - H1 )2 + C1 - 4 . 4(1 + bH1 )2

(13.41)

We see that the parameters of this invariant relation must satisfy the condition
2 (b - H1 )2 + C1 - 4 0,

(13.42)

and that the phase space of system (13.17)­(13.19) is stratified into the family of surfaces determined by Eq. (13.41) . Thus, by relation (13.35), the first equation of system (13.32) has the form where U1 (C1 ,u2 ) = 1 2 C1 ±
2 C1 - 4(1 + bH1 ) 1 - (b + H1 )u2 +(1+ bH1 )u2 2 2 1 - (b + H1 )u2 +(1+ bH1 )u2 - (1 + bH1 )U1 (C1 ,u2 ) du2 2 = , 2 (C ,u )+ u2 - bH u (1 - 2 ) d -u2 + b (1 - 2 )+ b 2 U1 1 2 12 2

(13.43)

(13.44)

and the integration constant C1 is defined by condition (13.42) or the form of the Bernoulli equation:
2 (b - (1 + bH1 )u2 ) - b 3 1 - U1 (C1 ,u2 ) - u2 - H1 u2 d 2 = 2 - (1 + bH )U 2 (C ,u ) . du2 1 - (b + H1 )u2 +(1+ bH1 )u2 1 12 1

(13.45)

Using (13.44), we can easily transform Eq. (13.45) into a nonhomogeneous linear equation
2 2((1 + bH1 )u2 - b)p +2b 1 - H1 u2 - u2 - U1 (C1 ,u2 ) dp 2 , = 2 du2 1 - (b + H1 )u2 +(1+ bH1 )u2 - (1 + bH1 )U1 (C1 ,u2 ) 2

p=

1 . 2

(13.46) 859


This means that there exists another transcendental first integral in explicit form (i.e., through a finite combination of quadratures). Moreover, the general solution of Eq. (13.46) dep ends on an arbitrary constant C2 . We omit the complete calculations but note that the general solution of the homogeneous linear equation obtained from (13.46) in the particular case |b - H1 | = 2, has the following solution: p = p0 (u2 ) = C [1 - A1 u2 ]
2/(1+A
4 1

C1 =

1 - A4 1 , 1+ A4 1

A1 =

1 (b + H1 ) 2

)

2 C1 - 4A2 (1 - A1 u2 )2 ± C 1 2 C1 - 4A2 (1 - A1 u2 )2 C 1

1 1

±A4 /(1+A 1

4 1

)

2(A1 - b) , â exp 1+ A4 A1 (A1 u2 - 1) 1

C = const . (13.47)

Remark 13.2. Formally, in the expression of the first integral thus found, we must substitute for C1 the left-hand side of the first integral (13.35). Then the additional first integral obtained has the following structure (similar to the transcendental first integral from planar dynamics): ln | sin | + G2 sin , Z Z3 , sin sin = C2 = const . (13.48)

Thus, for the integration of the sixth-order system (13.17)­(13.22) we already have two indep endent first integrals. For the complete integrability, as was noted ab ove, it suffices to find one first integral for the (p otentially separated) system (13.20) , (13.21) and an additional first integral that "attaches" Eq. (13.22). To find the first integral of the (p otentially separated) system (13.20) , (13.21), we establish a corresp ondence b etween it and the following nonautonomous first-order equation:
2 1+ Z cos 1 dZ = . d1 Z sin 1

(13.49)

After integration, this leads to the required invariant relation
2 1+ Z = C3 = const, sin 1

(13.50)

which in terms of the variables Z1 and Z2 has the form
2 2 Z1 + Z2 = C3 = const . Z1 sin 1

(13.51)

Further, to find an additional first integral that "attaches" Eq. (13.22), we establish a corresp ondence b etween Eqs. (13.22) and (13.20) and the following nonautonomous equation: dZ 2 = - 1+ Z cos 1 . d2 Since, by (13.50) ,
2 2 C3 cos 1 = ± C3 - 1 - Z ,

(13.52)

(13.53) (13.54)

we have 1 dZ 2 = 1+ Z d2 C3 860
2 2 C3 - 1 - Z .


