Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ics.org.ru/upload/iblock/10f/171-rolling-of-a-homogeneous-ball-over-a-dynamically-asymmetric-sphere_ru.pdf Äàòà èçìåíåíèÿ: Wed Oct 28 19:31:21 2015 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 00:20:27 2016 Êîäèðîâêà:

ISSN 1560-3547, Regular and Chaotic Dynamics, 2011, Vol. 16, No. 5, pp. 465­483. c Pleiades Publishing, Ltd., 2011.

Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere
Alexey V. Borisov* , Alexander A. Kilin** , and Ivan S. Mamaev*
Institute of Computer Science, Udmurt State University ul. Universitetskaya 1, Izhevsk 426034, Russia
Received Novemb er 11, 2010; accepted Decemb er 6, 2010
**

Abstract--We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of "clandestine" linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability. MSC2010 numbers: 37N15 DOI: 10.1134/S1560354711050042 Keywords: nonholonomic constraint, rolling motion, Chaplygin ball, integral, invariant measure

Contents
INTRODUCTION 1 ROLLING OF A HOMOGENEOUS BALL OVER A DYNAMICALLY ASYMMETRIC SPHERE IN ZERO-GRAVITY 2 THE GENERALIZED ROUTH PROBLEM 2.1 The Problem of a Homogeneous Ball Rolling over a Dynamically Asymmetric Spherical Shell with a Fixed Center The Problem of a Dynamically Asymmetric Spherical Shell Rolling over a Homogeneous Ball with a Fixed Center 466 467 470

471 472 473 473 474 474 474 475 476

2.2

3 PARTICULAR CASES OF MOTION 3.1 3.2 3.3 The case mK = mM The Case mK = 0 The Case mM = 0 The Axisymmetric Case The Case mK = mM ( = 1)

4 INTEGRABLE PARTIAL CASES 4.1 4.2
* ** ***

E-mail: borisov@rcd.ru E-mail: aka@rcd.ru E-mail: mamaev@rcd.ru

465


466 5 LIMI 5.1 5.2 5.3

BORISOV et al. TING AND RESTRICTED PROBLEMS The Case = 0 The Case r = 0 ( , d2 = D = const) Rolling of a Homogeneous Ball Over an Independently Rotating Dynamically Asymmetric Shell 5.4 Rolling of a Dynamically Asymmetric Shell over a Constantly Rotating Ball 5.5 Passage to a Ball Suspension ACKNOWLEDGMENTS REFERENCES 477 477 478 478 479 480 480 482

INTRODUCTION In his well-known pap er [1] S. A. Chaplygin considered a dynamically asymmetric balanced ball rolling on a plane without sliding. This problem turned out to b e integrable and b ecame part of the "golden fund" of analytical mechanics. In addition, it not only gave rise to many investigations on the general theory of non-holonomic systems and the development of methods of their integration but also initiated a search for its various extensions. One of the interesting extensions of the Chaplygin ball problem was prop osed by S. A. Chaplygin himself in [2], where he examined a system of nested balls rolling over each other for the existence of additional integrals of motion. In particular, he found a new system of integrals of the angular momentum (or the constant of areas) typ e which have no explicit physical interpretation. The integration of one of such problems was reduced by Chaplygin to quadratures, however, this problem is little known in the scientific community. Without claiming to give an exhaustive list of the extensions, we would like to mention the study of the motion of a rigid b ody in a spherical susp ension [3], the problem of motion of a ball with liquid-filled cavities [4], the problem of rolling motion of a ball over a ball [5­7], rolling of the Chaplygin ball along a straight line [8] and an imp ortant class of problems dealing with rolling of b odies of various shap es and thus forming an hierarchy of non-holonomic dynamical systems in resp ect to the existence of various tensor invariants [9, 10]. Noteworthy are also some formal generalizations like those of the multidimensional Chaplygin ball (see, e.g., [11, 12]). However, its integrability has not b een proved until now. As a matter of fact, the methods for integration of nonholonomic systems with a large numb er of degrees of freedom are almost totally undevelop ed (all the systems of this kind known to date are integrated by means of the theory of the last multiplier). The problems dealt with in the pap er are sp ecific examples of such systems whose remarkable feature is the existence of additional integrals of motion. Though the problem itself may not b e integrable in general, these additional integrals allow us to p oint out a series of particular integrable cases and remarkable motions. Among the ma jor works dedicated to advancing the dynamics of the Chaplygin ball itself, sp ecial mention may should b e made of [13, 14]. In particular, [13] is devoted to the study of b oth the motion of a reduced system and that of a ball in a fixed frame of reference. Owing to the advances in the design of mobile rob ots, one of the modifications of which is a rolling ball with an internal control system (based on gyroscop es or various p endulums) [15­19], the results of theoretical investigations can find interesting practical applications. In this case, of course, a controlled motion of the ball should b e examined, therefore, a more systematic study of the dynamics of b oth the Chaplygin ball and its various generalizations is b ecoming an urgent necessity. In this pap er we explore the free dynamics of a more general model involving the problem of the Chaplygin ball as a particular case. This model describ es the rolling of one (dynamically asymmetric) ball over another (homogeneous) one, however, as opp osed to [5] none of the balls is fixed in space. In particular, the model treated in the pap er can b e regarded as a description of the motion of a bundle of b odies in zero-gravity and can have a b earing on the study of the dynamics and controllability of satellites that contain rolling ob jects. What is also interesting is that a numb er of other problems, such as the problem of a ball rolling over a freely rotating sphere with a fixed center (the simplest case of the problem was considered by Routh [20]) reduces to the system in question. Sp ecifically, this allows a more advanced analysis of mechanical systems with susp ensions.
REGULAR AND CHAOTIC DYNAMICS Vol. 16 No. 5 2011


