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CONFERENCE on DYNAMICAL SYSTEMS THEORY AND APPLICATIONS
December 5-8, 2011. Lґ ґ Poland odz,

11

TH

Variety of the cases of integrability in dynamics of a 2D-, 3D-, and 4D-rigid b o dy interacting with a medium
Maxim V. Shamolin
A vast number of papers are devoted to studying the complete integrability of equations of fourdimensional rigid-body motion. Although in studying lowdimensional equations of motion of quite concrete (two- and three-dimensional) rigid bodies in a nonconservative force field, the author arrived at the idea of generalizing the equations to the case of a four-dimensional rigid body in an analogous nonconservative force field. As a result of such a generalization, he obtained the variety of cases of integrability in the problem of body motion in a resisting medium that fills the four-dimensional space in the presence of a certain tracing force that allows one to reduce the order of the general system of dynamical equations of motion in a methodical way. 1. Intro duction A huge number of works is devoted to studying the complete integrability cases of the equations of motion of a four-dimensional rigid body. In studying the "low-dimensional" equations of motion of quite concrete (two- and three-dimensional) rigid bodies in a nonconservative force field, he arrived at the idea to generalize the equations to the case of motion of a four-dimensional rigid body in an analogously constructed field. As a result of such a generalization, he obtained several cases of integrability in the problem of body motion in a resisting medium that fills a four-dimensional space under the presence of a certain tracing force, which allows one to methodologically reduce the order of the general system of dynamical equations of motion. Moreover, to the author opinion, the obtained results are original from the viewpoint that a pair of non-conservative force exists in the system. Previously, in [1­3], the author showed the complete integrability of the equation of plane-parallel body motion in a resisting medium under the streamline flow around conditions, when the system of dynamical equations has a first integral that is a transcendental


function (in the sense of theory of functions of one complex variable, having essentially singular points) of quasi-velocities.In this case, it was assumed that the whole interaction of the medium and the body is concentrated on a part of the body surface that has the form of a (one-dimensional) plate. Later the plane problem was generalized to the spatial (three-dimensional) case where the system of of dynamical equations has a complete tuple of transcendental first integrals. It was assumed here that the whole interaction of the medium and the body is concentrated on a part of the body surface that has the form of a plane (two-dimensional) disk. 2. Motion on Two-Dimensional Plane 2.1. A more general problem of motion with tracing force Let us consider the plane-parallel motion of a body with forward plane endwall in the resistance force field under the quasi-stationary conditions [1­3]. If (v , ) are the polar coordinates of a certain characteristic point of the rigid body, is its angular velocity"-- and I and m are the inertia-mass characteristics, then the dynamical part of the equations of motion (including the case of Chaplygin analytical functions of medium action; see below) takes the form v cos - v sin - v sin + 2 = Fx , v sin + v cos + v cos - = 0, I = yN (, , v )s(), where Fx = -s()v 2 /m, > 0. If we consider a more general problem on the body motion under the existence of a certain tracing force T passing through the center of masses and ensuring the fulfilment of the relation VC const during all the time of motion(V forces acts on the body. In the case of Chaplygin analytical functions, we take the dynamical functions s and y
N C

(1)

(2) is the velocity of the center of masses; then in system (1),

instead of Fx , we have a quantity identically equal to zero, since a non-conservative par of

in the form s() = B cos , yN (, , v ) = A sin - h1 /v , h1 > 0, A, B > 0, v = 0, which shows that in the system considered, there also exists an additional damping (and breaking in some domains of the phase space) non-conservative force moment. Owing to constraint (2), under certain condition, system (1) reduces to the following


system on the three-dimensional cylinder W1 = R1 {v } в S1 { mod 2 } в R1 { } : + v = v (, ), = - + n2 sin cos2 + 2 sin - 0 (3)

h1 B cos2 , I h1 B 2 h1 B = n2 sin cos - n2 sin2 cos + 3 cos + sin cos - cos , 0 0 I I (4) (, ) = - 2 cos + n2 sin2 cos - 0 where = v , n2 = AB /I , · = v 0 . h1 B sin cos , I

