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New cases of integrability in multidimensional dynamics in a nonconservative field

New cases of integrability in multidimensional dynamics in a nonconservative field
Maxim V. Shamolin shamolin@rambler.ru, shamolin@imec.msu.ru

Abstract Study of the dynamics of a multidimensional solid depends on the force-field structure. As reference results, we consider the equations of motion of low-dimensional solids in the field of a medium-drag force. Then it becomes possible to generalize the dynamic part of equations to the case of the motion of a solid, which is multidimensional in a similarly constructed force field, and to obtain the full list of transcendental first integrals. The obtained results are of importance in the sense that there is a nonconservative moment in the system, whereas it is the potential force field that was used previously.

1

Intro duction

We study nonconservative systems for which the methods for studying, for example, Hamiltonian systems is not applicable in general. Therefore, for such systems, it is necessary, in some sense, to "directly" integrate the main equation of dynamics. We generalize old cases and also obtain new cases of complete integrability in transcendental functions in two-, three-, and four-dimensional rigid body dynamics in a nonconservative force field. We obtain a whole spectrum of complete integrability cases for nonconservative dynamical systems having nontrivial symmetries. Moreover, in almost all cases, each of the first integrals is expressed through a finite combination of elementary functions, being one transcendental function of its variables. In this case, the transcendence is understood in the complex analysis sense, when after continuation of given functions to the complex domain, they have essentially singular points. The latter fact is stipulated by the existence of attracting and repelling sets in the system (for example, attracting and repelling foci) [1]. We introduce the class of autonomous dynamic systems having one periodic phase coordinate, and therefore, possessing the certain symmetries which are typical for the pendulum-like systems. We show that offered class of systems are embedded to the class of zero mean variable dissipation systems by natural way, i.e., on the average, for the period of the existing periodic coordinate, the sop and diffusing to energy balance to each other in certain sense. We offer the examples of pendulum-like systems on lower-dimension manifolds from dynamics of a rigid body in a nonconservative field of force [1, 2, 3, 4]. In [1, 5] the obtained results are systematized on study of the dynamic equations of the motion of symmetrical three-dimensional (3D-) rigid body which residing in a certain nonconservative field of the forces. Its type is also unoriginal from dynamics of the real rigid bodies interacting with a resisting medium on the laws of a jet flow, under which the nonconservative tracing force acts onto the body, and it either forces the value of the 435

Proceedings of XLI International Summer SchoolConference APM 2013 velocity of a certain typical point of the rigid body to remain as constant in all time of motion, that means the presence in system of nonintegrable servo-constraint. Therefore, in [5] three additional transcendental first integrals expressing through the finite combination of elementary functions are found to having analytical invariant relations (nonintegrable constraint and the integral on the equality to zero one of the component of angular velocity). The question on tensor of inertia of four-dimensional (4D-) rigid body is considered. It is proposed to study two possible cases logically on principal moments of inertia, i.e., when there exists two relations on the principal moments of inertia: (i) when there exist three equal principal moments of inertia (I2 = I3 = I4 ); (ii) when there exist two pairs of equal moments of inetria (I1 = I2 , I3 = I4 ). In this activity the obtained results are systematized on study of the dynamic equations of the motion of symmetrical four-dimensional (4D-) rigid body which residing in a certain nonconservative field of the forces for the case (ii). Its type is also unoriginal from dynamics of lower-dimensional real rigid bodies interacting with a resisting medium on the laws of a jet flow, under which the nonconservative tracing force acts onto the body, and it forces both the value of the velocity of a certain typical point of the rigid body and the certain phase variable to remain as constant in all time, that means the presence in system of nonintegrable servo-constraints. Many results of this work were regularly reported at numerous workshops, including the workshop "Actual Problems of Geometry and Mechanics" named after professor V. V. Trofimov led by D. V. Georgievskii and M. V. Shamolin.