Integrating this relation, we obtain the following quadrature: C3 dZ , C4 = const . (2 + C4 ) = 2 2 2 (1 + Z ) C3 - 1 - Z Another integration leads to the relation tan(2 + C4 ) = C3 Z
2 C - 1 - Z 2 3

(13.55)

,

C4 = const .

(13.56)

In terms of the variables Z1 and Z2 , this invariant relation b ecomes C3 Z2 , C4 = const . tan(2 + C4 ) = 2 2 2 C3 - 1 Z1 - Z2 Finally, we have the additional first integral that "attaches" Eq. (13.22): arctan or arctan C3 Z
2 C - 1 - Z 2 3

(13.57)

± 2 = C4 , ± 2 = C4 ,

C4 = const

(13.58)

C3 Z2
2 2 2 C3 - 1 Z1 - Z2

C4 = const .

(13.59)

Thus, in the case considered, the system of dynamical equations (11.3)­(11.6), (11.9)­(11.14) under condition (13.5) has eight invariant relations: the analytic nonintegrable constraint of the form (11.19) corresp onding to the analytic first integral (13.23) , the cyclic first integrals of the form (11.17) and (11.18) , the first integral of the form (13.36) ; moreover, there is a first integral that can b e found from Eq. (13.46) (it is a transcendental function of phase variables in the sense of complex analysis), and, finally, transcendental first integrals of the form (13.50) (or (13.51) ) and (13.58) (or (13.59)). Theorem 13.1. System (11.3)­(11.6) , (11.9)­(11.14) under conditions (11.19) , (13.5), (11.18) , and (11.17) possesses eight invariant relations (complete set ), four of which are transcendental functions (from the point of view of complex analysis ). Moreover, at least seven of these eight relations are expressed through finite combinations of elementary functions. 13.3. ogy. Top ological analogies. We show that there exists another mechanical and top ological anal-

Theorem 13.2. The first integral (13.36) of system (11.3) ­(11.6), (11.9)­(11.14) under conditions (11.19) , (13.5), (11.18) , and (11.17) is constant on phase trajectories of system (10.14) ­(10.19) . Proof. Indeed the first integral (13.36) can b e obtained by a change of coordinates by means of relation (13.35), whereas the first integral (10.35) can b e obtained by a change of coordinates by means of relation (10.34). But relations (13.35) and (10.34) coincide. The theorem is proved. Thus, we have the following top ological and mechanical analogies in the sense explained ab ove: (1) motion of a free rigid b ody in a nonconservative field with a tracing force (under a nonintegrable constraint); (2) motion of a fixed physical p endulum in a flow of a running medium (nonconservative field); (3) rotation of a rigid b ody ab out the center of mass that moves rectilinearly and uniformly in a nonconservative field. On more general top ological analogies, see also [91]. Acknowledgment. This work was partially supp orted by the Russian Foundation for Basic Research (Pro ject No. 12-01-00020-a). 861


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865


84.

85. 86.

87. 88.

89. 90.

91. 92. 93.

94.

95. 96.

97.

98.

99.