2.2. A complete list of first integrals From the system (3), (4), the independent second-order system (4) is separated. Theorem 1. The system (3), (4) has a complete tuple of first integrals; one of them is an analytic function, and the other is a transcendental function of phase variables expressing through a finite combination of elementary functions. It is necessary to make an important remark here. The matter is that from the viewpoint of elementary function theory,the obtained first integral is transcendental (i.e., nonalgebraic).In this case, the transcendence is understood in the sense of theory of functions of one complex variable, when after a formal continuation of a function to the complex domain, it has essentially singular points corresponding to attracting and repelling limit sets of the dynamical system considered. Indeed, by (2), the value of the center-of-masses velocity if a first integral of system (1) under the condition Fx 0, precisely the function of phase variables
2 0 (v , , ) = v 2 + 2 2 - 2 v sin = VC

(5)

is constant on phase tra jectories. By a nondegenerate change of the independent variable, the system (3), (4) also has an analytic integral, precisely, the function of phase variables
2 1 (v , , ) = v 2 (1 + 2 2 - 2 sin ) = VC

(6)

is constant on phase tra jectories. Relation (6) allows us to find the dependence of the velocity of a characteristic rigid body point on other phase variables not solving the system (3), (4); precisely, for VC = 0,
2 the following relation holds: v 2 = (VC )/(1 + 2 2 - 2 sin ).


Since the phase space W1 of the system (3), (4) is three-dimensional and there exist asymptotic limit sets in it,relation (6) defines a unique analytic (even continuous) first integral of the system (3), (4) on the whole phase space. Let us examine the problem on the existence of the second (additional) first integral of the system (3), (4) in more detail. Its phase space is foliated into surfaces {(v , , ) W1 : VC = const}. To justify the latter fact, let us introduce the dimensionless differentiation
2 0

n

0

and the additional dimensionless parameter H1 = h1 B /I n0 , n = AB /I , = n0 , = sin , and to the separated second-order system (4), let us put in correspondence the differential equation v = v (, ), = - + sin cos2 + 2 sin - H1 cos2 , = sin cos - sin2 cos + 3 cos + H 2 sin cos - H cos , 1 1 (, ) = - 2 cos + sin2 cos - H1 sin cos . The analytic first integral (6) obtained above joins Eq. (3) (or (7)). To find the additional transcendental first integral, to the separated system (8), we put in correspondence the differential equation - [ 2 - 2 ] + H1 [ - 1] d = . d - + + [ 2 - 2 ] - H1 [1 - 2 ] After introducing the homogeneous change of variables = t , d = td + dt, the integration of the latter equation reduces to the integration of the following Bernoulli equation: a1 (t)d /dt = a2 (t) + a3 (t) 3 , a1 (t) = -(1 + H1 )t2 + ( + H1 )t - 1, a2 (t) = (1 + H1 )t - , a3 (t) = - H1 t - t2 . Applying the classical change of variables p = 1/ 2 , we reduce the equation studied to the linear homogeneous equation dp = c1 (t)p + c2 (t), dt where c1 (t) = 2t(1 + H1 ) - 2 , (1 + H1 )t2 - ( + H1 )t + 1 c2 (t) = 2 - 2 H1 t - 2 t2 . (1 + H1 )t2 - ( + H1 )t + 1 (7) (8)

The solution p1 of the homogeneous part of the equation studied is represented in the following form (three cases are possible): 1. forD = ( - H1 )2 - 4 > 0, 2(1 + H1 )t - ( - H1 ) - D p1 = k[(1 + H1 )t - ( + H1 )t + 1] · 2(1 + H1 )t - ( - H1 ) + D
2
H1 - D

;


2. for D = ( - H1 )2 - 4 < 0, p1 = k[(1 + H1 )t2 - ( + H1 )t + 1] · exp arctan 3. for D = ( - H1 )2 - 4 = 0, p1 = k (1 + H1 )t2 - ( + H1 )t + 1 · exp 2 L1 1 + H1 ± 1 , L1 = ± 1. 1 + H1 2(1 + H1 )t - ( + H1 ) -D ;