2
2.1

Cases of integrability corresponding to a rigid b ody motion in four-dimensional space
More general problem of the motion with the tracing force

Let consider the motion of a homogeneous dynamically symmetric rigid body with "the front end-wall" (two-dimensional disk interacting with a medium which filling the fourdimensional space) in the field of force S of the resistance under the conditions of quasistationarity. Let (v , , 2 , 1 ) are the coordinates of the vector velocity of a certain typical point D of a rigid body (D is the center of two-dimensional disk) such that is the angle between the vector vD and the plane Dx1 x2 , 2 is the angle measured in the plane Dx1 x2 up to the pro jection of the vector vD on the plane Dx1 x2 , 1 is the angle mesured in the plane Dx3 x4 up to the pro jection of the vector vD on the plane Dx3 x4 , 0 -6 5 -3 0 -4 2 = 6 0 -1 -5 4 3 -2 1 0 is the angular velocity tensor of the body, Dx1 x2 x3 x4 is the coordinate system related to the body, herewith, the straight line C D lies in the plane Dx1 x2 (C is the center of mass), and the axes Dx3 , Dx4 lie in the disk plane, I1 , I2 = I1 , I3 , I4 = I3 , m are the inertiamass characteristics. Let accept the following decompositions in the pro jections on the axes of the coordinate system Dx1 x2 x3 x4 : DC = { sin , - cos , 0, 0}, 436

New cases of integrability in multidimensional dynamics in a nonconservative field vD = {v cos sin 2 , v cos cos 2 , v sin cos 1 , v sin sin 1 }. (1)

Herewith, in our case the decomposition will be also correct for the function of a medium interaction on four-dimensional body: S = {S1 , S2 , 0, 0}, S1 = S sin , S2 = -S cos , = const, i.e. in this case F = S, and the angle is measured in the plane Dx1 x2 . Then those part of dynamic equations of the body motion (including and in the case of Chaplygin analytical functions, see below) which describes the center of mass motion and corresponds to the space R4 under which the tangent forces to three-dimensional disk are absent, has the form: v cos sin 2 - v sin sin 2 + 2 v cos cos 2 - -6 v cos cos 2 + 5 v sin cos 1 - 3 v sin sin 1 -
2 2 2 - (6 + 5 + 3 ) sin - (4 5 + 2 3 ) cos + 6 cos =

S1 , m

(2)

v cos cos 2 - v sin cos 2 - 2 v cos sin 2 + +6 v cos sin 2 - 4 v sin cos 1 + 2 v sin sin 1 +
2 2 2 + (6 + 4 + 2 ) cos + (4 5 + 2 3 ) sin + 6 sin =

S2 , m

(3)

v sin cos 1 + v cos cos 1 - 1 v sin sin 1 - -5 v cos sin 2 + 4 v cos cos 2 - 1 v sin sin 1 + + (4 6 - 1 3 ) sin - (5 6 + 1 2 ) cos - 5 sin - 4 cos = 0, v sin sin 1 + v cos sin 1 + 1 v sin cos 1 + +3 v cos sin 2 - 2 v cos cos 2 + 1 v sin cos 1 - - (2 6 + 1 5 ) sin + (3 6 - 1 4 ) cos + 3 sin + 2 cos = 0, (5) (4)

where S = s()v 2 , = C D, v > 0. Those part of the dynamic equations of the body motion which describes the body motion around the center of mass, and corresponds to the Lie algebra so(4), has the form: (4 + 3 )1 + (3 - 4 )(3 5 + 2 4 ) = 0, (2 + 4 )2 + (2 - 4 )(3 6 - 1 4 ) = -x
4N

(6) , 1 , 2 , , 1 , 2 , , 1 , 2 , , 1 , 2 , v v v v s()v 2 cos , s()v 2 sin , s()v 2 cos , s()v 2 sin , (7) (8) (9) (10) (11)

(4 + 1 )3 + (4 - 1 )(2 6 + 1 5 ) = -x4 (3 + 2 )4 + (2 - 3 )(5 6 + 1 2 ) = x (1 + 3 )5 + (3 - 1 )(4 6 - 1 3 ) = x

N

3N

3N

(1 + 2 )6 + (1 - 2 )(4 5 + 2 3 ) = 0.

Thus, the following direct product of four-dimensional manifold on the Lie algebra so(4) is the phase space of the tenth order system (2)(5), (6)(11): R1 в S3 в so(4). 437

Proceedings of XLI International Summer SchoolConference APM 2013 We notice right now that the system (2)(5), (6)(11), by virtue of the having dynamical symmetry I1 = I2 , I3 = I4 , possesses the cyclic first integrals
0 0 1 1 = const, 6 6 = const.