Spaces, Approximation Theory, and Nonlinear Analysis" Devoted to the 100th Anniversary of S. M. Nikol'skii, Moscow, May 23­29, 2005 [in Russian], V. A. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow (2005), p. 244. M. V. Shamolin, "On a certain integrable case in dynamics on so(4) â R4 ," in: Abstracts of Reports of Al l-Russian Conference "Differential Equations and Their Applications," Samara, June 27­July 2, 2005 [in Russian], Univers-Grupp, Samara (2005), pp. 97­98. M. V. Shamolin, "On a certain integrable case of equations of dynamics in so(4) â R4 ," Usp. Mat. Nauk, 60, No. 6, 233­234 (2005). M. V. Shamolin, "A case of complete integrability in spatial dynamics of a rigid b ody interacting with a medium taking into account rotational derivatives of force moment in angular velocity," Dokl. Ross. Akad. Nauk, 403, No. 4, 482­485 (2005). M. V. Shamolin, "Comparison of Jacobi integrable cases of plane and spatial b ody motions in a medium under streamline flow around," Prikl. Mat. Mekh., 69, No. 6, 1003­1010 (2005). M. V. Shamolin, "Structural stable vector fields in rigid b ody dynamics," in: Proc. 8th Conf. "Dynamical Systems: Theory and Applications," Lodz, Poland, December 12­15, 2005, 1, Tech. Univ. Lodz, Lodz (2005), pp. 429­436. M. V. Shamolin, "On the problem of the motion of a rigid b ody in a resisting medium," Izv. Ross. Akad. Nauk, Mekh. Tverdogo Tela, 3, 45­57 (2006). M. V. Shamolin, "On a case of complete integrability in four-dimensional rigid b ody dynamics," in: Abstracts of Reports of International Conference "Differential Equations and Dynamical Systems," Vladimir, July 10­15, 2006 [in Russian], Vladimir State University, Vladimir (2006), pp. 226­228. M. V. Shamolin, Methods for Analysis of Variable Dissipation Dynamical Systems in Rigid Body Dynamics [in Russian], Ekzamen, Moscow (2007). M. V. Shamolin, "Some model problems of dynamics of a rigid b ody interacting with a medium," Prikl. Mekh., 43, No. 10, 49­67 (2007). M. V. Shamolin, "New integrable cases in dynamics of a four-dimensional rigid b ody interacting with a medium," in: Abstracts of Sessions of Workshop "Current Problems of Geometry and Mechanics," J. Math. Sci., 154, No. 4, 462­495 (2008). M. V. Shamolin, "On integrability of motion of four-dimensional b ody-p endulum situated in over-running medium flow," in: Abstracts of Sessions of Workshop "Current Problems of Geometry and Mechanics," J. Math. Sci., 154, No. 4, 462­495 (2008). M. V. Shamolin, "A case of complete integrability in dynamics on a tangent bundle of a twodimensional sphere," Usp. Mat. Nauk, 62, No. 5, 169­170 (2007). M. V. Shamolin, "Complete integrability of equations of motion for a spatial p endulum in medium flow taking into account rotational derivatives of moment of its action force," Izv. Ross Akad. Nauk, Mekh. Tverdogo Tela, 3, 187­192 (2007). M. V. Shamolin, "Case of complete integrability in dynamics of a four-dimensional rigid b ody in a nonconcervative force field," in: Abstracts of Reports of Int. Congress "Nonlinear Dynamical Analysis-2007," St. Petersburg, June 4­8, 2007 [in Russian], St. Petersburg State University, St. Petersburg (2007), p. 178. M. V. Shamolin, "Cases of complete integrability in dynamics of a four-dimensional rigid b ody in a nonconservative force field," In: Abstracts of Reports of Int. Conf. "Analysis and Singularities" dedicated to the 70th Anniversary of V. I. Arnold, August 20­24, 2007, Moscow [in Russian], MIAN, Moscow (2007), pp. 110­112. M. V. Shamolin, "A case of complete integrability in dynamics on a tangent bundle of twodimensional sphere," Usp. Mat. Nauk, 62, No. 5, 169­170 (2007).