It is clear that to find a particular solution of the equation studied, applying the variation-of-constant method, we need to assume that k is a function of t, which is certainly solvable in the class of elementary functions. In this work, we do not present the corresponding calculations. 3. Motion in Three-Dimensional Space 3.1. General problem of motion with tracing force Let us consider the spatial motion of a homogeneous axially-symmetric rigid body of mass m with forward round endwall in the resistance force field under the quasi-stationarity condition. If (v , , ) are the spherical coordinates of a certain characteristic point of the rigid body, {x , y , z } are components of its angular velocity, and I1 , I2 , and I2 are the principal moments of inertia in a certain coordinate system related to the body, then the dynamical part of the equations of motion in the case of Chaplygin functions [1] of medium action has the form v cos - v sin + y v sin sin - z v sin cos + (2 + 2 ) = Fx , y z v sin cos + v cos cos - v sin sin + z v cos - x v sin sin - - x y - z = 0, v sin sin + v cos sin + v sin cos + x v sin cos - y v cos - - x z + y = 0, x = 0, hy I2 y + (I1 - I2 )x z = -AB v 2 sin cos sin - , v hz I2 z + (I2 - I1 )x y = AB v 2 sin cos cos - , v (9)

where Fx = -B v 2 /m cos , A, B , , h > 0. If we consider a more general problem of body motion in a resisting medium under the existence of a certain tracing force T passing through the symmetry axis and ensuring the fulfilment of relation (2) during all the motion time, then in system (9), instead of Fx , we have the quantity (T - B cos )v 2 /m; moreover, owing to condition (2), under certain condition, system (9) reduces to a system of a lower order.


It is seen that the choice of phase variables allows us to consider the six-order system (9) of dynamical equations as an independent system. Moreover, as is seen from the equations of motion, the component of the longitudinal angular velocity component is conserved: x =
x0

= const.

(10)

In what follows, we restrict ourselves to the body motion without proper rotation, ie., to the case where
x0

= 0; moreover, for simplicity, let h = 0.

Introduce the following notation: z1 = y cos +z sin , z2 = -y sin +z cos , zi = Zi v , i = 1, 2, = v , = v , v = v v . Then system (9) in case (2) for x0 = 0 can be transformed into the following form: v = v (, Z1 , Z2 ), 2 2 = -Z2 + n2 sin cos2 + (Z1 + Z2 ) sin , 0 2 cos Z2 = n2 sin cos - Z2 (, Z1 , Z2 ) - Z1 , 0 sin cos Z1 = -Z1 (, Z1 , Z2 ) + Z1 Z2 , sin cos = Z1 , sin where
2 2 (, Z1 , Z2 ) = - (Z1 + Z2 ) cos + n2 sin2 cos , 0

(11)

(12)

(13)

n2 = 0

AB . I2

3.2. A complete list of first integrals As above, let us consider the problem of complete integrability (in elementary functions) for the dynamical system (11)­(13) with analytic right-hand sides. Since we consider the class of body motions for which property (2) holds, the fifth-order system (11)­(13) has (along with (10)) an analytic first integral. Indeed, in the coordinate system considered, we can represent the center-of-masses velocity in the form VC = v cos , v sin cos - z , v sin sin + relation is invariant for system (9) under conditions (10) (
2 2 2 v 2 - 2 v z2 sin + 2 (z1 + z2 ) = VC 0 . x0 y

. Then the following

= 0) and (2): (14)

Moreover, relation (14) in which the linear and angular velocities compose a homogeneous form of degree 2 allows us to write the polynomial integral in the above velocities for the system (11)­(13):
2 2 v 2 (1 - 2 Z2 sin + 2 (Z1 + Z2 )) = V 2 C0

,

(15)


and relation (15) allows us to explicitly find the dependence of v on the other quasi-velocities: v2 =
2 VC 0 . 2 2 1 - 2 Z2 sin + 2 (Z1 + Z2 )

(16)