ROLLING OF A HOMOGENEOUS BALL

467

The problem we consider in this pap er has more degrees of freedom as compared with the Chaplygin ball, and its analysis is considerably more complicated. In spite of this, the methods [9, 10, 21] which allowed us to develop a hierarchy of non-holonomic dynamical systems, also apply to this problem. In particular, we use them to show the presence of an invariant measure and additional integrals of motion. On the one hand, this provided a b etter insight into the dynamics and allowed us to apply computer methods to the general system, and on the other hand, to consider a series of interesting particular cases which can b e investigated completely. It is interesting that the system under consideration includes as particular or extreme cases quite a numb er of well-known non-holonomic systems related to various problems of rolling of balls. Besides, all these systems can b e regarded as another hierarchy of non-holonomic systems based not on the presence of tensor invariants but on various physical realizations and p ossible passages to the limit. A general diagram that generalizes particular and extreme cases of the problem considered in this pap er is presented at the end of this article in Fig. 5. 1. ROLLING OF A HOMOGENEOUS BALL OVER A DYNAMICALLY ASYMMETRIC SPHERE IN ZERO-GRAVITY Consider the problem of a homogeneous ball rolling without sliding on the surface of an unfixed dynamically asymmetric spherical shell. Dep ending on the dimensions of the ball and the shell and their mutual arrangement, there are three different ways in which they can roll over each other (Figs. 1a­1c).

Fig. 1. Possible ways in which a ball can roll on the surface of a shell. The dynamically asymmetric b ody -- shell -- is marked in gray here and in the sequel, the dynamically symmetric b ody -- ball -- is marked in white or is hatched.

Remark 1. As is seen from the illustrations, the names bal l and shel l are applicable only to the case shown in Fig. 1a; nevertheless, throughout this pap er we will use these terms for the cases shown in Fig. 1b, c as well, meaning by "shell" a dynamically asymmetric b ody and by "ball" a dynamically symmetric one. As shown b elow, in the three rolling cases the governing equations are identical up to the sign of some parameters. For definiteness, we will denote the coordinates and characteristics of a dynamically asymmetric shell by capital letters and those of a homogeneous ball by small letters. Consider in greater detail the derivation of equations for the case (a) in Fig. 1. We write the equations governing the evolution of the absolute velocities and angular velocities (i. e. referred to an inertial coordinate system, e.g., with resp ect to the system of the center of mass), assuming that all vectors are pro jected onto the axes of a fixed coordinate system, which are attached to the principal axes of the dynamically asymmetric shell: M (V + â V ) = -N , m(v + â v ) = N , I + â I = -Rc â N , i + i â = rc â N ; (1.1)

wher ball, ball, from

e M , m, V , v , and are the masses, velocities and angular velocities of the shell and the I = diag (I1 , I2 , I3 ) is the central tensor of inertia of the shell, i is the moment of inertia of the N is the reaction exerted by the shell on the ball and Rc = OC and rc = oC are the vectors the centers of the shell and the ball to the contact p oint (Fig. 2).
Vol. 16 No. 5 2011

REGULAR AND CHAOTIC DYNAMICS


468

BORISOV et al.

Fig. 2. Rolling of a ball over a shell.