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100. M. V. Shamolin, "4D rigid b ody and some cases of integrability," in: Abstr. ICIAM07, Zurich, ¨ Switzerland, June 16­20, 2007, ETH, Zurich (2007), pp. 311. ¨ 101. M. V. Shamolin, "The cases of integrability in a 2D-, 3D- and 4D-rigid b ody," in: Proc. Int. Conf. "Dynamical Methods and Mathematical Model ling," Val ladolid, Spain, September 18­22, 2007, ETSI I, Valladolid (2007), pp. 31. 102. M. V. Shamolin, "Cases of integrability in terms of transcendental functions in dynamics of a rigid b ody interacting with a medium," Proc. 9th Conf. "Dynamical Systems: Theory and Applications," Lodz, Poland, December 17­20, 2007, 1, Tech. Univ. Lodz, Lodz (2007), pp. 415­ 422. 103. M. V. Shamolin, "Dynamical systems with variable dissipation: approaches, methods, and applications," J. Dynam. Sci., 162, No. 6, 741­908 (2008). 104. M. V. Shamolin, "New integrable cases in the dynamics of a b ody interacting with a medium taking into account the dep endence of the moment of the resistance force on the angular velocity," Prikl. Mat. Mekh., 72, No. 2, 273­287 (2008). 105. M. V. Shamolin, "New integrable case in the dynamics of a four-dimensional rigid b ody in a nonconservative field," in: Proc. Voronezh Spring Math. School "Pontryagin Readings­XIX," Voronezh, May 2008, Voronezh State Univ., Voronezh (2008), pp. 231­232. 106. M. V. Shamolin, "New cases of complete integrability in the dynamics of a symmetric fourdimensional rigid b ody in a nonconservative field," in: Proc. Int. Conf. "Contemporary Problems of Mathematics, Mechanics, and Informatics" dedicated to the 85th Anniversary of L. A. Tolokonnikov, Tula, November 17­21, 2008, Grif, Tula (2008), pp. 317­320. 107. M. V. Shamolin, "On the integrability in elementary functions of some classes of dynamical systems," Vestn. Mosk. Univ., Ser. 1, Mat. Mekh., 3, 43­49 (2008). 108. M. V. Shamolin, "Three-parameter family of phase p ortraits in dynamics of a rigid b ody interacting with a medium," Dokl. Ross. Akad. Nauk, 418, No. 1. 46­51 (2008). 109. M. V. Shamolin, "Methods of analysis of dynamic systems with various dissipation in dynamics of a rigid b ody," Proc. ENOC-2008, June 30­July 4, 2008, St. Petersburg, Russia (2008). 110. M. V. Shamolin, "Some methods of analysis of dynamical systems with various dissipation in dynamics of a rigid b ody," PAMM, 8, 10137­10138 (2008). 111. M. V. Shamolin, "The various cases of complete integrability in dynamics of a rigid b ody interacting with a medium," in: CD-Proc. ECCOMAS Thematic Conf. "Multibody Dynamics," Warsaw, Poland, June 29­July 2. 2009, Polish Acad. Sci., Warsaw (2009). 112. M. V. Shamolin, "On the integrability in elementary functions of some classes of nonconservative dynamical systems," J. Math. Sci., 161, No. 5, 734­778 (2009). 113. M. V. Shamolin, "New cases of integrability in the dynamics of a four-dimensional rigid b ody in a nonconservative field," in: Proc. Semin. "Current Problems of Geometry and Mechanics," J. Math. Sci., 165, No. 6, 607­615 (2009). 114. M. V. Shamolin, "Cases of the complete integrability in the dynamics of a symmetric fourdimensional rigid b ody in a nonconservative field," in: Proc. Semin. "Current Problems of Geometry and Mechanics," J. Math. Sci., 165, No. 6, 607­615 (2009). 115. M. V. Shamolin, "Classification of cases of complete integrability in the dynamics of a symmetric four-dimensional rigid b ody in a nonconservative field," J. Math. Sci., 165, No. 6, 743­754 (2009). 116. M. V. Shamolin, "New cases of complete integrability in the dynamics of a dynamically symmetric four-dimensional rigid b ody in a nonconservative field," Dokl. Ross. Akad. Nauk, 425, No. 3, 338­342 (2009).

