дНЙСЛЕМР БГЪР ХГ ЙЩЬЮ ОНХЯЙНБНИ ЛЮЬХМШ. юДПЕЯ НПХЦХМЮКЭМНЦН ДНЙСЛЕМРЮ : http://shamolin2.imec.msu.ru/art-141-1.pdf
дЮРЮ ХГЛЕМЕМХЪ: Thu Dec 5 18:13:35 2013
дЮРЮ ХМДЕЙЯХПНБЮМХЪ: Thu Feb 27 21:40:49 2014
йНДХПНБЙЮ: ISO8859-5

International Journal of Structural Stability and Dynamics Vol. 13, No. 7 (2013) 1340011 (14 pages) c # World Scienti?c Publishing Company . DOI: 10.1142/S0219455413400117

VARIETY OF THE CASES OF INTEGRABILITY IN DYNAMICS OF A SYMMETRIC 2D-, 3D- AND 4D-RIGID ? BODY IN A NONCONSERVATIVE FIELD

MAXIM V. SHAMOLIN Institute of Mechanics, Lomonosov Moscow State University Michurinskii Ave., 1, Moscow 119899, Russian Federation shamolin@imec.msu.ru shamolin@rambler.ru Received 27 January 2012 Accepted 16 July 2012 Published 30 April 2013 A vast number of papers are devoted to studying the complete integrability of equations of fourdimensional rigid-body motion. Although in studying low-dimensional equations of motion of quite concrete (two- and three-dimensional) rigid bodies in a nonconservative force ?eld, the author arrived at the idea of generalizing the equations to the case of a four-dimensional rigid body in an analogous nonconservative force ?eld. As a result of such a generalization, the author obtained the variety of cases of integrability in the problem of body motion in a resisting medium that ?lls 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. Keywords: Many-dimensional rigid body; integrability; transcendental ?rst integral.

1. Introduction 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 ?eld, the author arrived at the idea to generalize the equations to the case of motion of a four-dimensional rigid body in an analogously constructed ?eld. As a result of such a generalization, the author obtained several cases of integrability in the problem of body motion in a resisting medium that ?lls 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, according to the author's opinion, the obtained results are original from the viewpoint that a pair of nonconservative force exists in the system.
* This paper has been presented at the 11th Conference on Dynamical Systems -- Theory and Applications (December 5п8, 2011, Lodz, Poland).

1340011-1

M. V. Shamolin

Previously, in Refs. 1п3, the author showed the complete integrability of the equation of plane-parallel body motion in a resisting medium under the streamline юow around conditions, when the system of dynamical equations has a ?rst 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 dynamical equations has a complete tuple of transcendental ?rst 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 (twodimensional) disk (see also Refs. 4 and 5).

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 ?eld under the quasi-stationary conditions.1 If ?v;Н 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 п v sin п v sin ? 2 л Fx ; v sin ? v cos ? v cos п л 0; I л yN ?; ; vНs?Н; where Fx л пs?Нv 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 ful?lment of the relation VC const ? 2Н
: : : : : :

? 1Н

during the entire time of motion (VC 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 nonconservative pair of forces act on the body. In the case of Chaplygin analytical functions, we take the dynamical functions s and yN in the form s?Н л B cos , yN ?; ; vН л A sin п h1 =v, h1 > 0, A, B > 0, v л 0, 6 which shows that in the system considered, there also exists an additional damping (and breaking in some domains of the phase space) nonconservative force moment. Owing to constraint (2), under certain condition, system (1) reduces to the following system on the three-dimensional cylinder W1 л R 1 fvgр S 1 f mod 2gр R 1 f!g : ? v 0 л vы?; !Н;
1340011-2

? 3Н

Variety of the Cases of Integrability

л п! ? n 2 sin cos 2 ? ! 2 sin п 0
:

:

h1B cos 2 ; I ? 4Н

! л n 2 sin cos п n 2 !sin 2 cos ? ! 3 cos 0 0 h 1 B 2 hB ! sin cos п 1 ! cos ; ? I I h B ы?; !Н л п! 2 cos ? n 2 sin 2 cos п 1 ! sin cos ; 0 I where л !v, n 2 л AB=I , hяi л vh 0 i. 0 2.2. A complete list of ?rst integrals