Eqs. (1.1) must b e complemented with the no-slip condition (i. e. the velocities of the p oint of contact for the ball and the shell are the same) and the geometric constraint V + â Rc = v + â rc Rc - rc = -(R0 - r0 ), (1.2) (1.3)

where R0 and r0 are the p osition vectors of the centers of the shell and the ball in a fixed coordinate system (Fig. 2). The vectors Rc and rc can b e written in terms of the unit vector n from the center of the shell to the center of the ball: Rc = Rn, rc = r n. (1.4) Differentiating constraint (1.3) using (1.4) and pro jecting it onto the moving coordinate axes, we obtain V - v = -(R - r )(n + â n), (1.5) n=- r n â ( - ). R-r (1.6)

Hence, using (1.2) we find

Mm where w = R - r , µ = M +m . We now introduce the following variables: R K = I + i , M = I + i + µ(R - r )n â (w â n). (1.8) r The vector M is the total angular momentum of the system of interest with resp ect to its center of mass. The vector K determines some "hidden" momentum, which has no clear physical treatment and was found by Chaplygin [2], who attempted to give it a mechanical interpretation in a related problem of a dynamically symmetric ball rolling on a plane and having a spherical cavity inside which another homogeneous ball is rolling. By using (1.7), it can b e shown that b oth these quantities are constant in a fixed coordinate system. The inverse transformation from momenta M and K to the angular velocities has the form d n â (i - L) â n, i = L + i+d (An, P ) = AP + D An , 1 - D (n, An)

Eliminating from Eqs. (1.1) the reaction force N and the velocities v and V using (1.2) and (1.6), we obtain 1 I + â I = -Rµn â (w + â w ) â n + w â (w â n) , R-r (1.7) 1 i + i â = r µn â (w + â w ) â n + w â (w â n) , R-r

REGULAR AND CHAOTIC DYNAMICS

Vol. 16

No. 5

2011


ROLLING OF A HOMOGENEOUS BALL

469

where the following notation is used L= M-K , (1 - ) N=
-1

A = (I + D )

M - K d , P =N + n â (L â n), ( - 1) i+d R id2 , = , d = mr 2 , D = . r i+d

(1.9)

Finally we obtain a closed system of nine equations governing the evolution of the vectors M, K and n in the movable coordinate system: r n â ( - ). (1.10) M = M â , K = K â , n = - R-r To b e able to determine the motion of the b odies' centers, one needs, in addition to the knowledge of K(t), M(t), n(t) (1.10), 1) the two following equations v - V = (R - r ) â n, mv + M V = P , (1.11)

where P is the integral of the total momentum of the system (in the system of the center of mass P = 0), and 2) the equations governing the motion of the shell-fixed coordinate system = â , = â , = â , (1.12)

where , , are orthogonal unit vectors for the shell-fixed coordinate system, which are written in pro jections onto the axes of the movable coordinate system. Now consider the two remaining cases (see Figs. 1b and 1c). The only difference of these cases from that considered ab ove is that the relations (1.4) are replaced with: Rc = Rn, Rc = -Rn, rc = -r n rc = -r n for case (b), for case (c). (1.13)

In the reduced system (1.10) such a change entails the change of signs in front of r and R. Note that Eqs. (1.10) are invariant with resp ect to R -R, r -r , therefore, the case (c) is exactly describ ed by Eqs. (1.10), and it differs from the case (a) in the change of sign of the velocities in quadratures (1.11) and in that in the case (a) R > r and in the case (c) R < r . In the case (b) the change of sign in front of r is equivalent to assuming r to b e negative. Thus, the following prop osition holds. The problem of a homogeneous bal l (spherical shel l) rol ling without sliding on a dynamical ly asymmetric shel l (sphere) in zero-gravity is described by the system of equations M = M â , K = I + i , 1 n â ( - ), 1- M = I + i + d( - 1)n â (( - ) â n), K = K â , n= (1.14)

where d = µr 2 and = R . For the three cases the values of the parameters and the constraints r imposed on the absolute velocities are as fol lows: (a) > 1, v - V = r ( - ) â n; (b) < 0, v - V = -r ( - ) â n; (c) 0 < < 1, v - V = -r ( - ) â n.
REGULAR AND CHAOTIC DYNAMICS Vol. 16 No. 5 2011


470

BORISOV et al.

Eqs. (1.14) admit five first integrals of motion M2 = const, K2 = const, (M, K) r 1 (M, w ) + E= 2(R - r ) 2(R - r 1 (K - M, ) + (M - = 2( - 1) = const, ) n2 = 1, (1.15)

(K, - )

K, ) = const

and have the invariant measure dM dK dn with density = 1 - D (n, An)
1/2

=

1-

i+d n, E + I id2

-1

1/2

n

.