(12)

(13)

Herewith, hereinafter we shall consider the dynamics of the system on zero level:
0 0 1 = 6 = 0.

(14)

And if there exists the more general problem of the body motion with the certain tracing force T, which acts on the plane Dx1 x2 and providing the fulfillment of the following equalities in all time of the motion v const, 2 const, (15)
1

that in the system (2)(5), (6)(11) the values T1 + S1 , T2 + S2 will stand instead of F and F2 accordingly. Let assign the following function:
v ,2

, 1 ,

v

=x

3N

, 1 , 2 ,

v

cos 1 + x

4N

, 1 , 2 ,

v

sin 1 .

(16)

It makes possible to look at this procedure from two positions. In first, the transformation of the system has occurred at presence of the tracing (control) force in the system which provides the consideration of interesting classes of the motion (15). In second, it makes possible to look at this like the procedure which allows to deflate the system. Really, the system (2)(5), (6)(11) as a result of that action generates an independent system of the sixth order of the following type: v cos cos 1 - 1 v sin sin 1 - 5 v cos sin 2 + +4 v cos cos 2 - 5 sin - 4 cos = 0, v cos sin 1 + 1 v sin cos 1 + 3 v cos sin 2 - -2 v cos cos 2 + 3 sin + 2 cos = 0, (I1 + I3 )2 = -x (I1 + I3 )3 = -x (I1 + I3 )4 = x3 (I1 + I3 )5 = x3
4N

(17)

(18) (19) (20) (21) (22)

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

v v

s()v 2 cos , s()v 2 sin , s()v 2 cos , s()v 2 sin ,

4N

N

N

in which the parameters v , 2 are added to the constant parameters specified above. 438

New cases of integrability in multidimensional dynamics in a nonconservative field

2.2

Two systems of the discourses on integrability

Remark 1 (on analytical first integrals). Obviously that the system (17)(22) possesses two analytical first integrals which are expressed in terms of the finite combination of the elementary functions: 2 sin - 3 cos = W1 = const, 4 sin - 5 cos = W2 = const. (23) (24)

First of all this means that the system (17)(22) can be reduced to the fourth order system on its own four-dimensional phase manifold. Hereafter, it makes possible to develop by the following ways under the study of the system (17)(22) (i.e. to accept the following systems of the discourses). I. In first, it makes possible "not to notice" the existence in the system the first integrals of the forms (23), (24). Then conducting the series of the equivalent transformations it can possible try to reduce the investigated system (17)(22) to the equivalent system in which the reduction to the systems of lower dimensionality will occur. Herewith, it is sufficient to get the quantity of the independent first integrals smaller then previous one on two units for the complete system integration, by virtue of (23), (24). I I. In second, it makes possible to use the first integrals (23), (24) expressing two interested phase variables from the list 2 , 3 , 4 , 5 . Herewith, we shall get just the fourth order system as the system which is the reduction of the system (17)(22) to the certain four-dimensional phase manifold. In the beginning we shall choose the system of the discourses I. Really, the system (17)(22) is equivalent to v cos - 5 v cos cos 1 sin 2 + 4 v cos cos 1 cos 2 + +3 v cos sin 1 sin 2 - 2 v cos sin 1 cos 2 - - 5 sin cos 1 - 4 cos cos 1 + 3 sin sin 1 + 2 cos sin 1 = 0, 1 v sin + 3 v cos cos 1 sin 2 - 2 v cos cos 1 cos 2 + +5 v cos sin 1 sin 2 - 4 v cos sin 1 cos 2 + + 3 sin cos 1 + 2 cos cos 1 + 5 sin sin 1 + 4 cos sin 1 = 0, 2 = - 3 = - 4 = 5 = v2 x I1 + I3 v2 x I1 + I3
4N

(25)

(26) (27) (28) (29) (30)

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

v v

s() cos , s() sin , s() cos , s() sin ,

4N

v2 x I1 + I3 v2 x I1 + I3

3N

3N

Let introduce new quasivelocities in the system. We shall transform the values 2 , 3 , 4 , 5 by means of the composition of the following rotations for this: z1 -z = T (-1 )
3 5

,

2

z3 -z

4

= T (-1 )

2 4

,

(31) 439

Proceedings of XLI International Summer SchoolConference APM 2013 where T (1 ) = and also w w
1 2

cos 1 - sin 1 sin 1 cos 1

,

(32)

= T (2 )

z z

3 1

,

w w

3 4

= T (-2 )

-z z2

4

.