From the system (3) and (4), the independent second-order system (4) is separated. Theorem 2.1. The system (3), (4) has a complete tuple of ?rst integrals; one of them is an analytic function, and the other is a transcendental function of phase variables expressing through a ?nite 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 ?rst 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 ?rst integral of system (1) under the condition Fx 0, precisely the function of phase variables ы0 ?v;; Н л v 2 ? 2 2 п 2v sin л V
2 C

? 5Н

is constant on phase trajectories. By a nondegenerate change of the independent variable, the system (3), (4) also has an analytic integral, precisely, the function of phase variables ы1 ?v;;!Н л v 2 ?1 ? 2 ! 2 п 2! sin Н л V
2 C

? 6Н

is constant on phase trajectories. Relation (6) allows us to ?nd the dependence of the velocity of a characteristic rigid body point on other phase variables not solving the system (3), (4); precisely, 2 for VC 6л 0, the following relation holds: v 2 л ?V C Н=?1 ? 2 ! 2 п 2! sin Н. Since the phase space W1 of the system (3), (4) is three-dimensional and there exist asymptotic limit sets in it, relation (6) de?nes a unique analytic (even continuous) ?rst integral of the system (3), (4) on the whole phase space. Let us examine the problem on the existence of the second (additional) ?rst integral of the system (3), (4) in more detail. Its phase space is foliated into surfaces f?v;;!Н 2 W1 : VC л constg. To justify the latter fact, let us introduce the dimensionless di?erentiation h 0 i7! n0 h 0 i and the additional dimensionless parameter H1 л h1 B=In0 , n 2 л AB=I , л n0 , 0
1340011-3

M. V. Shamolin

л sin , and to the separated second-order system (4), let us put in correspondence the di?erential equation 8: > л п! ? sin cos 2 ? ! 2 sin п H1 !cos 2 ; > < : ! л sin cos п !sin 2 cos ? ! 3 cos > > : ? H1 ! 2 sin cos п H1 ! cos ; ы?; !Н л п! 2 cos ? sin 2 cos п H1 ! sin cos : The analytic ?rst integral (6) obtained above joins Eq. (3) [or (7)]. To ?nd the additional transcendental ?rst integral, to the separated system (8), we put in correspondence the di?erential equation d! п !м! 2 п 2 ? H1 !м! п 1 л : d п! ? ? м! 2 п 2 п H1 !м1 п 2 ? 9Н v 0 л vы?; !Н; ? 7Н ? 8Н

After introducing the homogeneous change of variables ! л t , d! л td ? dt, the integration of the latter equation reduces to the integration of the following Bernoulli equation: a1 ?tНd=dt л a2 ?tН ? a3 ?tН 3 , a1 ?tН л п?1 ? H1 Нt 2 ?? ? H1 Н t п 1, a2 ?tН л ?1 ? H1 Нt п , a3 ?tН л п H1 t п t 2 . Applying the classical change of variables p л 1= 2 , we reduce the equation studied to the linear homogeneous equation dp л c1 ?tНp ? c2 ?tН; dt where c1 ?tН л c2 ?tН л 2t?1 ? H1 Нп 2 ; ?1 ? H1 Нt 2 п? ? H1 Нt ? 1 2 п 2 H 1 t п 2 t 2 : ?1 ? H1 Нt 2 п? ? H1 Нt ? 1 ?10Н

The solution p1 of the homogeneous part of the equation studied is represented in the following form (three cases are possible): (i) for D л ? п H1 Н 2 п 4 > 0, H1 п 2?1 ? H Нt п? п H Нп pffiffiffiffi pffiffi D D 1 1 pffiffiffiffi p1 л kм?1 ? H1 Нt п? ? H1 Нt ? 1я ; 2?1 ? H1 Нt п? п H1 Н? D
2

(ii) for D л ? п H1 Н 2 п 4 < 0,

2?1 ? H1 Нt п? ? H1 Н pffiffiffiffiffiffiffiffi p1 л kм?1 ? H1 Нt 2 п? ? H1 Нt ? 1я exp arctan ; пD

