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New Developments in Pure and Applied Mathematics

Certain Integrable Cases in Dynamics of a Multi-Dimensional Rigid Body in a Nonconservative Field
Maxim V. Shamolin
In basic part we recall general aspects of the dynamics of a free multi-dimensional rigid body: the notion of the tensor of angular velocity of the body, the joint dynamical equations of motion on the direct product Rn в so(n), and the Euler and Rivals formulas in the multi-dimensional case. We also consider the tensor of inertia of a f ve-dimensional (5D-) rigid body. In this work, we study one of two possible cases in which there exists two relations between the principal moments of inertia: (i) there are four equal principal moments of inertia (I2 = I3 = I4 = I5 ). Furthermore, we systematize results on the study of equations of motion of a f ve-dimensional (5D-) rigid body in a nonconservative force fiel for the case (i). The form of these equations is taken from the dynamics of realistic rigid bodies of lesser dimension that interact with a resisting medium by laws of jet fl w when the is influence by a nonconservative tracing force. Under the action of this force, the following two cases are possible. In this case, the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo-constraint (see also [10], [11]). The results relate to the case where all interaction of the medium with the body part is concentrated on a part of the surface of the body, which has the form of a four-dimensional disk, and the action of the force is concentrated in the direction perpendicular to this disk. These results are systematized and are preserved in the invariant form. Moreover, we introduce an extra dependence of the moment of the nonconservative force on the angular velocity. This dependence can be further extended to cases of the motion in spaces of higher dimension. Many results of this paper were regularly presented on scientifi seminars, including the seminar Actual problems of geometry and mechanics named after Prof. V. V. Trofim v under the supervision of D. V. Georgievskii and M. V. Shamolin. January 18, 2015 II. GENERAL DISCOURSE A. Cases of dynamical symmetry of a five-dimensional body Let a f ve-dimensional rigid body of mass m with smooth four-dimensional boundary be under the influenc of a nonconservative force field this can be interpreted as a motion of the body in a resisting medium that fill up f ve-dimensional domain of Euclidean space E5 . We assume that the body

Abstract--This paper is a survey of integrable cases in dynamics of a five-dimensional rigid body under the action of a nonconservative force field. We review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean. The problem of the search for complete sets of transcendental first integrals of systems with dissipation is quite actual; a large number of works are devoted to it. We introduce a new class of dynamical systems that have a periodic coordinate. Due to the existence of a nontrivial symmetry groups of such systems, we can prove that these systems possess variable dissipation with zero mean, which means that on the average for a period with respect to the periodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either the energy pumping or dissipation can occur. Based on facts obtained, we analyze dynamical systems that appear in dynamics of a five-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions that can be expresses through a finite combination of elementary functions. Index Terms--Case of integrability, dynamic part of motion equations, multidimensional rigid body.

I . I N T RO D U C T I O N HIS This paper is a survey of integrable cases in dynamics of a f ve-dimensional rigid body under the action of a nonconservative force field We review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean. We study nonconservative systems for which usual methods of the study of Hamiltonian system is inapplicable. Thus, for such systems, we must directly integrate the main equation of dynamics (see also [1], [2], [3], [4], [5], [6]). We generalize previously known cases and obtain new cases of the complete integrability in transcendental functions of the equation of dynamics of a f ve-dimensional rigid body 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 difficul task. In a number of cases, where the systems considered have additional symmetries, we succeed in findin firs integrals through finit combinations of elementary functions [6], [7], [8], [9].

T

Maxim V. Shamolin is with the Institute of Mechanics, Lomonosov Moscow State University, Moscow, 119192, Russian Federation; e-mail: shamolin@rambler.ru, shamolin@imec.msu.ru (see also http://shamolin2.imec.msu.ru).

ISBN: 978-1-61804-287-3

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New Developments in Pure and Applied Mathematics

is dynamically symmetric. If the body has two independent principal moments of inertia, then in some coordinate system Dx1 x2 x3 x4 x5 attached to the body, the operator of inertia has the form diag{I1 , I2 , I2 , I2 , I2 }, (1) or the form diag{I1 , I1 , I3 , I3 , I3 }. (2)

where 1 , 2 , . . . , 10 are the components of the tensor of angular velocity corresponding to the projections on the coordinates of the Lie algebra so(5). Obviously, the following relations hold: i - j = I j - I
i

(7)

In the firs case, the body is dynamically symmetric in the hyperplane Dx2 x3 x4 x5 . B. Dynamics on so(5) and R The configuratio space of a free, n-dimensional rigid body is the direct product Rn в SO(n) (3) of the space Rn , which define the coordinates of the center of mass of the body, and the rotation group SO(n), which define rotations of the body about its center of mass and has dimension n(n + 1) n(n - 1) = . n+ 2 2 Therefore, the dynamical part of 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 angular velocity of a f vedimensional rigid body (it is a second-rank tensor, see [12], [13], [14], [15], [16]), so(5), then the part of dynamical equations of motion corresponding to the Lie algebra so(5) has the following form (see [17], [18]): + + [, + ] = M , where = diag{1 , 2 , 3 , 4 }, (4) (5)
5

for any i, j = 1, . . . , 5. For the calculation of the moment of an external force acting to the body, we need to construct the mapping R5 в R5 - so(5), than maps a pair of vectors (DN, F) R5 в R5 (9) (8)

from R5 в R5 to an element of the Lie algebra so(5), where DN = {0, x2N , x3N , x4N , x F = {F1 , F2 , F3 , F4 , F5 },
5N

},

(10)

and F is an external force acting to the body. For this end, we construct the following auxiliary matrix 0 F1 x2 F
N

x3

N

x

4N

x5 F

N

2

F3

F4

5

.

(11)

Then the right-hand side of system (4) takes the form M = {x4N F5 - x5N F4 , x x
2N 5N

F3 - x3N F5 , F4 - x4N F3 ,

F5 - x5N F2 , x
2N

5N

F1 , x

3N

x4N F2 - x

F4 , -x

4N

F1 , x2N F3 - x3N F2 ,
2N

x3N F1 , -x

F1 }.

(12)

-I1 + I2 + I3 + I4 + I5 , 1 = 2 I1 - I2 + I3 + I4 + I5 , 2 = 2 I1 + I2 - I3 + I4 + I5 , 3 = 2 I1 + I2 + I3 - I4 + I5 , 4 = 2 I1 + I2 + I3 + I4 - I5 , 5 = 2 M = MF is the natural projection of the moment of external forces F acting to the body in R5 on the natural coordinates of the Lie algebra so(5), and [ ] is the commutator in so(5). The skew-symmetric matrix corresponding to this second-rank tensor so(5) we represent in the form 0 -10 9 -7 4 0 -8 6 -3 10 -9 8 0 -5 2 , (6) -6 5 0 -1 7 -4 3 -2 1 0
ISBN: 978-1-61804-287-3

Dynamical systems studied in the following, generally speaking, are not conservative; they are dynamical systems with variable dissipation with zero mean (see [12]). We need to examine by direct methods a part of the main system of dynamical equations, namely, the Newton equation, which plays the role of the equation of motion of the center of mass, i.e., the part of the dynamical equations corresponding to the space R5 : mwC = F, (13) where wC is the acceleration of the center of mass C of the body and m is its mass. Moreover, due to the higherdimensional Rivals formula (it can be obtained by the operator method) we have the following relations: wC = wD + 2 DC + E DC, w
D

= vD + vD , E = , (14)

where wD is the acceleration of the point D, F is the external force acting on the body (in our case, F = S), and E is the tensor of angular acceleration (second-rank tensor). So, the system of equations (4) and (13) of fifteent order on the manifold R5 в so(5) is a closed system of dynamical equations of the motion of a free f ve-dimensional rigid body under the action of an external force F. This system have been separated from the kinematic part of the equations of motion on the manifold (3) and can be examined independently.