(33)

Thus, the following relations are correct: z1 = 3 cos 1 + 5 sin 1 , z2 = 3 sin 1 - 5 cos 1 , z3 = 2 cos 1 + 4 sin 1 , z4 = 2 sin 1 - 4 cos 1 , w1 = -z1 sin 2 + z3 cos 2 , w2 = z3 sin 2 + z1 cos 2 , w3 = z2 sin 2 - z4 cos 2 , w4 = z4 sin 2 + z2 cos 2 . As is seen from (25)(30), on the manifold O2 = (, 1 , 2 , 3 , 4 , 5 ) R6 : = k, k Z 2 (35) (34)

it is impossible to resolve the system uniquely relatively to , 1 . Thus, the violation of the uniqueness theorem is happened on the manifold (35) formally. Moreover, in first, the indefiniteness is happened for even or odd k by the reason of degeneration of the coordinates (v , , 1 , 2 ) which are parameterized the three-dimensional sphere (but are not the classical spherical coordinates), and, in second, it is happened the evident violation of the uniquiness theorem for odd k because of the first equation of (25) degenerates for this case. It follows that the system (25)(30) outside of and only outside of the manifold (35) is equivalent to the system = -w3 + v s() · I1 + I3 cos w4 = - +w w
v ,2

, 1 ,

v

, v

(36)

v2 s() sin(2 + ) · I1 + I3
v ,2

v ,2

, 1 ,

+ (37)

2

1

v s() cos - · sin I1 + I3 sin w3 =

, 1 ,

v
v ,2

, v

v2 s() cos(2 + ) · I1 + I3
v ,2

, 1 , ,

- (38)

-w

1

w

1

cos v s() - · sin I1 + I3 sin w2 =

, 1 ,

v
v ,2

v2 s() sin(2 + ) · I1 + I3
v ,2

, 1 , ,

v

- (39)

-w 440

4

w

1

cos v s() - · sin I1 + I3 sin

, 1 ,

v

New cases of integrability in multidimensional dynamics in a nonconservative field v2 s() cos(2 + ) · I1 + I3
,2

w1 = +w w

v ,2

, 1 , ,

v

+ (40) (41)

3

1

cos v s() - · v sin I1 + I3 sin

, 1 , , 1 ,

v ,

cos v s() 1 = w1 - · sin I1 + I3 sin where
v ,2

v ,2

v

, 1 ,

v
v ,2

= -x

4N

, 1 ,

v

cos 1 + x

3N

, 1 ,

v

sin 1 ,

(42)

and the function

, 1 , /v is represented in the form (16).

2.3
2.3.1

Case of the dep endence of the moment of the nonconservative forces on the angular velo city
Intro duction on the dep endence on the angular velo city

Let x = (x1N , x2N , x3N , x4N ) are the coordinates of the point N of the action of the nonconservative force (of a medium interaction) to two-dimensional disk, Q = (Q1 , Q2 , Q3 , Q4 ) are the components not depending on the angular velocity tensor. We shall introduce the dependence of the functions (x1N , x2N , x3N , x4N ) on the angular velocity tensor by the linear form only since given introduction itself is not obvious a priori. And so, let accept the following dependence: x = Q + R, where R = (R1 , R2 , R3 , R4 ) is the vector-function containing the components of angular velocity tensor. Herewith, the dependence of the function R on the angular velocity tensor is gyroscopic: R1 0 -6 5 -3 h1 1 0 -4 2 h2 R R= 2 =- 6 (43) . 0 -1 h3 v -5 4 R3 R4 3 -2 1 0 h4 Here (h1 , h2 , h3 , h4 ) are the certain positive parameters. And now, with the reference to our problem, since x x 2.3.2
3N

1N

x

2N

0, then (44)

= Q3 -

h1 (4 - 5 ), x v

4N

= Q4 -

h1 (3 - 2 ). v

Reduced system

Similarly to the choice of the Chaplygin analytical functions Q3 = A sin cos 1 , Q4 = A sin sin 1 , A > 0, we shall accept the dynamic functions s, x
3N