1340011-4

Variety of the Cases of Integrability

(iii) for D л ? п H1 Н 2 п 4 л 0,

2 L1 p1 л kм?1 ? H1 Нt 2 п? ? H1 Нt ? 1я exp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 ? H1 ж 1

L1 л pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ж 1: 1 ? H1 It is clear that to ?nd 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 ?eld under the quasistationarity condition. If ?v;; Н are the spherical coordinates of a certain characteristic point of the rigid body, fx ; y ; z g 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 functions1 of medium action has the form v cos п v sin ? y v sin sin п z v sin cos ? ? 2 ? 2 Н л Fx ; y z v sin cos ? v cos cos п v sin sin ? z v cos v sin sin ? v cos sin ? v sin cos ? x v sin cos x л 0;
: : : : : : : :

пx v sin sin п x y п z л 0;
: :

:

I2 y ??I1 п I2 Нx z л пABv 2 sin cos sin п h z ; v

:

пy v cos п x z ? y л 0; h y ; v

:

?11Н

I2 z ??I2 п I1 Нx y л ABv 2 sin cos cos п

where Fx л пBv 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 ful?lment of relation (2) during all the motion time, then in system (11), instead of Fx , we have the quantity ?T п B cos Нv 2 =m; moreover, owing to condition (2), under certain condition, system (11) 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 (11) of dynamical equations as an independent system. Moreover, as is seen from the equations of motion, the component of the longitudinal angular velocity
1340011-5

M. V. Shamolin

component is conserved: x л
x0

л const:

?12Н

In what follows, we restrict ourselves to the body motion without proper rotation, i.e., to the case where x0 л 0; moreover, for simplicity, let h л 0. Introduce the following notation: z1 л y cos ? z sin , z2 л пy sin ? z : : : cos , zi л Zi v, i л 1; 2; л v 0 ; л v 0 ; v л vv 0 . Then system (11) in case (2) for x0 л 0 can be transformed into the following form: 8 2 2 > 0 л пZ2 ? n 2 sin cos 2 ? ?Z 1 ? Z 2 Н sin ; 0 > > > <0 2 cos ; Z 2 л n 2 sin cos п Z2 ы?; Z1 ; Z2 Нп Z 1 0 sin > > >0 >Z л пZ ы?; Z ; Z Н? Z Z cos ; :1 1 1 2 12 sin cos 0 л Z1 ; sin where
2 2 ы?; Z1 ; Z2 Н л п?Z 1 ? Z 2 Н cos ? n 2 sin 2 cos ; 0

v 0 л vы?; Z1 ; Z2 Н;

?13Н

?14Н

?15Н

n2 л 0

AB : I2

3.2. A complete list of ?rst integrals As above, let us consider the problem of complete integrability (in elementary functions) for the dynamical system (13)п(15) with analytic right-hand sides. Since we consider the class of body motions for which property (2) holds, the ?fth-order system (13)п(15) has [along with (12)] an analytic ?rst integral. Indeed, in the coordinate system considered, we can represent the center-ofmasses velocity in the form VC л fv cos ; v sin cos п z ; v sin sin ? y g. Then the following relation is invariant for system (11) under conditions (12) (x0 л 0) and (2): v 2 п 2vz2 sin ? 2 ?z 2 ? z 2 Н л V 1 2
2 C0

:

?16Н

Moreover, relation (16) 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 (13)п(15):
2 2 v 2 ?1 п 2Z2 sin ? 2 ?Z 1 ? Z 2 НН л V 2 C0

;

?17Н

and relation (17) allows us to explicitly ?nd the dependence of v on the other quasi-velocities: v2 л
2 V C0 2 2: 1 п 2Z2 sin ? 2 ?Z 1 ? Z 2 Н

?18Н

1340011-6

Variety of the Cases of Integrability

It is seen that relation (18) allows us to consider the problems of integrability in elementary functions of the system (13)п(15), which is just of lower order, the fourth order. Let us rewrite the third-order system (14) in the form
2 2 0 л пZ2 ? b sin cos 2 ? b?Z 1 ? Z 2 Н sin ; 0 2 2 Z 2 л sin cos ? bZ2 ?Z 1 ? Z 2 Н cos п bZ2 sin 2 cos п Z 0 2 2 Z 1 л bZ1 ?Z 1 ? Z 2 Н cos п bZ1 sin 2 cos ? Z1 Z 2 1

cos ; sin

?19Н

2

cos ; sin

where b л n0 and the new dimensionless di?erentiation h 0 i7!n0 h 0 i is also introduced. Furthermore, applying the substitution л sin , which is often used, or reduce system (19) to the following form with algebraic right-hand sides:
2 2 2 dZ2 ? bZ2 ?Z 1 ? Z 2 Нп bZ2 2 п Z 1 = л 2 2; d пZ2 ? b ?1 п 2 Н? b ?Z 1 ? Z 2 Н 2 2 dZ1 bZ1 ?Z 1 ? Z 2 Нп bZ1 2 ? Z1 Z2 = л : 2 2 d пZ2 ? b ?1 п 2 Н? b ?Z 1 ? Z 2 Н