329

New Developments in Pure and Applied Mathematics

I I I . G E N E R A L P RO B L E M O N T H E M OT I O N U N D E R A T R AC I N G F O R C E Consider a motion of a homogeneous, dynamically symmetric (case (1)), rigid body with front end face (a fourdimensional disk interacting with a medium that fill the f vedimensional space) in the fiel of a resistance force S under the quasi-stationarity conditions. Let (v , , 1 , 2 , 3 ) be the (generalized) spherical coordinates of the velocity vector of the center of the fourdimensional disk lying on the axis of symmetry of the body, 0 -10 9 -7 4 0 -8 6 -3 10 -9 8 0 -5 2 = -6 5 0 -1 7 -4 3 -2 1 0 be the tensor of angular velocity of the body, Dx1 x2 x3 x4 x5 be the coordinate system attached to the body such that the axis of symmetry C D coincides with the axis Dx1 (recall that C is the center of mass), and the axes Dx2 , Dx3 , Dx4 , Dx5 lie in the hyperplane of the disk, and I1 , I2 , I3 = I2 , I4 = I2 , I5 = I2 , m are characteristics of inertia and mass. We adopt the following expansions in the projections to the axes of the coordinate system Dx1 x2 x3 x4 x5 : DC = {-, 0, 0, 0, 0}, v
D

-2 v sin sin 1 sin 2 - 9 v cos + 8 v sin cos 1 - -5 v sin sin 1 sin 2 cos 3 + +2 v sin sin 1 sin 2 sin 3 - - (8 10 - 5 7 - 2 4 ) + 9 = 0, +1 v sin cos 1 sin 2 cos 3 + +2 v sin sin 1 cos 2 cos 3 - -3 v sin sin 1 sin 2 sin 3 + 7 v cos - 6 v sin cos 1 + +5 v sin sin 1 cos 2 - 1 v sin sin 1 sin 2 sin 3 + + (6 10 + 5 9 - 1 4 ) - 7 = 0, +1 v sin cos 1 sin 2 sin 3 + +2 v sin sin 1 cos 2 sin 3 + +3 v sin sin 1 sin 2 cos 3 - 4 v cos + 3 v sin cos 1 - -2 v sin sin 1 cos 2 + 1 v sin sin 1 sin 2 cos 3 - - (3 10 + 2 9 + 1 7 ) + 4 = 0, where S = s()v 2 , = C D, v > 0. (21) (22) (20) (19)

v sin sin 1 sin 2 cos 3 + v cos sin 1 sin 2 cos 3 +

v sin sin 1 sin 2 sin 3 + v cos sin 1 sin 2 sin 3 +

= {v cos , v sin cos 1 , v sin sin 1 cos 2 ,

v sin sin 1 sin 2 cos 3 , v sin sin 1 sin 2 sin 3 }. (15) In the case (1) we additionally have the expansion for the function of the influenc of the medium on the f vedimensional body: S = {-S, 0, 0, 0, 0}, (16) i.e., in this case F = S. Then the part of the dynamical equations of motion (including the analytic Chaplygin functions; see below) that describes the motion of the center of mass and corresponds to the space R5 , in which tangent forces of the influenc of the medium on the four-dimensional disk vanish, takes the form v cos -v sin -10 v sin cos 1 +9 v sin sin 1 cos 2 - -7 v sin sin 1 sin 2 cos 3 +4 v sin sin 1 sin 2 sin 3 +
2 2 2 2 + (10 + 9 + 7 + 4 ) = -

Further, the auxiliary matrix (11) for the calculation of the moment of the resistance force has the form 0 -S x2N 0 x3N 0 x4N 0 x5N 0 , (23)

then the part of the dynamical equations of motion that describes the motion of the body about the center of mass and corresponds to the Lie algebra so(5), becomes (4 + 5 )1 + (4 - 5 )(4 7 + 3 6 + 2 5 ) = 0, (24) (3 + 5 )2 + (5 - 3 )(1 5 - 3 8 - 4 9 ) = 0, (25) (2 + 5 )3 + (2 - 5 )(4 10 - 2 8 - 1 6 ) = 0, (26) (1 + 5 )4 + (5 - 1 )(3 10 + 2 9 + 1 7 ) = = -x
5N

, 1 , 2 , 3 ,

v

s()v 2 ,

(27)

S , m

(3 + 4 )5 + (3 - 4 )(7 9 + 6 8 + 1 2 ) = 0, (28) (2 + 4 )6 + (4 - 2 )(5 8 - 7 10 - 1 3 ) = 0, (29) (1 + 4 )7 + (1 - 4 )(1 4 - 6 10 - 5 9 ) = =x
4N

(17)

v sin cos 1 + v cos cos 1 - 1 v sin sin 1 + +10 v cos - 8 v sin sin 1 cos 2 + +6 v sin sin 1 sin 2 cos 3 - -3 v sin sin 1 sin 2 sin 3 - - (9 8 + 6 7 + 3 4 ) - = 0, 10 v sin sin 1 cos 2 + v cos sin 1 cos 2 + +1 v sin cos 1 cos 2 -
ISBN: 978-1-61804-287-3 330

, 1 , 2 , 3 ,

v

s()v 2 ,

(30)

(2 + 3 )8 + (2 - 3 )(9 10 + 5 6 + 2 3 ) = 0, (31) (18) (1 + 3 )9 + (3 - 1 )(8 10 - 5 7 - 2 4 ) = = -x
3N

, 1 , 2 , 3 ,

v

s()v 2 ,

(32)

(1 + 2 )10 + (1 - 2 )(8 9 + 6 7 + 3 4 ) =

New Developments in Pure and Applied Mathematics

=x

2N

, 1 , 2 , 3 ,

v

s()v 2 .

(33)

+10 v cos - = 0, 10 v cos sin 1 cos 2 + 1 v sin cos 1 cos 2 - -2 v sin sin 1 sin 2 - 9 v cos + 9 = 0, +2 v sin sin 1 cos 2 cos 3 -

(42) (43)

Thus, the phase space of system (17)(21), (24)(33) of fifteent order is the direct product of the f ve-dimensional manifold and the Lie algebra so(5): R в S в so(5).
1 4

(34)

v cos sin 1 sin 2 cos 3 + 1 v sin cos 1 sin 2 cos 3 + -3 v sin sin 1 sin 2 sin 3 + 7 v cos - 7 = 0, (44) v cos sin 1 sin 2 sin 3 + 1 v sin cos 1 sin 2 sin 3 + +2 v sin sin 1 cos 2 sin 3 + +3 v sin sin 1 sin 2 cos 3 - 4 v cos + 4 = 0, (45) 3I2 4 = -x 3I2 7 = x
5N

We note that system (17)(21), (24)(33), due to the existing dynamical symmetry I2 = I3 = I4 = I5 , possesses cyclic firs integrals
0 0 0 0 0 0 1 1 , 2 2 , 3 3 , 5 5 , 6 6 , 8 8 . (36) In the sequel, we consider the dynamics of the system on zero levels: 0 0 0 0 0 0 1 = 2 = 3 = 5 = 6 = 8 = 0.