It is seen that relation (16) allows us to consider the problems of integrability in elementary functions of the system (11)­(13), which is just of lower order, the fourth order. Let us rewrite the third-order system (12) in the form
2 2 = -Z2 + b sin cos2 + b(Z1 + Z2 ) sin , 2 2 2 Z2 = sin cos + bZ2 (Z1 + Z2 ) cos - bZ2 sin2 cos - Z1

cos , Z1 = bZ1 (Z + Z ) cos - bZ1 sin cos + Z1 Z2 sin
2 1 2 2 2

cos , sin

(17)

where b = n0 and the new dimensionless differentiation (17)) to the following form with algebraic right-hand sides:
2 2 2 + bZ2 (Z1 + Z2 ) - bZ2 2 - Z1 / dZ2 = 2 2, d -Z2 + b (1 - 2 ) + b (Z1 + Z2 ) 2 2 bZ1 (Z1 + Z2 ) - bZ1 2 + Z1 Z2 / dZ1 = 2 2. d -Z2 + b (1 - 2 ) + b (Z1 + Z2 )

n0

is also introduced.

Furthermore, applying the substitution = sin , which is often used, er reduce system

(18)

Let us pass to homogeneous coordinates uk , k = 1, 2, by the formulas uk = Zk . Then system (18) reduces to the form du2 1 - bu2 + u2 - u2 2 1 = , d -u2 + b 2 (u2 + u2 ) + b(1 - 2 ) 1 2 2u1 u2 - bu1 du1 = . d -u2 + b 2 (u2 + u2 ) + b(1 - 2 ) 1 2

(19)

To system (19), we can put in correspondence the following first-order equation: du2 1 - bu2 + u2 - u2 2 1 = . du1 2u1 u2 - bu1 (20)

This equation is integrated in elementary functions, since we integrate the following identity obtained from Eq. (20): d 1 - bu2 + u u1
2 2

+ du1 = 0,

and in the coordinates ( , Z1 , Z2 ), it corresponds to the transcendental first integral of the following form
2 2 Z1 + Z2 - bZ2 + 2 = const. Z1

(21)


Using relation (21), we conclude that system (12) has the following transcendental first integral, which is expressed through a finite combination of elementary functions:
2 2 Z1 + Z2 - bZ2 sin + sin2 = const. Z1 sin

(22)

Now, using the just found first integral (22), we write the first equation of system (19) in the form 2 - 2bu2 + 2u2 - C1 U1 (C1 , u2 ) du2 2 = , d -u2 + b - 2b 2 + b 2 (C1 U1 (C1 , u2 ) + bu2 ) U1 (C1 , u2 ) = C1 ±
2 C1 - 4(u2 - bu2 + 1) 2 , 2

(23)

or in the form of the Bernoulli equation (b - u2 ) + b 3 (C1 U1 (C1 , u2 ) + bu2 - 2) d = . du2 2 - 2bu2 + 2u2 - C1 U1 (C1 , u2 ) 2 Equation (24) (by using (23)) easily reduces to the linear inhomogeneous equation 2(u2 - b)p - 2b(C1 U1 (C1 , u2 ) + bu2 - 2) dp = , du2 2 - 2bu2 + 2u2 - C1 U1 (C1 , u2 ) 2 p= 1 . 2 (25) (24)

The latter fact means that we can find one more transcendental first integral in explicit form (i.e., through a combination of qyadratures). Moreover, the general solution of Eq. (25) depends on an arbitrary constant C2 ; we do not present complete calculations. To find the last additional integral of the system (11)­(13) (i.e., the integral, which
2 connects the equation for the angle ) we note that since d /d = (Z1 / )/(-Z2 + b (Z1 + 2 Z2 ) + b (1 - 2 )), it follows that to the relation

d u1 = d -u2 + b 3 (u2 + u2 ) + b (1 - 2 ) 1 2 the relation du1 2u1 u2 - bu1 = d -u2 + b 2 (u2 + u2 ) + b(1 - 2 ) 1 2

(26)

(27)

taken from system (19) is added. The obtained system (26), (27) allows us to write the following equation for obtaining the desired integral: du1 = 2u1 - . d (28)

Now, using the first integral of Eq. (20) (C1 is its constant of integration) and Eq. (28), we can obtain that du1 =± d b2 - 4(u2 - C1 u1 + 1); 1 (29)


hence, by (29), the desired quadrature takes the form ± du1 = + C3 , - 4(u2 - C1 u1 + 1) 1 C3 = const. (30)

b2

The left-hand side of (30) (without sign) has the form 1 arcsin 2 u1 -
2 C1 + ( b2 2 b2 -

4)

.