?20Н

Let us pass to homogeneous coordinates uk , k л 1; 2; by the formulas uk л Zk . Then system (20) reduces to the form du2 1 п bu2 ? u 2 п u 2 2 1 л ; d пu2 ? b 2 ?u 2 ? u 2 Н? b?1 п 2 Н 1 2

du 2u1 u2 п bu1 1л : d пu2 ? b 2 ?u 2 ? u 2 Н? b?1 п 2 Н 1 2

?21Н

To system (21), we can put in correspondence the following ?rst-order equation: du2 1 п bu2 ? u 2 п u 2 2 1 л : du1 2u1 u2 п bu1 ?22Н

This equation is integrated in elementary functions, since we integrate the following identity obtained from Eq. (22): 1 п bu2 ? u 2 2 d ?23Н ? du1 л 0; u1 and in the coordinates ?; Z1 ; Z2 Н, it corresponds to the transcendental ?rst integral of the following form
2 2 Z 1 ? Z 2 п bZ2 ? 2 л const: Z1

?24Н

Using relation (24), we conclude that system (24) has the following transcendental ?rst integral, which is expressed through a ?nite combination of elementary
1340011-7

M. V. Shamolin

functions:
2 2 Z 1 ? Z 2 п bZ2 sin ? sin 2 л const: Z1 sin

?25Н

Now, using the just found ?rst integral (25), we write the ?rst equation of system (21) in the form du2 2 п 2bu2 ? 2u 2 п C1 U1 ?C1 ; u2 Н 2 л ; пu2 ? b п 2b 2 ? b 2 ?C1 U1 ?C1 ; u2 Н? bu2 Н d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 C1 ж C 1 п 4?u 2 п bu2 ? 1Н 2 ; U1 ?C1 ; u2 Н л 2 or in the form of the Bernoulli equation d ?b п u2 Н ? b 3 ?C1 U1 ?C1 ; u2 Н? bu2 п 2Н л : du2 2 п 2bu2 ? 2u 2 п C1 U1 ?C1 ; u2 Н 2 dp 2?u2 п bНp п 2b?C1 U1 ?C1 ; u2 Н? bu2 п 2Н ; л du2 2 п 2bu2 ? 2u 2 п C1 U1 ?C1 ; u2 Н 2 ?27Н

?26Н

Equation (27) [by using (26)] easily reduces to the linear inhomogeneous equation pл 1 : 2 ?28Н

The latter fact means that we can ?nd one more transcendental ?rst integral in explicit form (i.e. through a combination of qyadratures). Moreover, the general solution of Eq. (28) depends on an arbitrary constant C2 ; we do not present complete calculations. To ?nd the last additional integral of the system (13)п(15) (i.e. the integral, which connects the equation for the angle ) we note that since d=d л ?Z1 = Н= 2 2 ?пZ2 ? b ?Z 1 ? Z 2 Н? b ?1 п 2 НН, it follows that to the relation d u1 л d пu2 ? b 3 ?u 2 ? u 2 Н? b ?1 п 2 Н 1 2 the relation du1 2u1 u2 п bu1 л 2 ?u 2 ? u 2 Н? b?1 п 2 Н d пu2 ? b 1 2 ?30Н ?29Н

taken from system (21) is added. The obtained system (29), (30) allows us to write the following equation for obtaining the desired integral: du1 л 2u1 п : d ?31Н

Now, using the ?rst integral of Eq. (22) (C1 is its constant of integration) and Eq. (31), we can obtain that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du1 л ж b 2 п 4? u 2 п C 1 u 1 ? 1Н ; ?32Н 1 d
1340011-8

Variety of the Cases of Integrability

hence, by (32), the desired quadrature takes the form Z du1 ж qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi л ? C3 ; b 2 п 4? u 2 п C 1 u 1 ? 1Н 1 The left-hand side of (33) (without sign) has the form
b ? u1 п 2 Н 2 1 arcsin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 2 C 1 ??b 2 п 4Н