(35)

, 1 , 2 , 3 , , 1 , 2 , 3 ,

v

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

(46) (47) (48) (49)

(37)

4N

v v v

If one considers a more general problem on the motion of a body under a tracing force T that lies on the straight line C D = Dx1 and provides the fulfillmen of the relation v const (38) throughout the motion, then instead of F1 system (17)(21), (24)(33) contains T - s()v 2 , = DC. (39) Choosing the value T of the tracing force appropriately, one can achieve the equality (38) throughout the motion. Indeed, expressing T due to system (17)(21), (24)(33), we obtain for cos = 0 the relation
2 2 2 2 T = Tv (, 1 , 2 , 3 , ) = m (4 + 7 + 9 + 10 )+

3I2 9 = -x

3N

, 1 , 2 , 3 , , 1 , 2 , 3 ,

3I2 10 = x2

N

which, in addition to the permanent parameters specifie above, contains the parameter v . System (42)(49) is equivalent to the system v cos + v cos {10 cos 1 + +[(7 cos 3 - 4 sin 3 ) sin 2 - 9 cos 2 ] sin 1 }+ + {- cos 1 + [9 cos 2 - 10 -(7 cos 3 - 4 sin 3 ) sin 2 ] sin 1 } = 0, 1 v sin + v cos {[(7 cos 3 - -4 sin 3 ) sin 2 - 9 cos 2 ] cos 1 - 10 sin 1 }+ + {[9 cos 2 - (7 cos 3 - -4 sin 3 ) sin 2 ] cos 1 + sin 1 } = 0, 10 (51) 2 v sin sin 1 + v cos {[7 cos 3 - -4 sin 3 ] cos 2 + 9 sin 2 }+ + {- [7 cos 3 - 4 sin 3 ] cos 2 - 9 sin 2 } = 0, (52) 3 v sin sin 1 sin 2 + v cos {-4 cos 3 - 7 sin 3 } + (50)

+s()v where

2

1-

m sin v , 1 , 2 , 3 , 3I2 cos v
v

,

(40)

, 1 , 2 , 3 , v v v

=

= x5 +x

N

, 1 , 2 , 3 , , 1 , 2 , 3 ,
3N

sin 1 sin 2 sin 3 + sin 1 sin 2 cos 3 + sin 1 cos 2 + v cos 1 ; (41)

4N

+x

, 1 , 2 , 3 , v
2N

+x

, 1 , 2 , 3 ,

+ {4 cos 3 + 7 sin 3 } = 0, 4 = - 7 = v x5 3I2
2 N

(53) (54) (55) (56) (57)

here we used conditions (36)(38). This procedure can be interpreted in two ways. First, we have transformed the system using the tracing force (control) that provides the consideration of the class (38) of motions interesting for us. Second, we can treat this as an order-reduction procedure. Indeed, system (17)(21), (24)(33) generates the following independent system of eighth order: v cos cos 1 - 1 v sin sin 1 +
ISBN: 978-1-61804-287-3 331

, 1 , 2 , 3 , , 1 , 2 , 3 ,

v

s(), s(), s(), s().

v2 x4 3I2

N

v v v

9 = - 10 =

v2 x3 3I2

N

, 1 , 2 , 3 , , 1 , 2 , 3 ,

v2 x2 3I2

N

New Developments in Pure and Applied Mathematics

Introduce the new quasi-velocities. For this, we transform 4 , 7 , 9 , 10 by three rotations: z1 z2 z3 = z4 = T3,4 (-1 ) T2,3 (-2 ) T1,2 (-3 ) where 4 7 9 10 , (58)

z4 =

v2 s() 3I2

v

, 1 , 2 , 3 , cos + sin

v

-

2 2 2 -(z1 + z2 + z3 )

+

v s() {-z3 3I2 sin +z2
v ,2

v ,1

, 1 , 2 , 3 , v -

v

+

, 1 , 2 , 3 , , 1 , 2 , 3 ,

-z1 z3 = z3 z4 +

v ,3

}, v

(62)

cos 2 2 cos cos 1 + (z1 + z2 ) + sin sin sin 1
v ,1

T3

,4

0 0 1 ( ) = 0 0 cos 0 0 sin ( ) = 1 0 0 cos 0 sin 0 0

10

0

0 , - sin cos 0 0 0 1 0 0 0 1 , . + +

0

v s() {z4 3I2 sin -z2
v ,2

, 1 , 2 , 3 , v v

v

-

, 1 , 2 , 3 , , 1 , 2 , 3 ,
,1

cos 1 + sin 1 cos 1 }- sin 1 v , (63)

T2

,3

0 - sin cos 0 0 0 1 0

+z1 -

v ,3

v2 s()v 3I2
4

, 1 , 2 , 3 ,

T1

,2

cos sin ( ) = 0 0

- sin cos 0 0

z2 = z2 z

cos cos 1 cos - z2 z3 - sin sin sin 1 cos 1 cos 2 + sin sin 1 sin 2 v v -z4 + z3 -z1 cos 1 sin 1 + + (64)

-z v s() 3I2 sin
v ,2

2 1

, 1 , 2 , 3 , , 1 , 2 , 3 ,
,2

Therefore, the following relations hold: z1 = 4 cos 3 + 7 sin 3 , z2 = (7 cos 3 - 4 sin 3 ) cos 2 + 9 sin 2 , z3 = [(-7 cos 3 + 4 sin 3 ) sin 2 + + 9 cos 2 ] cos 1 + 10 sin 1 , z4 = [(7 cos 3 - 4 sin 3 ) sin 2 - - 9 cos 2 ] sin 1 + 10 cos 1 .

v s() 3I2 sin

v ,3

1 cos 2 sin 1 sin 2 v ,

(59)

+

v2 s()v 3I2
4

, 1 , 2 , 3 ,

z1 = z1 z

cos cos 1 cos - z1 z3 + sin sin sin 1 cos 1 cos 2 + sin sin 1 sin 2
v ,3

As we see from (50)(57), we cannot solve the system with respect to , 1 , 2 , 3 on the manifold O1 = {(, 1 , 2 , 3 , 4 , 7 , 9 , 10 ) R8 : k , 1 = l1 , 2 = l2 , k , l1 , l2 Z}. (60) 2 Therefore, on the manifold (60) the uniqueness theorem formally is violated. Moreover, for even k and any l1 , l2 , an indeterminate form appears due to the degeneration of the spherical coordinates (v , , 1 , 2 , 3 ). For odd k , the uniqueness theorem is obviously violated since the firs equation (50) degenerates. This implies that system (50)(57) outside (and only outside) the manifold (60) is equivalent to the system = = -z4 + v s() 3I2 cos
v

+z1 z2 +

v s() 3I2 sin
3

, 1 , 2 , 3 ,

v

в - , v , (65) (66)

в z4 - z -

1 cos 2 cos 1 + z2 sin 1 sin 1 sin 2
,3

v2 s()v 3I2

, 1 , 2 , 3 ,
v ,1

v

v s() cos + 1 = z3 sin 3I2 sin 2 = -z +
2

, 1 , 2 , 3 ,

cos + sin sin 1
v ,2

, 1 , 2 , 3 ,

v

,

(61)
332

s() v 3I2 sin sin 1

, 1 , 2 , 3 ,

v

,

(67)

ISBN: 978-1-61804-287-3

New Developments in Pure and Applied Mathematics

3 = z + where = -x +x3 +x
4N v ,1

1

cos + sin sin 1 sin 2
v ,3

= L 1 , 2 , 3 , v , (68) Note that |L| < + if and only if
/2

v

.

(70)

s() v 3I2 sin sin 1 sin 2

, 1 , 2 , 3 , v

lim

v , 1 , 2 ,

3

v

s()

< +.

(71)

, 1 , 2 , 3 , , 1 , 2 , 3 , v

= sin 1 +

2N

v

N

, 1 , 2 , 3 , v v

cos 1 cos 2 +

, 1 , 2 , 3 ,

cos 1 sin 2 cos 3 + cos 1 sin 2 sin 3 , v = sin 2 + (69)

+x

5N

, 1 , 2 , 3 ,
v ,2

For = /2, the required value of the tracing force is define by the equation , 1 , 2 , 3 , = T = Tv 2 m Lv 2 2 2 2 2 . (72) = m (4 + 7 + 9 + 10 ) - 2I2 where 4 , 7 , 9 , 10 are arbitrary. On the other hand, maintaining the rotation about some point W by the tracing force, we must choose this force according to the relation mv 2 , 1 , 2 , 3 , = , (73) 2 R0 where R0 is the distance C W . Relations (72) and (73) define in general, different values of the tracing force T for almost all points of the manifold (60), which proves our assertion. T = Tv I V. C A S E W H E R E T H E M O M E N T O F A N O N C O N S E RVAT I V E FORCE IS INDEPENDENT OF THE ANGULAR VELOCITY A. Reduced system Similarly to the choice of Chaplygin analytic functions, we take the dynamical functions s, x2N , x3N , x4N , and x5N in the following form: s() = B cos , v v v v