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117. M. V. Shamolin, "Cases of integrability of the equations of motion of a four-dimensional rigid b ody in a nonconservative field," in: Proc. Int. Conf. "Contemporary Problems of Mathematics, Mechanics, and Their Applications" dedicated to the 70th Anniversary of Prof. V. A. Sadovnichy, Moscow, March 30­April 2, 2009, Univ. Kniga, Moscow (2009), p. 233. 118. M. V. Shamolin, "Dynamical systems with variable dissipation: methods and applications," in: Proc. 10th Conf. "Dynamical Systems: Theory and Applications," Poland, Lodz, December 7­10, 2009, Tech. Univ. Lodz, Lodz (2009), pp. 91­104. 119. M. V. Shamolin, "New cases of integrability in dynamics of a rigid b ody with the cone form of its shap e interacting with a medium," PAMM, 9, 139­140 (2009). 120. M. V. Shamolin, "Dynamical systems with various dissipation: background, methods, applications," in: CD-Proc. XXXVIII Summer School-Conf. "Advanced Problems in Mechanics," St. Petersburg (Repino), Russia, July 1­5, 2010, IPME, St. Petersburg (2010), pp. 612­621. 121. M. V. Shamolin, "Integrability and nonintegrability in terms of transcendental functions in dynamics of a rigid b ody," PAMM, 10, 63­64 (2010). 122. M. V. Shamolin, "New cases of the integrability in the spatial dynamics of a rigid b ody," Dokl. Ross. Akad. Nauk, 431, No. 3, 339­343 (2010). 123. M. V. Shamolin, "Spatial motion of a rigid b ody in a resisting medium," Prikl. Mekh., 46, No. 7, 120­133 (2010). 124. M. V. Shamolin, "Cases of complete integrability of the equations of motion of a dynamically symmetric four-dimensional rigid b ody in a nonconservative field," in: Proc. Int. Conf. "Differential Equations and Dynamical Systems," Suzdal', July 2­7, 2010, Vladimir State Univ., Vladimir (2010), pp. 195. 125. M. V. Shamolin, "A case of complete integrability in the dynamics of a four-dimensional rigid b ody in a nonconservative field," Usp. Mat. Nauk, 65, No. 1, 189­190 (2010). 126. M. V. Shamolin, "Motion of a rigid b ody in a resisting medium," Mat. Model., 23, No. 12, 79­104 (2011). 127. M. V. Shamolin, "On a multi-parameter family of phase p ortraits in the dynamics of a rigid b ody interacting with a medium," Vestn. Mosk. Univ., Ser. 1, Mat. Mekh., 3, 24­30 (2011). 128. M. V. Shamolin, "A new case of integrability in the dynamics of a four-dimensional rigid b ody in a nonconservative field," Dokl. Ross. Akad. Nauk, 437, No. 2, 190­193 (2011). 129. M. V. Shamolin, "A new case of complete integrability of dynamical equations on the tangent bundle of the three-dimensional sphere," Vestn. Samar. State Univ., Estestvennonauch. Ser., Miscel lanious, 5, 187­189 (2011). 130. M. V. Shamolin, "Complete lists of first integrals in the dynamics of a four-dimensional rigid b ody in a nonconservative field," in: Proc. Int. Conf. dedicated to the 110th birthday of Prof. I. G. Petrovsky, Moscow State Univ., Moscow (2011), pp. 389­390. 131. M. V. Shamolin, "Complete list of first integrals in the problem on the motion of a fourdimensional rigid b ody in a nonconservative field under a linear damping," Dokl. Ross. Akad. Nauk, 440, No. 2, 187­190 (2011). 132. M. V. Shamolin, "Comparison of complete integrability cases in dynamics of two-, three-, and four-dimensional rigid b odies in a nonconservative field," in: Proc. XV Int. Conf. "Dynamical System Model ling and Stability Investigation," May 25­27, 2011, Kiev (2011), pp. 139. 133. M. V. Shamolin, "Cases of complete integrability in transcendental functions in dynamics and certain invariant indices," in: CD-Proc. 5th Int. Sci. Conf. "Physics and Control" (PHYSCON 2011), Leon, September 5­8, 2011, Leon, Spain (2011).