C3 л const:

?33Н

?34Н

After substitutions, from (34), we obtain the desired invariant relation cos 2 м2? ? C3 Н л
b ? u2 п 2 Н 2 u 2 1 ; G1

?35Н

where G1 л мu 2 п bu2 2 ? 2мu 2 п bu2 мu 2 ? 1? мu 2 ? 1 2 ? b 2 u 2 . 2 2 1 1 1 In particular, if b л 2, then relation (35) takes the form cos 2 м2? ? C3 Н л ?Z2 п sin НZ1 : 2 ?Z2 п sin Н 2 ? Z 1 ?36Н

The right-hand side, as an odd function of л ?Z2 п sin Н=?Z1 Н has a global maximum for л 1, which is equal to 1=2. Therefore, we have proved the following assertion. Theorem 3.1. The system (13)п(15) has a complete tuple of ?rst 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 ?rst 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 body Let a four-dimensional rigid body р of mass m with smooth three-dimensional boundary @ р move in a resisting medium that ?lls 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 diagfI1 ; I2 ; I2 ; I2 g; or the form diagfI1 ; I1 ; I3 ; I3 g: ?38Н In the second case, the two-dimensional planes Dx1 x2 and Dx3 x4 are planes of body dynamical symmetry.
1340011-9

?37Н

M. V. Shamolin

4.2. Physical assumptions and equations on so?4Н Assume that the distance from the point N of application of a nonconservative force S to a point D is a function of only one parameter, the angle : DN л R?Н (in the case of motion in the three-dimensional space, this is the angle of attack. In case (37), this angle is measured between the velocity vD of the point D and the axis Dx1 . In case (38), the meaning of the angle will be clear from the equations. The value of the nonconservative (resistance) force S is S л s?Нsign cos я v 2 , jvD j л v, where s is a certain function, which is characterized as as scattering or pumping of energy in the system. To obtain the explicit form of the dynamical part of the equations of motion, let us de?ne two functions R and S using the information about the motion of threedimensional bodies as follows (in this case, we also use the known analytical result of Chaplygin): R л R?Н л A sin , S л Sv ?Н л Bv 2 cos ; A; B > 0. If is the angular velocity tensor of the four-dimensional rigid body, 2 so?4Н, then the part of the equations of motions, which corresponds to the algebra so?4Н, has the following form: с ? с ?м; с ? с л M ;
: :

?39Н

where с л diagf1 ;2 ;3 ;4 g, 1 л?пI1 ? I2 ? I3 ? I4 Н=2; .. . ;4 л?I1 ? I2 ? I3 п I4 Н=2, M is the exterior force moment acting on the body in R 4 and projected on the natural coordinates in the algebra so?4Н, and мя; я is the commutator in so?4Н. A skewsymmetric matrix 2 so?4Н is represented in the form 0 1 0 п !6 ! 5 п ! 3 B ! 0 п! ! C 4 2C B6 ?40Н B C; @ п !5 !4 0 п !1 A !3 п !2 !
1

0 angular velocity tensor in projections this case, it is obvious that for any п j л Ij п Ii . is necessary to construct the mapping ?41Н

where !i , i л 1; ... ; 6, are components of the on the coordinates in the algebra so?4Н. In i; j л 1; ... ; 4, the following relations hold: i In calculating the exterior force moment, it

R 4 р R 4 ! so?4Н;

which transforms a pair of vectors from R 4 into a certain element of the algebra so?4Н. 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 is represented in the form mw C л F ;
1340011-10

?42Н

Variety of the Cases of Integrability

where, by the many-dimensional Rivals formula, wC л wD ? 2 DC ? E DC; wD л vD ? vD ; E л ;
:

?43Н

F is the exterior force acting on the body (in our case, F л S), E is the angular acceleration tensor. 4.4. Generalized problem of body motion under tracing force action In this work, we consider only the case (37) of distribution of principal moments of inertia. Let us slightly extend the problem. Assume that along the line Dx1 (in case (37)), 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 fourdimensional rigid body in the case (2), i.e. its center of masses moves rectilinear and uniformly. 4.5. Case (37) By a completely de?nite choice of the tracing force,the ful?lment of condition (2) can be achieved. If ?0; x2N ; x3N ; x4N Н are coordinates of the point N in the system Dx1 x2 x3 x4 and fпS ; 0; 0; 0g are coordinates of the resistance force vector in the same system, then to ?nd the force moment, we construct the auxiliary matrix ! 0 x2N x3N x4N ; ?44Н пS 0 0 0 which allows us to obtain the resistance force moment in the projections on the coordinates in the algebra so?4Н: f0; 0; x4N S ; 0; пx3N S ; x2N S g 2 R 6 ffi M 2 so?4Н. Here, it is necessary to take into account that if ?v;;1 ;2 Н are the spherical coordinates in R 4 , then x2N л R?Н cos 1 , x3N л R?Н sin 1 cos 2 , x4N л R?Нр sin 1 sin 2 . Taking into account all what was said, we can write Eq. (35) in the form ?4 ? 3 Н!1 ??3 п 4 Н?!3 !5 ? !2 !4 Н л 0; ?2 ? 4 Н!2 ??2 п 4 Н?!3 !6 п !1 !4 Н л 0; ?4 ? 1 Н!3 ??4 п 1 Н?!2 !6 ? !1 !5 Н л x
: : :
4N

: :

S; S;

?3 ? 2 Н!4 ??2 п 3 Н?!5 !6 ? !1 !2 Н л 0; ?1 ? 3 Н!5 ??3 п 1 Н?!4 !6 п !1 !3 Н л пx ?1 ? 2 Н!6 ??1 п 2 Н?!4 !5 ? !2 !3 Н л x
1340011-11

?45Н

:

3N

2N

S:

M. V. Shamolin

Obviously, in the case (37), Eqs. (45) have three cyclic ?rst integrals !1 л ! 0 ; 1 !2 л ! 0 ; 2 !4 л ! 0 : 4 ?46Н

For simplicity, let us consider the motions on zero levels ! 0 л ! 0 л ! 0 л 0. The 1 2 4 remaining equations on the algebra so(4) take the following form (here, n 2 л 0 : : : AB=2I2 Н: !3 л n 2 v 2 sin cos sin 1 sin 2 , !5 л пn 2 v 2 sin cos sin 1 cos 2 , !6 л 0 0 n 2 v 2 sin cos cos 1 . 0 If we introduce the change of angular velocities by the formulas z1 л !3 cos 2 ? !5 sin 2 , z2 л п!3 sin 2 cos 1 ? !5 cos 2 cos 1 ? !6 sin 1 , z3 л !3 sin 2 sin 1 п !5 cos 2 sin 1 ? !6 cos 1 , then the \compatible" equations of motion on the tangent bundle T S 3 of the three-dimensional sphere (after taking into account four conditions (2) and (46),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 ( л DC ): v л cos мn 2 v 2 sin 2 п?z 2 ? z 2 ? z 3 Н; 0 1 2 3 л пz3 ? n 2 v sin cos 2 ? sin ?z 2 ? z 2 ? z 3 Н=v; 0 1 2 3 z3 л n 2 v 2 sin cos п?z 2 ? z 2 Нctg; 0 1 2 z2 л z2 z3 ctg ? z 2 ctgctg 1 ; 1 z1 л z1 z3 ctg п z1 z2 ctgctg1 ; 1 л z2 ctg;
: : : : : : :

?47Н

?48Н

2 л пz1 ctg csc 1 :

?49Н

From the complete system of the seventh order (47)п(49), the independent system (48), (49), of the sixth order is separated, and, in turn, it has an independent subsystem (48) of the ?fth order. To completely integrate this system, we need, in general, six independent ?rst integrals.ffiffiffiffiffiffiffiffiffiffiffiffiffiever, after changes of variables and ffi p How ffiffi introducing a new di?erentiation z л z 2 ? z 2 , zс л z2 , z л n0 vZ , zk л n0 vZk , k л 1 2 z1 1; 2; 3, zс л Zс , n0 v 0 7! 0 , the system (47)п(49) reduces to the following form (b л n0 , мb л 1):
2 v 0 л vы?; Z ; Z3 Н; ы?; Z ; Z3 Н л b cos мsin 2 п?Z 2 ? Z 3 Н; 8 2 > 0 л пZ3 ? b sin cos 2 ? b sin ?Z 2 ? Z 3 Н; > <