, 1 , 2 , 3 , , 1 , 2 , 3 , v v

= -x +x4 +x

3N

v

N

, 1 , 2 , 3 ,

cos 2 cos 3 + cos 2 sin 3 , v = sin 3 + cos 3 ,

5N

, 1 , 2 , 3 ,
v ,3

, 1 , 2 , 3 , , 1 , 2 , 3 , , 1 , 2 , 3 ,

= -x +x

4N

v

x = x2
N0

2N

, 1 , 2 , 3 ,

=

5N

v

(, 1 , 2 , 3 ) = A sin cos 1 ,
3N

and the function v (, 1 , 2 , 3 , /v ) can be represented in the form (41). Here and in the sequel, the dependence on the group of variables (, 1 , 2 , 3 , /v ) is meant as the composite dependence on (, 1 , 2 , 3 , z1 /v , z2 /v , z3 /v , z4 /v ) due to (59). The uniqueness theorem for system (50)(57) on the manifold (60) for odd k violates in the following sense: for odd k through almost all points of the manifold (60), passes a nonsingular phase trajectory of system (50)(57) intersecting the manifold (60) at right angle and there exists a phase trajectory that at any time instants completely coincides with the point specified However, physically these trajectories are different since they correspond to different values of the tracing force. Prove this. As was shown above, to maintain the constraint of the form (38), we must take a value of T for cos = 0 according to (40). Let s() v , 1 , 2 , 3 , = lim v /2 cos
ISBN: 978-1-61804-287-3

x = x3
N0

, 1 , 2 , 3 ,

= (74)

(, 1 , 2 , 3 ) = A sin sin 1 cos 2 , x
4N

, 1 , 2 , 3 ,

=

= x4

N0

(, 1 , 2 , 3 ) = A sin sin 1 sin 2 cos 3 , x
5N

, 1 , 2 , 3 ,
N0

=

= x5

(, 1 , 2 , 3 ) =

= A sin sin 1 sin 2 sin 3 , A, B > 0, v = 0. We see that in the system considered, the moment of nonconservative forces in independent of the angular velocity (but depends on the angles , 1 , 2 , 3 ). Herewith, the functions v (, 1 , 2 , 3 , /v ) , v,s (, 1 , 2 , 3 , /v ) , s = 1, 2, 3, in system (61) (68), take the following form:
v

, 1 , 2 , 3 ,

v

= A sin ,

333

New Developments in Pure and Applied Mathematics

v ,s

, 1 , 2 , 3 ,

v

0, s = 1, 2, 3.

(75)

Then, due to the nonintegrable constraint (38), outside the manifold (60), the dynamical part of the equations of motion (system (61)(68)) has the form of the following analytic system: AB v sin , = -z4 + (76) 3I2 z4 = AB v 2 2 2 2 cos sin cos - (z1 + z2 + z3 ) , 3I2 sin z3 = z3 z4 cos 2 2 cos cos 1 + (z1 + z2 ) , sin sin sin 1 cos cos 1 cos - z2 z3 - sin sin sin 1
2 1

(77) (78)

cos . (92) sin sin 1 sin 2 We see that the eighth-order system (85)(92) (which can be considered as a system on the tangent bundle T S4 of the fourdimensional sphere S4 , see below) contains the independent seventh-order system (85)(91) on its own seven-dimensional manifold. For the complete integration of system (85)(92), in general, we need seven independent firs integrals. However, after the change of variables z4 w4 z3 w3 z2 w2 , 3 = z
1

z2 = z2 z4 -z

z1 (79)

w1

cos 1 cos 2 , sin sin 1 sin 2

cos cos 1 cos - z1 z3 + z1 = z1 z4 sin sin sin 1 cos 1 cos 2 , sin sin 1 sin 2 cos , 1 = z3 sin cos , 2 = -z2 sin sin 1 cos . 3 = z1 sin sin 1 sin 2 +z1 z2 (80) (81) (82) (83)

2 2 2 w4 = z4 , w3 = z1 + z2 + z3 , z2 z3 , w2 = , w1 = 2 2 z1 z1 + z2 system (85)(92) splits as follows:

(93)

= -w4 + b sin , 2 cos w4 = sin cos - w3 sin cos , w3 = w3 w4 sin 1 w2 = d2 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 ) 2 = d2 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 ),

,

(94) (95) (96)

2 + w2 cos 2 , w2 sin 2 (97)

Further, introducing the dimensionless variables, parameters, and the differentiation as follows:
2 zk n0 v zk , k = 1, 2, 3, 4, n0 =

AB , 3I2 (84)

2 1 + w1 cos 1 , w1 sin 1 (98) 1 = d1 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 ),

w1 = d1 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 )

3 = d3 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 ), where d1 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 ) = cos , = Z3 (w4 , w3 , w2 , w1 ) sin d2 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 ) = cos , = -Z2 (w4 , w3 , w2 , w1 ) sin sin 1 d3 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 ) = cos , = Z1 (w4 , w3 , w2 , w1 ) sin sin 1 sin 2 herewith zk = Zk (w4 , w3 , w2 , w1 ), k = 1, 2, 3,

(99)

b = n0 , < · >= n0 v < >, we reduce system (76)(83) to the form = -z4 + b sin , cos , sin cos 2 2 cos cos 1 + (z1 + z2 ) , z3 = z3 z4 sin sin sin 1
2 2 2 z4 = sin cos - (z1 + z2 + z3 )

(85) (86) (87)

(100)

z2 = z2 z4 -z z1 = z1 z4

cos cos 1 cos - z2 z3 - sin sin sin 1
2 1

cos 1 cos 2 , sin sin 1 sin 2

(88)

(101)

cos cos 1 cos - z1 z3 + sin sin sin 1 (89) (90) (91)

+z1 z2

cos 1 cos 2 , sin sin 1 sin 2 cos , 1 = z3 sin cos , 2 = -z2 sin sin 1

are the functions due to the change of variables (93). We see that the eighth-order system splits into independent subsystems of lower order: system (94)(96) has order three and systems (97), (98) (after the change of the independent variable) have order two. Thus, for the complete integration of system (94)(99) it suffice to specify two independent firs integrals of system (94)(96), one firs integral of each system (97), (98), and an additional firs integral that attaches Eq. (99). Note that system (94)(96) can be considered on the tangent bundle T S2 of the two-dimensional sphere S2 .

ISBN: 978-1-61804-287-3

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New Developments in Pure and Applied Mathematics

B. Complete list of invariant relations System (94)(96) has the form of a system that appears in the dynamics of a three-dimensional (3D-) rigid body in a fiel of nonconservative forces. First, to the third-order system (94)(96), we put in correspondence the nonautonomous second-order system
2 sin cos - w3 cos / sin dw4 = , d -w4 + b sin w3 w4 cos / sin dw3 = . d -w4 + b sin

It possesses two analytic firs integrals of the form
2 2 w4 + w3 + sin2 = C1 = const, w3 sin = C2 = const.