(31)

After substitutions, from (31), we obtain the desired invariant relation cos2 [2( + C3 )] =
b u2 - 2 G1 2

u

2 1

,

(32)

where G1 = [u2 - bu2 ]2 + 2[u2 - bu2 ][u2 + 1] + [u2 + 1]2 + b2 u2 . 2 2 1 1 1 In particular, if b = 2, then relation (32) takes the form cos2 [2( + C3 )] = (Z2 - sin )Z1 2. (Z2 - sin )2 + Z1

The right hand side, as an odd function of = (Z2 - sin )/(Z1 ) has a global maximum for = 1, which is equal to 1/2. Therefore, we have proved the following assertion. Theorem 2. The system (11)­(13) has a complete tuple of first integrals; one of them is an analytic function, and two other are elementary transcendental functions of their phase own variables. In conclusion, we note that for searching for first integrals of the systems considered, we need to reduce them to the corresponding systems with polynomial right-hand sided; the form of the latter ones determines the possibility of integrating the initial system in elementary functions. 4. Motion in Four-Dimensional Space 4.1. Two case of dynamical symmetry of a four-dimensional b o dy Let a four-dimensional rigid body of mass m with smooth three-dimensional boundary move in a resisting medium that fills a four-dimensional domain of the Euclidean space. Assume that it is dynamically symmetric; in this case, there exist two logical possibilities of representation of its tensor of inertia: either in a certain coordinate system Dx1 x2 x3 x4 related to the body, the tensor of inertia has the form diag{I1 , I2 , I2 , I2 }, (33)


or the form diag{I1 , I1 , I3 , I3 }. (34)

In the second case, the two-dimensional planes Dx1 x2 and Dx3 x4 are planes of body dynamical symmetry. 4.2. Physical assumptions and equations on so(4) Assume that the distance from the point N of application of a non-conservative force S to a point D is a function of only one parameter, the angle : DN = R() (in the case of motion in the three-dimensional space, this is the angle of attack. In case (33), this angle is measured between the velocity v
D

of the point D and the axis Dx1 . In case (34), the |v D | =

meaning of the angle will be clear from the equations. The value of the non-conservative (resistance) force S is S = s() sgn cos · v 2 , in the system. To obtain the explicit form of the dynamical part of the equations of motion,let us define two functions R and S using the information about the motion of three-dimensional bodies as follows (in this case, we also use the known analytical result of S. A. Chaplygin): R = R() = A sin , S = Sv () = B v 2 cos ; A, B > 0. If is the angular velocity tensor of the four-dimensional rigid body, so(4), then the part of the equations of motions,which corresponds to the algebra so(4), has the following form: + + [, + ] = M , where = diag{1 , 2 , 3 , 4 }, (35) v , where s is a certain function , which is characterized as as scattering or pumping of energy

1 = (-I1 + I2 + I3 + I4 )/2, . . . , 4 = (I1 + I2 + I3 - I4 )/2,

M is the exterior force moment acting on the body in R4 and pro jected on the natural coordinates in the algebra so(4), and [·, ·] is the commutator in so(4). A skew-symmetric matrix so(4) is represented in the form 0 - 6 0 6 - 5 4 3 - 2 -
5 4

- -

3 2 1

0
1

,

0

where i , i = 1, . . . , 6, are components of the angular velocity tensor in pro jections on the coordinates in the algebra so(4). In this case, it is obvious that for any i, j = 1, . . . , 4, the following relations hold: i - j = Ij - Ii .