(45) as the following form:

and x

4N

s() = B cos , B > 0, h = h1 > 0, v = 0, h = h2 > 0, x
3N

, 1 , 2 ,

v
4N

= A sin cos 1 - , 1 , 2 , v

h (4 - 5 ), v h (3 - 2 ), v

(46)

x

= A sin sin 1 -

441

Proceedings of XLI International Summer SchoolConference APM 2013 which convinces us that the additional dependence of the damping moment of the nonconservative forces (and the dispersing one in some domains of the phase space) is also present in considered system (i.e. the dependence of the moment on the angular velocity tensor is present). Moreover, h1 = h2 , h3 = h4 by virtue of the dynamical symmetry (12) of the body. Later on, let accept the system of discourses I which takes into account and the system of discourses I I (see above). We shall arouse to introduce the following variables in this section: u1 = 2 - 3 , u2 = 4 - 5 , u3 = 2 cos 2 - 3 sin 2 , u4 = 4 cos 2 - 5 sin 2 . (47)

Really, the assigned coordinates are defined correctly for cos 2 = sin 2 , and Jacobian of the mapping is equal to -(cos 2 - sin 2 )-2 , herewith, the inverse transformation is assigned as follows: 2 = u3 cos u4 4 = cos - u1 2 - - u2 2 - sin sin sin sin 2 u3 - u1 cos , 3 = 2 cos 2 - sin 2 u4 - u2 cos , 5 = 2 cos 2 - sin 2 , 2 2 , 2

(48)

and the particular case cos 2 = sin 2 , which simplifies the dynamic equations can be considered separately. Then the equations (25)(30) under the condition (46) outside of and only outside of the manifold O3 = (, 1 , 2 , 3 , 4 , 5 ) R6 : = + k, k Z 2 (49)

transform to the following equations: - u3 sin 1 + u4 cos 1 - n2 v sin + H1 [-u1 sin 1 + u2 cos 1 ] = 0, 0 1 sin - cos [u3 cos 1 + u4 sin 1 ] - H1 cos [u1 cos 1 + u2 sin 1 ] = 0, u1 = -n2 v 2 r1 sin cos sin 1 - 0 u2 = n2 v 2 r1 sin cos cos 1 - 0 B vh r1 u1 cos , I1 + I3 (50) (51) (52) (53) (54) (55)

B vh r1 u2 cos , I1 + I3 B vh u1 cos cos( + 2 ), I1 + I3

u3 = -n2 v 2 sin cos sin 1 cos( + 2 ) - 0 u4 = n2 v 2 sin cos cos 1 cos( + 2 ) - 0

B vh u2 cos cos( + 2 ), I1 + I3

where r1 = cos - sin = 0, n2 = AB /(I1 + I3 ), H1 = B h/(I1 + I3 ). 0 Let introduce the following phase variables by the formulas: v1 = -u1 sin 1 + u2 cos 1 , v2 = u1 cos 1 + u2 sin 1 , v3 = -u3 sin 1 + u4 cos 1 , v4 = u3 cos 1 + u4 sin 1 . then outside of and only outside of the manifold O4 = (, 1 , u1 , u2 , u3 , u4 ) R6 : 1 = k , k Z 442 (57) (56)

New cases of integrability in multidimensional dynamics in a nonconservative field the system (50)(55) has the form = -v3 - bH1 v1 + b sin , cos 1 = [v4 + bH1 v2 ] , sin v1 = n2 v 2 r1 sin cos - H1 v r1 v1 cos - v2 · [v4 + bH1 v2 ] 0 v2 = -H1 v r1 v2 cos + v1 · [v4 + bH1 v2 ] cos , sin cos , sin (58) (59) (60) (61) cos , (62) sin (63)

v3 = n2 v 2 sin cos cos( + 2 ) - H1 v v1 cos cos( + 2 ) - v4 · [v4 + bH1 v2 ] 0 cos , sin where we introduce as before the dimensionless parameters as follows: v4 = -H1 v v2 cos cos( + 2 ) + v3 · [v4 + bH1 v2 ] n2 = 0 H Bh AB , b = n0 , [b] = 1, H1 = 1 = , [H1 ] = 1. I1 + I3 n0 (I1 + I3 )n0

(64)