(112) (113)

(102)

Obviously, the ratio of two firs integrals (112), (113) is also a firs integral of system (111). But for b = 0, each of the functions 2 2 w4 + w3 - bw4 sin + sin2 (114) and (113) is not a firs integral of system (94)(96). However, but their ratio is a firs integral for any b. Further, we fin the explicit form of the additional firs integral of the third-order system (94)(96). For this, we transform the invariant relation (109) for u1 = 0 as follows: u2 - b 2
2

Applying the substitution = sin , we rewrite system (102) in the algebraic form
2 - w3 / dw4 = , d -w4 + b w3 w4 / dw3 = . d -w4 + b

(103)

+ u1 -

C1 2

2

=

2 b 2 + C1 - 1. 4

(115)

Later on, introducing the homogeneous variables by the formulas w3 = u1 , w4 = u2 , (104) we reduce system (103) 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
2 1 - u2 + u2 - bu2 du2 1 = , d -u2 + b 2u1 u2 - bu1 du1 = . d -u2 + b

We see that the parameters of this invariant relation satisfy the condition 2 b2 + C1 - 4 0, (116) and the phase space of system (94)(96) is stratifie into the family of surfaces define by Eq. (115). Thus, by relation (109), the firs equation of system (106) has the form where 2(1 - bu2 + u2 ) - C1 U1 (C1 , u2 ) du2 2 = , d -u2 + b (117)

(105)

(106)

To the second-order system (106), we put in correspondence the nonautonomous first-orde equation 1 - u2 + u2 - bu2 du2 1 2 = , du1 2u1 u2 - bu1 u2 + u2 - bu2 + 1 1 2 u1 (107)

1 2 {C1 ± C1 - 4(u2 - bu2 + 1)}; (118) 2 2 the integration constant C1 is define by condition (116). Therefore, the quadrature for the search for the additional firs integral of system (94)(96) becomes U1 (C1 , u2 ) = d = = (b - u2 )du 2A - C1 {C1 ±
0

which can be easily reduced to the exact-differential form: d = 0. (108)

C - 4A0 }/2

2 2 1

,

(119)

A0 = 1 - bu2 + u2 . 2 Obviously, the left-hand side (up to an additive constant) equals ln | sin |. (120) b 2 = r1 , b2 = b2 + C1 - 4, 1 2 then the right-hand side of Eq. (119) has the form u2 - - 1 4
2 d(b2 - 4r1 ) 1 2 (b - 4r1 ) ± C 2 1 1

Thus, Eq. (107) has the following firs integral: u2 + u2 - bu2 + 1 2 1 = C1 = const, u1 which in the previous variables has the form
2 2 w4 + w3 - bw4 sin + sin2 = C1 = const. w3 sin

(109)

If

(121)

(110)

Remark 1. Consider system (94)(96) with variable dissipation with zero mean, that becomes conservative for b = 0:
2 cos , w4 = sin cos - w3 sin cos . w3 = w3 w4 sin

b2 - 4r 1 b2 - 4r 1

2 1

- = (122)

= -w4 ,

-b (111)
335

dr

1 1 2 1

2 (b2 - 4r1 ) ± C 1

1 = - ln 2

2 b2 - 4r1 b 1 ± 1 ± I1 , C1 2

ISBN: 978-1-61804-287-3

New Developments in Pure and Applied Mathematics

where I1 = dr3
2 b2 - r3 (r3 ± C1 ) 1

, r3 =

2 b2 - 4r1 . 1

(123)

In the calculation of integral (123), the following three cases are possible. I. b > 2. 2 b2 - 4 + b2 - r3 C1 1 1 ln + ± I1 = - r3 ± C1 2 b2 - 4 b2 - 4 1 ln + 2 b2 - 4 2 b2 - 4 - b2 - r3 1 r 3 ± C1 +const. II. b < 2. I1 = III. b = 2. I1 =
2 b2 - r3 1 + const. C1 (r3 ± C1 ) 2 ±C1 r3 + b1 1 + const. arcsin b1 (r3 ± C1 ) 4 - b2

(97), (98), and an additional firs integral that attaches Eq. (99). To fin a firs integral for each (potentially separated) system (97), (98), we put in correspondence the following nonautonomous first-orde equation:
2 1 + ws cos s dws = , s = 1, 2. ds ws sin s

(132)

After integration, this leads to the invariant relation
2 1 + ws = Cs sin s +2

C1 + b2 - 4 (124)

= const, s = 1, 2.

(133)

Further, for the search for an additional firs integral that attaches Eq. (99), to Eqs. (99) and (97) we put in correspondence the following nonautonomous equation: dw2 2 = -(1 + w2 ) cos 2 . d3 Since, by (133), (134)

(125)

(126) we have

2 2 C4 cos 2 = ± C4 - 1 - w2 ,

(135)

Returning to the variable b w4 -, (127) r1 = sin 2 we obtain the fina expression for I1 : I. b > 2. C1 b2 - 4 ± 2r1 1 ± ln + I1 = - 2 2 b2 - 4 b2 - 4 b2 - 4r1 ± C1 1 C b2 - 4 2r1 1 1 ln +const. (128) + 2 - 4r 2 ± C 2-4 2b b2 - 4 b1 1 1 II. b < 2.
2 ±C1 b2 - 4r1 + b2 1 1 1 + const. (129) arcsin I1 = 2 - 4r 2 ± C ) 2 4-b b1 ( b1 1 1

dw2 = d3

1 2 2 2 (1 + w2 ) C4 - 1 - w2 . C4

(136)

Integrating the last relation, we arrive at the following quadrature: (3 + C5 ) = C4 dw (1 +
2 w2 2 4 2 2 ) C - 1 - w2

, C5 = const. (137) (138)

Integrating this relation we obtain tg(3 + C5 ) = C4 w
2 4 2 2 C - 1 - w2

, C5 = const.

III. b = 2. I1 = 2r
2 1 1 2 r1

Finally, we have the following form of the additional firs integral that attaches Eq. (99): arctg C4 w2
2 C - 1 - w2 2 4

C1 ( b - 4

± C1 )

+ const.

(130)

± 3 = C5 , C5 = const.

(139)

Thus, we have found an additional firs integral for the third-order system (94)(96) and we have the complete set of firs integrals that are transcendental functions of their phase variables. Remark 2. We must substitute the left-hand side of the firs integral (109) in the expression of this firs integral instead C1 . Then the additional firs integral obtained has the following structure (similar to the transcendental firs integral in planar dynamics): w3 w4 , = C2 = const. (131) ln | sin | + G2 sin , sin sin Thus, for the integration of the eighth-order system (94) (99), we have found two independent firs integrals. For the complete integration, as was mentioned above, it suffice to fin one firs integral for each (potentially separated) system
ISBN: 978-1-61804-287-3

Thus, in the case considered, the system of dynamical equations (17)(21), (24)(33) under condition (74) has twelve invariant relations: the nonintegrable analytic constraint of the form (38), the cyclic firs integrals of the form (36), (37), the firs integral of the form (110), the firs integral expressed by relations (124)(131), which is a transcendental function of the phase variables (in the sense of complex analysis) expressed through a finit combination of elementary functions, and, finall , the transcendental firs integrals of the form (133) and (139). Theorem 1. System (17)(21), (24)(33) under conditions (38), (74), (37) possesses twelve invariant relations (complete set), five of which transcendental functions from the point of view of complex analysis. Herewith, all relations are expressed through finite combinations of elementary functions.

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New Developments in Pure and Applied Mathematics

C. Topological analogies Consider the following seventh-order system: Ё + b cos + sin cos - - [1 2 + 2 2 sin2 1 + 3 2 sin2 1 sin2 2 ] sin = 0, cos

2 1 + cos - 1 + b 1 cos + 1 Ё cos sin - (2 2 + 3 2 sin2 2 ) sin 1 cos 1 = 0, 2 1 + cos + 2 + b 2 cos + 2 Ё cos sin cos 1 2 - 3 sin 2 cos 2 = 0, + 21 2 sin 1 2 1 + cos + 3 + b 3 cos + 3 Ё cos sin cos 1 cos 2 + 22 3 = 0, b > 0, + 21 3 sin 1 sin 2

Let x = (x1N , x2N , x3N , x4N , x5N ) be the coordinates of the point N of application of a nonconservative force (influ ence of the medium) acting on the four-dimensional disk and Q = (Q1 , Q2 , Q3 , Q4 , Q5 ) be the components independent of the tensor of the angular velocity. We consider only linear dependence of the functions (x1N , x2N , x3N , x4N , x5N ) on the tensor of angular velocity since this introduction itself is not obvious. We adopt the following dependence: (140) x = Q + R, (144) where R = (R1 , R2 , R3 , R4 , R5 ) is a vector-valued function containing the components of the tensor of angular velocity. The dependence of the function R on the components of the tensor of angular velocity is gyroscopic: R1 R 2 R = R3 = R 4 R5 1 =- v 0 10 -9 7 -4 -10 0 8 -6 3 9 -8 0 5 -2 -7 6 -5 0 1 4 -3 2 -1 0 h h h h h
1 2 3 4 5

which describes a fi ed f ve-dimensional pendulum in a fl w of a running medium for which the moment of forces is independent of the angular velocity, i.e., a mechanical system in a nonconservative field In general the order of such a system is equal to 8, but the phase variable 3 is a cyclic variable, which leads to the stratificatio 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 , 3 , , 1 , 2 , 3 } (141)