In calculating the exterior force moment, it is necessary to construct the mapping R4 в R4 - so(4), which transforms a pair of vectors from R4 into a certain element of the algebra so(4). 4.3. Dynamics in R
4

As for the equation of motion of the center of masses C of the four-dimensional rigid body, then it ie represented in the form m wC = F , where, by the many-dimensional Rivals formula, wC = wD + 2 DC + E DC, w
D

(36)

= vD + vD ,

E = ,

F is the exterior force acting on the body (in our case, F = S), E is the angular acceleration tensor. 4.4. Generalized problem of b o dy motion under tracing force action In this work, we consider only the case (33) of distribution of principal moments of inertia. Let us slightly extend the problem. Assume that along the line Dx1 (in case (33)), a certain tracing force acts whose line of action passes through the center of masses C . The introduction of such a force is used for consideration of classes of motions interesting for us; as a result of which the order of the dynamical system can be reduced. as in the previous sections, let us consider the class of motion of the four-dimensional rigid body in the case (2)), i. e., its center of masses moves rectilinear and uniformly. 4.5. Case (33) By a completely definite choice of the tracing force,the fulfilment of condition (2) can be achieved. If (0, x
2N

,x

3N

,x

4N

) are coordinates -- of the point N in the system Dx1 x2 x3 x4 and

{-S, 0, 0, 0} are coordinates of the resistance force vector in the same system, then to find the force moment, we construct the auxiliary matrix 0 x2N x3N x4N , -S 0 0 0 which allows us to obtain the resistance force moment in the pro jections on the coordinates in the algebra so(4): {0, 0, x4N S, 0, -x3N S, x2N S } R6 = M so(4). Here,it is necessary


to take into account that if (v , , 1 , 2 ) are the spherical coordinates in R4 , then x2N = R() cos 1 , x
3N

= R() sin 1 cos 2 ,

x

4N

= R() sin 1 sin 2 .

Taking into account all what was said, we can write Eq. (35) in 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 ) = x4N S, (3 + 2 )4 + (2 - 3 )(5 6 + 1 2 ) = 0, (1 + 3 )5 + (3 - 1 )(4 6 - 1 3 ) = -x3N S, (1 + 2 )6 + (1 - 2 )(4 5 + 2 3 ) = x2N S. Obviously, in the case (33), equations (37) have three cyclic first integrals
0 1 = 1 , 0 2 = 2 , 0 4 = 4 . 0 1

(37)

(38) =
0 2

For simplicity, let us consider the motions on zero levels

=

0 4

= 0. The

remained equations on the algebra so(4) take the following form (here, n = AB /2I2 ): 3 = n2 v 2 sin cos sin 1 sin 2 , 5 = -n2 v 2 sin cos sin 1 cos 2 , 6 = n2 v 2 sin cos cos 1 . 0 0 0 If we introduce the change of angular velocities by the formulas z1 = 3 cos 2 + 5 sin 2 , z2 = -3 sin 2 cos 1 + 5 cos 2 cos 1 + 6 sin 1 , z3 = 3 sin 2 sin 1 - 5 cos
3 2

2 0

sin 1 + 6 cos 1 , then the "compatible" equations of motion on the tangent bundle T S of the three-dimensional sphere (after taking into account four conditions (2) and (38),which help us to reduce the order of the general system of dynamical equations of the tenth order to the sixth order) take the following symmetric form ( = DC ):
2 2 3 v = cos n2 v 2 sin2 - (z1 + z2 + z3 ) , 0 2 2 3 = -z3 + n2 v sin cos2 + sin (z1 + z2 + z3 )/v , 0 2 2 z3 = n2 v 2 sin cos - (z1 + z2 )ctg, 0 2 z2 = z2 z3 ctg + z1 ctgctg1 ,

(39)

(40)

z1 = z1 z3 ctg - z1 z2 ctgctg1 , 1 = z2 ctg, 2 = -z1 ctg csc 1 . (41)

From the complete system of the seventh order (39)­(41), the independent system (40), (41), of the sixth order is separated,and, in turn, it has an independent subsystem (40) of the fifth order. To completely integrate this system, we need, in general, six independent


first integrals. However, after changes of variables and introducing a new differentiation z=
2 2 z1 + z2 , z = z2 z1