Let also introduce one more auxiliary change of the part of the phase variables, as follows: s1 = v3 + bH1 v1 , s2 = v4 + bH1 v2 . (65)

Then the investigated system (58)(63) after the introduction of dimensionless variables and differentiability vk n0 v vk , k = 1, . . . , 4, < · >= n0 v < >, will rewrite as the form = -s1 + b sin , 1 = s
2

(66) (67) cos - R1 H1 v1 cos , sin (68)

cos , sin

s1 = R1 sin cos - s2 2 s2 = s1 s
2

cos - R1 H1 v2 cos , (69) sin cos v1 = R2 sin cos - s2 v2 - H1 R2 v1 cos , (70) sin cos v2 = s2 v1 - H1 R2 v2 cos , (71) sin where R1 = bH1 (cos - sin ) + cos( + 2 ), R2 = r1 = cos - sin . Obviously, that for H1 = 0 formally the independent fourth order subsystem (66)(69) stands out in the system (66)(71) on the tangent stratification T S2 to two-dimensional sphere S2 {0 < < , 0 1 < 2 }, in which, in turn, it can be stand out the independent third order subsystem (66), (68), (69) on its own three-dimensional phase manifold. And in the given case it is great for us that H1 = 0. Therefore, we transform the having analytical first integrals (23), (24). We have the evident type of its in the different variables: u3 - u1 cos 2 u3 - u1 sin 2 sin - cos = W1 = const, cos - 2 - sin 2 cos - 2 - sin 2 u4 - u2 sin 2 u4 - u2 cos 2 sin - cos = W2 = const. cos - 2 - sin 2 cos - 2 - sin 2 (72) (73) 443

Proceedings of XLI International Summer SchoolConference APM 2013 If we consider the case (15) (i.e., in particular, when the value 2 is the identical constant along the phase tra jectories), then the following analytical functions are constant on the phase tra jectories of the considered system:
0 u3 (sin - cos ) + u1 cos( + 2 ) = W1 = const, 0 u4 (sin - cos ) + u2 cos( + 2 ) = W2 = const.

(74) (75)

In another variables the latter two invariant relations have the forms
0 R1 v2 cos 1 - R1 v1 sin 1 + R2 [s1 sin 1 - s2 cos 1 ] = W1 = const, 0 R1 v2 sin 1 + R1 v1 cos 1 - R2 [s1 cos 1 + s2 sin 1 ] = W2 = const,

(76) (77)

where R1 = cos( + 2 ) + bH1 (cos - sin ), R2 = cos - sin as above. Later on, let express from the relations (76), (77) the values v1 , v2 . We have:
0 0 0 0 v2 R1 = R2 s2 + 1 (1 , W1 , W2 ), v1 R1 = R2 s1 + 2 (1 , W1 , W2 ),

(78)

where
0 0 0 0 1 (1 , W1 , W2 ) = W1 cos 1 + W2 sin 1 , 0 0 0 0 2 (1 , W1 , W2 ) = W2 cos 1 - W1 sin 1 .

(79)

Then the system (66)(69) has the form of the independent fourth order system: = -s1 + b sin , s1 = R1 sin cos - s s2 = s1 s
2 2 2

(80) cos 0 0 - R2 H1 s1 cos - H1 2 (1 , W1 , W2 ) cos , sin (81)

cos 0 0 - R2 H1 s2 cos - H1 1 (1 , W1 , W2 ) cos , (82) sin cos 1 = s2 . (83) sin The system (80)(83) can be considered as the system (66)(69) which is reduced to 0 0 the levels (W1 , W2 ) of the analytical first integrals (76), (77). Obviously, that 1 (1 , 0, 0) 2 (1 , 0, 0) 0. Therefore, we shall consider the system (80)(83) on the zero levels of the analytical first integrals (76), (77):
0 0 W1 = W2 = 0,

(84)

which has the form = -s1 + b sin , s1 = R1 sin cos - s s2 = s1 s
2 2 2

(85) cos - R2 H1 s1 cos , sin (86) (87) (88)

cos - R2 H1 s2 cos , sin cos 1 = s2 . sin

The given system can be considered on the tangent stratification T S2 to two-dimensional sphere S2 {0 < < , 0 1 < 2 }, in which, in turn, it can be stand out the independent third order subsystem (85)(87) on its own three-dimensional phase manifold. 444