,

of the four-dimensional sphere S4 { , 1 , 2 , 3 }. The equation that transforms system (140) the system on the tangent bundle of the three-dimensional sphere 3 0, and the equations of great circles 1 0, 2 0, 3 0 (143) (142)

defin families of integral manifolds. It is easy to verify that system (140) is equivalent to the dynamical system with variable dissipation with zero mean on the tangent bundle (141) of the four-dimensional sphere. Moreover, the following theorem holds. Theorem 2. System (17)(21), (24)(33) under conditions (38), (74), (37) is equivalent to the dynamical system (140). Indeed it suffice to set = , 1 = 1 , 2 = 2 , 3 = 3 , b = -b . V. C A S E W H E R E T H E M O M E N T O F A N O N C O N S E RVAT I V E FORCE DEPENDS ON THE ANGULAR VELOCITY A. Introduction of the dependence on the angular velocity This chapter is devoted to the dynamics of a f vedimensional rigid body in the f ve-dimensional space. Since the present section is devoted to the study of the motion in the case where the moment of forces depends on the tensor of angular velocity, we introduce this dependence in a more general situation. This also allows us to introduce this dependence for multi-dimensional bodies.
ISBN: 978-1-61804-287-3

(145) where (h1 , h2 , h3 , h4 , h5 ) are some positive parameters. Since x1N 0, we have 10 , x2N = Q2 - h1 v 9 x3N = Q3 + h1 , v (146) 7 x4N = Q4 - h1 , v 4 x5N = Q5 + h1 . v B. Reduced system Similarly to the choice of the Chaplygin analytic functions Q2 = A sin cos 1 , Q3 = A sin sin 1 cos 2 , Q4 = A sin sin 1 sin 2 cos 3 , Q5 = A sin sin 1 sin 2 sin 3 , A > 0, we take the dynamical functions s, x2N , x3N , x4N , and x5 in the following form: s() = B cos , B > 0, x2 x3 x
N N N

(147)

, 1 , 2 , v v

v

= A sin cos 1 - h

10 , v 9 , (148) v 7 , v

, 1 , 2 , , 1 , 2 ,

= A sin sin 1 cos 2 + h

4N

= A sin sin 1 sin 2 cos 3 - h

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New Developments in Pure and Applied Mathematics

= A sin sin 1 sin 2 sin 3 + v 4 +h , h = h1 > 0, v = 0. v This shows that in the problem considered, there is an additional damping (but accelerating in certain domains of the phase space) moment of a nonconservative force (i.e., there is a dependence of the moment on the components of the tensor of angular velocity). Moreover, h2 = h3 = h4 = h5 due to the dynamical symmetry of the body. In this case, the functions v (, 1 , 2 , 3 , /v ) , v,s (, 1 , 2 , 3 , /v ) , s = 1, 2, 3, in system (61)(68) have the following form: x5
N

, 1 , 2 ,

1 =

1+

B h 3I2

z3 z2

cos , sin

(155) (156) (157)

B h 2 = - 1 + 3I2 3 = 1+ B h 3I2 z1

cos , sin sin 1

cos . sin sin 1 sin 2

Introducing the dimensionless variables, parameters, and the differentiation as follows: AB , zk n0 v zk , k = 1, 2, 3, 4, n2 = 0 3I2 (158) Bh , < · >= n0 v < >, b = n 0 , H1 = 3I2 n0 we reduce system (150)(157) to the form = - (1 + bH1 ) z4 + b sin , (159)

v

, 1 , 2 , 3 ,
v ,1

v

= A sin -

h z4 , v (149)

, 1 , 2 , 3 , v , 1 , 2 , 3 , v v

h = z3 , v h = - z2 , v = h z1 . v

v ,2

v ,3

, 1 , 2 , 3 ,

Then, due to the nonintegrable constraint (38), outside the manifold (60) the dynamical part of the equations of motion (system (61)(68)) takes the form of the analytic system =- 1+ z4 = - 1+ B h 3I2 B h 3I2 z4 + AB v sin , 3I2 (150)

z4 = sin cos - 2 2 2 cos - H1 z4 cos , - (1 + bH1 ) (z1 + z2 + z3 ) sin cos + z3 = (1 + bH1 ) z3 z4 sin 2 2 cos cos 1 - H1 z3 cos , + (1 + bH1 ) (z1 + z2 ) sin sin 1 cos - z2 = (1 + bH1 ) z2 z4 sin cos cos 1 - - (1 + bH1 ) z2 z3 sin sin 1 - (1 + bH1 ) z cos 1 cos 2 - H1 z2 cos , sin sin 1 sin 2 cos - z1 = (1 + bH1 ) z1 z4 sin cos cos 1 + - (1 + bH1 ) z1 z3 sin sin 1
2 1

(160)

(161)

AB v 2 sin cos - 3I2 cos B hv - z4 cos , (151) sin 3I2 z3 z
4

(162)

2 2 2 (z1 + z2 + z3 )

z3 = B h + 1+ 3I2

1+

B h 3I2

cos + sin

B hv 2 2 cos cos 1 (z1 + z2 ) - z3 cos , (152) sin sin 1 3I2 z2 = B h 1+ 3I2 B h 3I2 z2 z
3

+ (1 + bH1 ) z1 z2

cos z2 z4 - sin cos cos 1 - sin sin 1

- 1+ - 1+ B h 3I2
2 z1

cos 1 cos 2 - H1 z1 cos , sin sin 1 sin 2 cos , 1 = (1 + bH1 ) z3 sin cos , 2 = - (1 + bH1 ) z2 sin sin 1 cos . 3 = (1 + bH1 ) z1 sin sin 1 sin 2

(163) (164) (165) (166)

cos 1 cos 2 B hv - z2 cos , (153) sin sin 1 sin 2 3I2 1+ B h 3I2 B h 3I2 z1 z
2 3

z1 = - 1+ + 1+

z1 z

4

cos - sin

cos cos 1 + sin sin 1

B h 3I2 -

z1 z

cos 1 cos 2 - sin sin 1 sin 2 (154)

B hv z1 cos , 3I2

We see that the eighth-order system (159)(166) (which can be considered on the tangent bundle T S4 of the fourdimensional sphere S4 ), contains an independent seventh-order system (159)(165) on its own seven-dimensional manifold. For the complete integration of system (159)(166), we need, in general, seven independent firs integrals. However, after the change of variables z4 w4 z3 w3 z2 w2 , z1 w1

ISBN: 978-1-61804-287-3

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New Developments in Pure and Applied Mathematics
2 2 2 z1 + z2 + z3 ,

w4 = z4 , w3 = w2 = z2 , w1 = z1

z3
2 z1

First, to the third-order system (168)(170), we put in correspondence the nonautonomous second-order system (167) dw4 = d 2 sin cos - (1 + bH1 )w3 cos / sin - H1 w4 cos = , -(1 + bH1 )w4 + b sin dw3 = d (1 + bH1 )w3 w4 cos / sin - H1 w3 cos = . -(1 + bH1 )w4 + b sin (176) Using the substitution = sin , we rewrite system (176) in the algebraic form:
2 - (1 + bH1 )w3 / - H1 w4 dw4 = , d -(1 + bH1 )w4 + b (1 + bH1 )w3 w4 / - H1 w3 dw3 = . d -(1 + bH1 )w4 + b

+z

2 2

,

system (159)(166) splits as follows: = -(1 + bH1 )w4 + b sin ,
2 w4 = sin cos - (1 + bH1 )w3

(168)

cos - H1 w4 cos , (169) sin (170)

w3 = (1 + bH1 )w3 w4

cos - H1 w3 cos , sin

w2 = d2 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 )в 2 1 + w2 cos 2 , в w2 sin 2 2 = d2 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 ), w1 = d1 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 )в 2 1 + w1 cos 1 , в w1 sin 1 1 = d1 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 ), 3 = d3 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 ), where