, z = n0 v Z, zk = n0 v Zk , k = 1, 2, 3, z = Z , n0 v



, the

system (39)­(41) reduces to the following form (b = n0 , [b] = 1): v Z Z Z = v (, Z, Z3 ),
2 (, Z, Z3 ) = b cos [sin2 - (Z 2 + Z3 )],

(42)

2 = -Z3 + b sin cos2 + b sin (Z 2 + Z3 ), 3

= sin cos - Z 2 ctg - Z3 (, Z, Z3 ), = Z Z3 ctg - Z (, Z, Z3 ),

(43)



=Z

2 1 + Z ctgctg1 ,

1 = 2 = -

ZZ

2

1+Z Z1

ctg, ctg csc 1 .

(44)

1+Z

2

(45)

It is seen that the fifth-order system (40) splits into independent subsystems of lower order: system (43) is of the third order and system (44) (of course, after the change of the independent variable ) is of the second order. Therefore, for the complete integrability of the system studied, it suffices to find two independent integrals of the system (43), one for system (44) and additional integrals "connecting" Eqs. (42) and (45). Moreover, we note that system (43) can be considered on the tangent bundle T S2 of the two-dimensional sphere. 4.6. Complete list of first integrals The complete system (42)­(45) has an analytic first integral of the form
2 2 v 2 (1 - 2bZ3 sin + (Z 2 + Z3 )) = VC ,

(46)

since property (2) holds. The latter invariant relation allows us to find v . System (43) belongs to the class of systems arising in the three-dimensional rigid body dynamics and has two independent integrals, which are transcendental functions of their phase variables (in the sense of definitions of complex analysis) and are expressed through a finite combination of elementary functions:
2 Z 2 + Z3 - bZ3 sin + sin2 = C1 = const, Z sin

(47) (48)

G(Z, Z3 , sin ) = C2 = const. System (44) has a first integral of the form
2 1 + Z = C3 = const sin 1

(49)


and, in turn, it has an additional first integral, which allows us to find 2 ; it has the form ± cos 1 = sin{C3 (2 + C4 )}, 2 C3 - 1 C4 = const. (50)

Also, it is necessary to note the fact that the denominators of the presented systems contain the functions sin and sin 1 , which reflect only the information about the fact that the coordinates (v , , 1 , 2 ) are spherical, and for sin = 0 and sin (kinematically) degenerate. Theorem 3. The dynamical system (42)­(45) has a complete list of first integrals (46)­ (50); one of them is an analytic function, and the other are transcendental functions of their variables (after their formal continuation to the complex domain). 4.7. Conclusion This work complements the previous studies and also opens a new series of works, since previously, only those motions of a four-dimensional body were considered in which the exterior force moment is identically equal to zero (M 0) or the exterior force field is potential; unfortunately, we cannot mention all the authors). In the present work, we continue the direction developed by the author in studying the equations of motion of rigid body on so(4) в R4 under the presence of a non-conservative exterior force moment. The results listed above and also studies of related fields were already reported at the workshop "Actual Problems of Geometry and Mechanics" named after professor V. V. Trofimov led by D. V. Georgievskii and M. V. Shamolin at Department of Mechanics and Mathematics of M. V. Lomonosov Moscow State University. References 1. Shamolin M.V.: Methods for Analysis of variable Dissipation Dynamical Systems in Rigid Body Dynamics[in Russian], Moscow, Ekzamen, 2007. 2. Shamolin M.V.: New Jacobi integrable cases in dynamics of a rigid body interacting with a medium, Dokl. Ross. Akad. Nauk, 364(5), 1999, 627­629. 3. Shamolin M.V.: On an integrable case in spatial dynamics of a rigid body interacting with a medium, Izv. Ross. Akad. Nauk. Mekhanika Tverdogo Tela, 2, 1997, 65­68. Maxim V. Shamolin, Professor: Lomonosov Moscow State University, Institute of Mechanics, Michurinskii Ave., 1, 119899 Moscow, Russian Federation (shamolin@rambler.ru).
1

= 0 they