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134. M. V. Shamolin, "Variety of cases of integrability in dynamics of a 2D-, 3D-, and 4D-rigid b ody interacting with a medium," in: Proc. 11th Conf. "Dynamical Systems: Theory and Applications," Lodz, Poland, December 5­8, 2011, Tech. Univ. Lodz, Lodz (2011), pp. 11­24. 135. M. V. Shamolin, "Variety of cases of integrability in dynamics of a 2D- and 3D-rigid b ody interacting with a medium," in: CD-Proc. 8th ESMC 2012, Graz, Austria, July 9­13, 2012, Graz (2012). 136. M. V. Shamolin, "Cases of integrability in dynamics of a rigid b ody interacting with a resistant medium," CD-Proc. 23th Int. Congr. "Theoretical and Applied Mechanics," August 19­24, 2012, Beijing, China, : China Science Literature Publishing House, Beijing (2012). 137. M. V. Shamolin, "Problem on the motion of a b ody in a resisting medium taking into account the dep endence of the moment of the resistance on the angular velocity," Mat. Model., 24, No. 10, 109­132 (2012). 138. M. V. Shamolin, "Comparison of complete integrability cases in dynamics of a two-, three-, and four-dimensional rigid b ody in a nonconservative field," J. Math. Sci., 187, No. 3, 346­359 (2012). 139. M. V. Shamolin, "Some questions of qualitative theory in dynamics of systems with variable dissipation," J. Math. Sci., 189, No. 2, 314­323 (2013). 140. M. V. Shamolin, "A new case of integrability in the spatial dynamics of a rigid b ody interacting with a medium taking into account linear damping," Dokl. Ross. Akad. Nauk, 442, No. 4, 479­481 (2012). 141. M. V. Shamolin, "A new case of integrability in the dynamics of a four-dimensional rigid b ody in a nonconservative field taking into account linear damping," Dokl. Ross. Akad. Nauk, 444, No. 5, 506­509 (2012). 142. M. V. Shamolin, "Cases of integrability in the dynamics of a four-dimensional rigid b ody in a nonconservative field," Proc. Int. Conf. "Voronezh Winter Mat. School of G. G. Krein," Voronezh, January 25­30, 2012, Voronezh State Univ., Voronezh (2012), pp. 213­215. 143. M. V. Shamolin, "Review of cases of integrability in the dynamics of lower- and higherdimensional rigid b odies in nonconservative fields," in: Proc. Int. Conf. "Differential Equations and Dynamical Systems," Suzdal', June 29­July 4, 2012, Vladimir State Univ., Vladimir (2012), pp. 179­180. 144. M. V. Shamolin, "Complete list of first integrals of dynamical equations of motion of a rigid b ody in a resisting medium taking into account linear damping," Vestn. Mosk. Univ., Ser. 1, Mat. Mekh., 4, 44­47 (2012). 145. V. A. Steklov, On the Motion of a Rigid Body in a Fluid [in Russian], Khar'kov (1893). 146. G. K. Suslov, Theoretical Mechanics [in Russian], Gostekhizdat, Moscow (1946). 147. V. V. Trofimov, "Euler equations on finite-dimensional solvable Lie groups," Izv. Akad. Nauk SSSR, Ser. Mat., 44, No. 5, 1191­1199 (1980). 148. V. V. Trofimov and A. T. Fomenko, " A methodology for constructing Hamiltonian flows on symmetric spaces and integrability of certain hydrodynamic systems," Dokl. Akad. Nauk SSSR, 254, No. 6, 1349­1353 (1980). 149. V. V. Trofimov and M. V. Shamolin, "Geometric and dynamical invariants of integrable Hamiltonian and dissipative systems," J. Math. Sci., 180, No. 4, 365­530 (2012). 150. S. V. Vishik and S. F. Dolzhanskii, "Analogs of Euler­Poisson equations and magnetic electrodynamic related to Lie groups," Dokl. Akad. Nauk SSSR, 238, No. 5, 1032­1035. 151. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. With an Introduction to the Problem of Three Bodies, At the University Press, Cambridge (1960).

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152. N. E. Zhukovsky, "On the fall of a light oblong b ody rotating ab out its longitudinal axis," in: Complete Works [in Russian], Vol. 5, Fizmatlit, Moscow (1937), pp. 72­80, 100­115. M. V. Shamolin Institute of Mechanics of the M. V. Lomonosov Moscow State University, Moscow, Russia E-mail: shamolin@imec.msu.ru

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