(171)

(177)

(172)

Further, introducing the homogeneous variables by the formulas w3 = u1 , w4 = u2 , (178) we reduce system (177) 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

(173)

d1 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 ) = cos , = (1 + bH1 )Z3 (w4 , w3 , w2 , w1 ) sin d2 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 ) = cos (174) , = -(1 + bH1 )Z2 (w4 , w3 , w2 , w1 ) sin sin 1 d3 (w4 , w3 , w2 , w1 ; , 1 , 2 , 3 ) = cos , = (1 + bH1 )Z1 (w4 , w3 , w2 , w1 ) sin sin 1 sin 2 herewith, zk = Zk (w4 , w3 , w2 , w1 ), k = 1, 2, 3, (175)

(179)

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

(180)

To the second-order system (180), we put in correspondence the nonautonomous first-orde equation 1 - (1 + bH1 )(u2 - u2 ) - (b + H1 )u2 du2 1 2 = , du1 2(1 + bH1 )u1 u2 - (b + H1 )u1 (181)

are the functions, due to the change of variables (167). We see that the eighth-order system splits into independent subsystems of lower orders: system (168)(170) of order 3 and each of system (171), (172) (certainly, after a choice of the independent variables) of order 2. Thus, for the complete integration of system (168)(173), it suffice to fin two independent firs integrals of system (168)(170), one firs integral of each system (171), (172), and an additional firs integral that attaches Eq. (173). Note that system (168)(170) can be considered on the tangent bundle T S2 of the two-dimensional sphere S2 . C. Complete list of invariant relation System (168)(170) has the form of a system of equations that appears in the dynamics of a three-dimensional (3D-) rigid body in a nonconservative field
ISBN: 978-1-61804-287-3

which can be easily reduce to the exact-differential form: d (1 + bH1 )(u2 + u2 ) - (b + H1 )u2 + 1 2 1 u1 = 0. (182)

Thus, Eq. (181) has the following firs integral: (1 + bH1 )(u2 + u2 ) - (b + H1 )u2 + 1 2 1 = u1 = C1 = const, which in the original variables has the form
2 2 (1 + bH1 )(w4 + w3 ) - (b + H1 )w4 sin + sin2 = w3 sin

(183)

= C1 = const.

(184)

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New Developments in Pure and Applied Mathematics

Remark 3. Consider system (168)(170) with variable dissipation with zero mean, which becomes conservative for b = H1 :
2 cos - bw4 cos , w4 = sin cos - (1 + b2 )w3 sin cos - bw3 cos . w3 = (1 + b2 )w3 w4 sin It possesses the following two analytic firs integrals:

A1 = 1 - (b + H1 )u2 + (1 + bH1 )u2 . 2 Obviously, the left-hand side (up to an additive constant) is equal to ln | sin |. (194) If u2 - b + H1 2 = r1 , b2 = (b - H1 )2 + C1 - 4, (195) 1 2(1 + bH1 )
2 2 d(b1 - 4(1 + bH1 )r1 ) 2 (b2 - 4(1 + bH1 )r1 ) ± C 1 1 2 b2 - 4(1 + bH1 )r1 1

= -(1 + b2 )w4 + b sin ,

(185)

then the right-hand side of Eq. (193) becomes (187) - 1 4 -

2 2 (1 + b2 )(w4 + w3 ) - 2bw4 sin + sin2 = C1 = const, (186) w3 sin = C2 = const.

Obviously, the ratio of the two firs integrals (186), (187) is also a firs integral of system (185). But for b = H1 none of the functions
2 2 (1 + bH1 )(w4 + w3 ) - (b + H1 )w4 sin + sin2

-(b - H1 )(1 + bH1 )в в dr (b - 4(1 + bH
2 1 2 1 )r1 1 1 2 b2 - 4(1 + bH1 )r1 1

)±C

= (196)

(188)

and (187) is a firs integral of system (168)(170). However, the ratio of the functions (188), (187) is a firs integral of system (168)(170) for any b, H1 . We fin the explicit form of the additional firs integral of the third-order system (168)(170). First, we transform the invariant relation (183) for u1 = 0 as follows: b + H1 u2 - 2(1 + bH1 ) =
2

1 = - ln 2 where

2 b2 - 4(1 + bH1 )r1 b - H1 1 ±1 ± I1 , C1 2

I1 = r3 =

dr b2 1

3

2 - r3 (r3 ± C1 )

, (197)

C1 + u1 - 2(1 + bH1 )

2

2 b2 - 4(1 + bH1 )r1 . 1

= (189)

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

In the calculation of integral (197), the following three cases are possible: I. |b - H1 | > 2. I1 = - в ln 1 2 (b - H1 )2 - 4 b2 - r 1 1 2 (b - H1 )2 - 4 b2 - r 1 +const. II. |b - H1 | < 2. I1 = 1 4 - (b - H1 )2 arcsin ±C1 r3 + b2 1 + const. b1 (r3 ± C1 ) (199)
2 3 2 3

We see that the parameters of this invariant relation must satisfy the condition (b - H1 ) + C - 4 0,
2 2 1

в C
1

(190)

(b - H1 )2 - 4 + r3 ± C1 +

± в

(b - H1 )2 - 4

+

and the phase space of system (168)(170) is stratifie into the family of surfaces define by Eq. (189). Thus, due to relation (183), the firs equation of system (180) has the form du2 = d = where 2(1 + bH1 )u2 - 2(b + H1 )u2 + 2 - C1 U1 (C1 , u2 ) 2 , b - (1 + bH1 )u2 (191) (192)

в ln

(b - H1 )2 - 4 - r3 ± C1

C

1

(b - H1 )2 - 4

+

(198)

1 U1 (C1 , u2 ) = {C1 ± U2 (C1 , u2 )}, 2(1 + bH1 ) U 2 ( C1 , u 2 ) = =

III. |b - H1 | = 2. I1 =
2 b2 - r3 1 + const. C1 (r3 ± C1 )

2 C1 - 4(1 + bH1 )(1 - (b + H1 )u2 + (1 + bH1 )u2 ), 2

(200)

and the integration constant C1 is define by condition (190). Therefore, the quadrature for the search for an additional firs integral of system (168)(170) becomes d = = (b - (1 + bH1 )u2 )du2 , 2A - C1 {C1 ± U2 (C1 , u2 )}/(2(1 + bH1 ))
1

Returning to the variable r1 = b + H1 w3 - , sin 2(1 + bH1 ) (201)

we have the following fina form of I1 : I. |b - H1 | > 2. (193)
340

I1 = -

1 2 (b - H1 )2 - 4

в

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New Developments in Pure and Applied Mathematics

в ln

(b - H1 )2 - 4 ± 2(1 + bH1 )r b2 1 - 4(1 + +
2 bH1 )2 r1

1

±C

± в

C

1

1

(b - H1 )2 - 4

+

by (207), we have dw2 = d3 1 2 2 2 (1 + w2 ) C4 - 1 - w2 . C4 (210)

1 2 (b - H1 )2 - 4 2(1 + bH1 )r ±C
1 1 2 bH1 )2 r1

в ln

(b - H1 )2 - 4 b2 1 - 4(1 +

Integrating this relation, we arrive at the following quadrature: C1 C4 dw2 + , (3 + C5 ) = 2 2 2 (b - H1 )2 - 4 (1 + w2 ) C4 - 1 - w2 (202) C5 = const. Integration leads to the relation C4 w
2 4

+const. II. |b - H1 | < 2. I1 = в arcsin ±C
1

(211)

1 4 - (b - H
1

)2

в + const. (203)

tg(3 + C5 ) =

2

2 C - 1 - w2

, C5 = const.