New cases of integrability in multidimensional dynamics in a nonconservative field And so, for the integration of the sixth order system at the beginning we used the system of discourses I (see above), when we did not yet take into account the existence of two independent analytical first integrals of the forms (23), (24). In consequence we have limited (reduced) the considered sixth order system on the levels (in consequence zero) of the assigned first integrals, i.e. the system of discourses I I was used (see above). Theorem 1. The system (2)(5), (6)(11) under the conditions (15), (46), (14), (84) possesses nine invariant relations (the complete tuple), three of which are the transcendental functions from the complex analysis view of point. Herewith, al l the relations express in terms of the finite combination of the elementary functions. And at proof of the theorem 1 the system of discourses I I is used (see above) which implies the reduction of investigated system on (zero) levels of the analytical first integrals (23), (24). The latter fact takes into account in principal the complete tuple of the having first integrals. 2.3.3 Top ological analogies

Let consider the following third order system of the equations: sin Ё + (b - H1 ) cos + R3 sin cos - 1 2 + cos 0 0 + H1 [W1 sin 1 - W2 cos 1 ] = 0,
2 1 + cos + 1 + (b - H1 )1 cos + 1 Ё cos sin 0 0 + H1 [W1 cos 1 + W2 sin 1 ] = 0, b > 0, H

(89) > 0, medium velocity, previous variable and the

1

describing the fixed spherical pendulum which is placed in a flow of the filling under the presence of the dependence of the moment of the forces on the angular i.e. the mechanical system in the nonconservative field of the forces. Unlike activities [1, 5], the order of such system is equal to 4 (but not 3) since the phase 1 is not the cyclic, that does not reduce to the stratification of the phase space deflation. Its phase space is the tangent stratification T S2 { , 1 , , 1 }

(90)

to two-dimensional sphere S2 { , 1 }, herewith, the equation of the big circles 1 0 0 0 integral manifolds for W1 = W2 = 0 only. It is not difficult to make sure that the system (89) is equivalent to the dynamic system with the (zero mean) variable dissipation on the tangent stratification (90) to twodimensional sphere. Moreover, the following theorem is equitable. Theorem 2. The system (2)(5), (6)(11) under the conditions (15), (46), (14) is equivalent to the dynamic system (89). Really, it is sufficient to accept = , 1 = 1 , b = -b , H1 = H1 , R2 H1 = , R - bR H = R . -H1 1 21 3

3

Conclusion

In the previous studies of the author, the problems on the motion of the four-dimensional solid were already considered in a nonconservative force field in the presence of the following force. This study opens a new cycle of works on integration of a multidimensional solid in 445

Proceedings of XLI International Summer SchoolConference APM 2013 the nonconservative field because previously, as was already specified, we considered only such motions of a solid when the field of external forces was the potential.

4

Acknowledgements

This work was supported by the Russian Foundation for Basic Research, project no. 1201-00020-a.

References
[1] M. V. Shamolin, Methods of analysis of dynamical systems with various disssipation in rigid body dynamics, Moscow, Russian Federation: Ekzamen, 2007. [2] M. V. Shamolin, New cases of integrability in dynamics of a rigid body with the cone form of its shape interacting with a medium, PAMM (Proc. Appl. Math. Mech.), 9, 139140 (2009). [3] M. V. Shamolin, The various cases of complete integrability in dynamics of a rigid body interacting with a medium, Multibody Dynamics, ECCOMAS Thematic Conf. Warsaw, Poland, 29 June2 July 2009, CD-Proc.; Polish Acad. Sci., Warsaw, 2009, 20 p. [4] M. V. Shamolin, Dynamical systems with various dissipation: background, methods, applications // CD-Proc. of XXXVIII Summer School-Conf. "Advances Problems in Mechanics" (APM 2010), July 15, 2010, St. Petersburg (Repino), Russia; St. Petersburg, IPME, 2010, p. 612621. [5] M. V. Shamolin, Variety of the cases of integrability in dynamics of a 2D-, 3D-, and 4D-rigid body interacting with a medium, Proc. of 11th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2011), Lodz, Poland, Dec. 58, 2011; Tech. Univ. Lodz, 2011, p. 1124.
Maxim V. Shamolin, Lomonosov Moscow State University, Russian Federation

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