?50Н

Z 0 л sin cos п Z 2 ctg п Z3 ы?; Z ; Z3 Н; >3 > :0 Z л ZZ3 ctg п Z ы?; Z ; Z3 Н; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 80 2 >Z с л Z 1 ? Z с ctgctg1 ; < ZZс > 01 л pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ctg; : 2 1 ? Zс

?51Н

?52Н

1340011-12

Variety of the Cases of Integrability

Z1 02 л п pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ctg csc 1 : 2 1 ? Zс

?53Н

It is seen that the ?fth-order system (48) splits into independent subsystems of lower order: system (51) is of the third order and system (52) (of course, after the change of the independent variable) is of the second order. Therefore, for the complete integrability of the system studied, it suаces to ?nd two independent integrals of the system (51), one for system (52) and additional integrals \connecting" Eqs. (50) and (53). Moreover, we note that system (51) can be considered on the tangent bundle T S 2 of the two-dimensional sphere. 4.6. Complete list of ?rst integrals The complete system (50)п(53) has an analytic ?rst integral of the form
2 2 v 2 ?1 п 2bZ3 sin ??Z 2 ? Z 3 НН л V C ;

?54Н

since property (2) holds. The latter invariant relation allows us to ?nd v. System (51) 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 de?nitions of complex analysis) and are expressed through a ?nite combination of elementary functions:
2 Z 2 ? Z 3 п bZ3 sin ? sin 2 л C1 л const; Z sin

?55Н ?56Н

G?Z ; Z3 ; sin Н л C2 л const: System (52) has a ?rst integral of the form pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ? Zс л C3 л const sin 1

?57Н

and, in turn, it has an additional ?rst integral, which allows us to ?nd 2 ; it has the form cos 1 ж pffiffiffiffiffiffiffiffiffiffiffiffiffiffi л sinfC3 ?2 ? C4 Нg; 2 C3 п 1 C4 л const: ?58Н

Also, it is necessary to note the fact that the denominators of the presented systems contain the functions sin and sin 1 , which reюect only the information about the fact that the coordinates ?v;;1 ; 2 Н are spherical, and for sin л 0 and sin 1 л 0 they (kinematically) degenerate. Theorem 4.1. The dynamical system (50)п(53) has a complete list of ?rst integrals (54)п(58); one of them is an analytic function, and the other are transcendental functions of their variables (after their formal continuation to the complex domain).

1340011-13

M. V. Shamolin

5. 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 ?eld is potential; unfortunately, we cannot mention all the authors (see, for instance, Refs. 6п8). In the present work, we continue the direction developed by the author in studying the equations of motion of rigid body on so?4Нр R 4 under the presence of a non-conservative exterior force moment. The results listed above and also studies of related ?elds were already reported at the workshop \Actual Problems of Geometry and Mechanics" named after professor V. V. Tro?mov led by D. V. Georgievskii and M. V. Shamolin at Department of Mechanics and Mathematics of M. V. Lomonosov Moscow State University. References
1. M. V. Shamolin, Methods for Analysis of Variable Dissipation Dynamical Systems in Rigid Body Dynamics (Moscow, Ekzamen, 2007) [in Russian]. 2. M. V. Shamolin, New Jacobi integrable cases in dynamics of a rigid body interacting with a medium, Dokl. Ross. Akad. Nauk. 364(5) (1999) 627п629. 3. M. V. Shamolin, On an integrable case in spatial dynamics of a rigid body interacting with a medium, in Izv. Ross. Akad. Nauk. Mekhanika Tverdogo Tela 2 (1997) 65п68. 4. J. Awrejcewicz, Classical Mechanics: Statics and Kinematics (Springer, New York, 2012). 5. J. Awrejcewicz, Classical Mechanics: Dynamics (Springer, New York, 2012). 6. O. I. Bogoyavlenskii, Euler equations on ?nite-dimensional Lie coalgebras, arising in problems of mathematical physics, Russian Math. Surveys 47(1) (1992) 117п189. 7. O. I. Bogoyavlenskii, Integrable problems of the dynamics of coupled rigid bodies, Russian Acad. Sci., Izvestiya Math. 41(3) (1993) 395п416. 8. A. P. Veselov, Parametric resonance and geodesics on an ellipsoid, Funct. Anal. Appl. 26(3) (1992) 211п213.

1340011-14