(212)

2 2 b2 - 4(1 + bH1 )2 r1 + b1 1

2 b1 ( b2 - 4(1 + bH1 )2 r1 ± C1 ) 1

Finally, we have the following additional firs integral that attaches Eq. (173): arctg C4 w2
2 C - 1 - w2 2 4

III. |b - H1 | = 2. I1 = 2(1 + bH1 )r1 C1 ( b2 1
2 - 4(1 + bH1 )2 r1 ± C1 )

± 3 = C5 , C5 = const.

(213)

+ const.

(204)

Thus, we have found an additional firs integral for the thirdorder system (168)(170) and we have the complete set of firs integrals that are transcendental functions of their phase variables. Remark 4. Formally, in the expression of the found firs integral, we must substitute instead of C1 the left-hand side of the firs integral (183). Then the obtained additional firs integral has the following structure (similar to the transcendental firs integral from planar dynamics): w3 w4 , = C2 = const. (205) ln | sin | + G2 sin , sin sin Thus, to integrate the eighth-order system (168)(173), we have already found two independent firs integrals. For the complete integration, as was mentioned above, it suffice to fin one firs integral for each (potentially separated) system (171), (172), and an additional firs integral that attaches Eq. (173). To fin a firs integral of each (potentially separated) system (171), (172), we put in correspondence the following nonautonomous first-orde equation:
2 1 + ws cos s dws = , s = 1, 2. ds ws sin s

Thus, in the case considered, the system of dynamical equations (17)(21), (24)(33) under condition (148) has twelve invariant relations: the analytic nonintegrable constraint of the form (38), the cyclic firs integrals of the form (36) and (37), the firs integral of the form (184), the firs integral expressed by relations (198)(205), which is a transcendental function of the phase variables (in the sense of complex analysis) expressed through a finit combination of functions, and the transcendental firs integrals of the form (207) and (213). Theorem 3. System (17)(21), (24)(33) under conditions (38), (148), (37) possesses twelve invariant relations (complete set); five of them are transcendental functions from the point of view of complex analysis. All relations are expressed through finite combinations of elementary functions. D. Topological analogies Consider the following seventh-order system: Ё + (b - H1 ) cos + sin cos - - [1 2 + 2 2 sin2 1 + 3 2 sin2 1 sin2 2 ]
2 1 + cos Ё 1 + (b - H1 )1 cos + 1 cos sin - (2 2 + 3 2 sin2 2 ) sin 1 cos 1 = 0, 2 1 + cos 2 + (b - H1 )2 cos + 2 Ё cos sin cos 1 2 - 3 sin 2 cos 2 = 0, + 21 2 sin 1 2 1 + cos 3 + (b - H1 )3 cos + 3 Ё cos sin cos 1 cos 2 + 22 3 = 0, + 21 3 sin 1 sin 2 b > 0, H1 > 0.

sin = 0, cos

- + (214)

(206)

After integration we obtain the required invariant relation
2 1 + ws =C sin s s+2

= const, s = 1, 2.

(207)

Further, to obtain an additional firs integral that attaches Eq. (173), to Eqs. (173) and (171) we put in correspondence the following nonautonomous equation: dw2 2 = -(1 + w2 ) cos 2 . d3 Since C4 cos 2 = ±
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2 2 C4 - 1 - w2 ,

+

(208) (209)

This system describes a fi ed f ve-dimensional pendulum in a fl w of a running medium for which the moment of forces depends on the angular velocity, i.e., a mechanical system in a nonconservative field Generally speaking, the order of this

341

New Developments in Pure and Applied Mathematics

system must be equal to 8, but the phase variable 3 is a cyclic variable, which leads to the stratificatio of the phase space and reduced the order of the system. The phase space of this system is the tangent bundle T S3 { , 1 , 2 , 3 , , 1 , 2 , 3 } (215) of the four-dimensional sphere S4 { , 1 , 2 , 3 }. The equation that transforms system (140) into the system on the tangent bundle of the three-dimensional sphere 3 0, and the equations of great circles 1 0, 2 0, 3 0 (217) defin families of integral manifolds. It is easy to verify that system (214) is equivalent to the dynamical system with variable dissipation with zero mean on the tangent bundle (215) of the four-dimensional sphere. Moreover, the following theorem holds. Theorem 4. System (17)(21), (24)(33) under conditions (38), (148), (37) is equivalent to the dynamical system (214). Indeed, it suffice to set = , 1 = 1 , 2 = 2 , 3 = 3 , b = -b , H1 = -H1 . VI. CONCLUSION In the previous studies of the author, the problems on the motion of the lower-dimensional solid were already considered in a nonconservative force fiel in the presence of the following force. This study opens a new cycle of works on integration of a multidimensional solid in the nonconservative fiel because previously, as was already specified we considered only such motions of a solid when the fiel of external forces was the potential. AC K N O W L E D G M E N T This work was supported by the Russian Foundation for Basic Research, project no. 12-01-00020-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, Some questions of the qualitative theory of ordinary differential equations and dynamics of a rigid body interacting with a medium, Journal of Mathematical Sciences, Vol. 110, No. 2, 2002, p. 25262555. [3] M. V. Shamolin, Foundations of differential and topological diagnostics, Journal of Mathematical Sciences, Vol. 114, No. 1, 2003, p. 9761024. [4] M. V. Shamolin, New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium, Journal of Mathematical Sciences, Vol. 114, No. 1, 2003, p. 919975. [5] M. V. Shamolin, Classes of variable dissipation systems with nonzero mean in the dynamics of a rigid body, Journal of Mathematical Sciences, Vol. 122, No. 1, 2004, p. 28412915. [6] M. V. Shamolin, Structural stable vector fields in rigid body dynamics, Proc. of 8th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2005), Lodz, Poland, Dec. 1215, 2005; Tech. Univ. Lodz, 2005, Vol. 1, p. 429436. [7] M. V. Shamolin, The cases of integrability in terms of transcendental functions in dynamics of a rigid body interacting with a medium, Proc. of 9th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2007), Lodz, Poland, Dec. 1720, 2007; Tech. Univ. Lodz, 2007, Vol. 1, p. 415422.

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[8] M. V. Shamolin, Methods of analysis of dynamic systems with various dissipation in dynamics of a rigid body, ENOC-2008, CD-Proc., June 30July 4, 2008, Saint Petersburg, Russia, 6 p. [9] M. V. Shamolin, Some methods of analysis of the dynami systems with various dissipation in dynamics of a rigid body, PAMM (Proc. Appl. Math. Mech.), 8, 1013710138 (2008) / DOI 10.1002/pamm.200810137. [10] M. V. Shamolin, Dynamical systems with variable dissipation: methods and applications, Proc. of 10th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2009), Lodz, Poland, Dec. 710, 2009; Tech. Univ. Lodz, 2009, p. 91104. [11] 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) / DOI 10.1002/pamm. 200910044. [12] 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. [13] M. V. Shamolin, Dynamical systems with various dissipation: background, methods, applications // CD-Proc. of XXXVIII Summer SchoolConf. "Advances Problems in Mechanics" (APM 2010), July 15, 2010, St. Petersburg (Repino), Russia; St. Petersburg, IPME, 2010, p. 612621. [14] M. V. Shamolin, Integrability and nonintegrability in terms of transcendental functions in dynamics of a rigid body, PAMM (Proc. Appl. Math. Mech.), 10, 6364 (2010) / DOI 10.1002/pamm.201010024. [15] M. V. Shamolin, Cases of complete integrability in transcendental functions in dynamics and certain invariant indices, CD-Proc. 5th Int. Sci. Conf. on Physics and Control PHYSCON 2011, Leon, Spain, September 58, 2011. Leon, Spain, 5 p. [16] 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. [17] M. V. Shamolin, Cases of integrability in dynamics of a rigid body interacting with a resistant medium, CD-proc., 23th International Congress of Theoretitical and Applied Mechanics, August 1924, 2012, Beijing, China; Beijing, China Science Literature Publishing House, 2012, 2 p. [18] M. V. Shamolin, Variety of the cases of integrability in dynamics of a 2D-, and 3D-rigid body interacting with a medium, 8th ESMC 2012, CD-Materials (Graz, Austria, July 913, 2012), Graz, Graz, Austria, 2012, 2 p.

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