(1.16)

Thus, in the general case the flow defined by (1.13) preserves the invariant measure on a fourdimensional integral manifold given by (1.14) for this system. Consider p ossible "mechanisms" of integrability amd methods of integration of this system. In non-holonomic mechanics it is customary to integrate n-dimensional systems using the Euler­ Jacobi method or the last multiplier theory [10, 21, 22], which requires the presence of n - 2 first integrals and of an invariant measure, the nonsingular paths representing (rectilinear) cables of two-dimensional tori. In this case, for system (1.13) to b e integrable in the Euler­Jacobi sense two more integrals are lacking, and in the general case this method is not applicable to the system involved. Indeed, as shown in Section 5.1, for = 0 Eqs. (1.13) define an integrable Hamiltonian system with three degrees of freedom, and its nonsingular paths are cables of three-dimensional tori (i. e. in this case instead of two additional first integrals there is only one integral and one symmetry field). The problem of integrability of a system dep ending on the presence of various tensor invariants was discussed in V. V. Kozlov's works [23­25]. There exists a "naive" criterion of integrability: for a system to b e integrable, n - 1 indep endent tensor invariants must exist. Remark 2. Recall that the tensor field is a tensor invariant ("conservation law") if its Lie derivative along the vector field of the system vanishes. Examples of tensor invariants include first integrals, symmetry fields, invariant measures, Poisson structures etc. We conclude this section by presenting additional first integrals of the full system that includes the equations governing the motion of the center of mass (1.11) and evolution of moving axes (1.12): (K, ) = const, (K, ) = const, (K, ) = const, (M, ) = const, (M, ) = const, (M, ) = const. The presence of these integrals considerably facilitates the control and motion planning for such systems [26].

2. THE GENERALIZED ROUTH PROBLEM Consider two sp ecial cases of the problem considered ab ove: assume that the center of either the shell or the homogeneous ball is fixed (i. e. in the former case we fix the center of the dynamically asymmetric b ody and in the latter case the center of the symmetric b ody). Only one scenario of rolling will b e considered in detail (the others result in the same manner as in Section 1). These problems are a natural extension of the simplest integrable problem of a homogeneous ball rolling over another dynamically symmetric ball with a fixed center, which was considered by Routh in [20].
REGULAR AND CHAOTIC DYNAMICS Vol. 16 No. 5 2011


ROLLING OF A HOMOGENEOUS BALL

471

Fig. 3. Rolling of a ball over a spherical shell with a fixed center.

2.1. The Problem of a Homogeneous Ball Rolling over a Dynamically Asymmetric Spherical Shell with a Fixed Center Consider the rolling scenario shown in Fig. 3 and use the same notation as in the previous section. Write the equations of motion for the system in the coordinate frame attached to the principal axes of the spherical shell: I + â I = -Rc â N , i + i â = rc â N , m(v + â v ) = N . (2.1)

Choosing the origin of coordinates of the fixed system at the center of the shell gives R0 0. In this case the no-slip conditions and geometric constraints have the form â Rc = v + â rc , Rc - rc = r0 , Rc = Rn, rc = r n. (2.2)

As in Section 1, eliminating the reaction N and the velocity v from Eqs. (2.1) and introducing the variables R (2.3) K = I + i , M = I + i + m(R - r )n â ((R - r ) â n), r we obtain a reduced system of equations r M = M â , K = K â , n = - n â ( - ). (2.4) R-r The vector M is the total angular momentum of the system with resp ect to the fixed center of the spherical shell. The equation relating the linear and angular velocities to each other b ecomes v = (R - r ) â n. (2.5)

Note that the system of equations (2.3) and (2.4) b ecomes identical with system (1.14) up on the change µ m; thus, the following prop osition holds Prop osition 1. The problem of sliding-free rol ling motion of a homogeneous bal l (spherical shel l ) of mass m over a dynamical ly asymmetric spherical shel l (bal l ) of mass M in zero-gravity is equivalent (up to additional quadratures) to the problem of rol ling of a homogeneous bal l of reduced Mm mass µ = M +m over a spherical shel l whose center is fixed. This prop osition resembles the famous theorem from celestial mechanics which states that the 2-b ody problem reduces to the problem of motion of a single b ody in the field of an attracting center. Remark. Eqs. (2.4) can b e obtained by letting M in Eqs. (1.14) and switching to a coordinate system that moves with constant velocity V .
REGULAR AND CHAOTIC DYNAMICS Vol. 16 No. 5 2011


472

BORISOV et al.

2.2. The Problem of a Dynamically Asymmetric Spherical Shell Rolling over a Homogeneous Ball with a Fixed Center We will deal with the case shown in Fig. 4. As ab ove, we make use of the notation of Section 1. In the coordinate system fixed to the movable shell, the equations of motion have the form I + â I = -Rc â N , i + i â = rc â N , M (V + â V ) = -N . (2.6)

Fig. 4. Rolling of a spherical shell over a ball with a fixed center.

Let the origin of coordinates of a fixed coordinate system b e at the center of the ball r0 0. Then the no-slip condition and the geometric constraint take the form V + â Rc = â rc , Rc - rc = -R0 , Rc = Rn, rc = r n. By analogy with the previous problems introduce the variables R i , M = I + i + M (R - r )n â ((R - r ) â n) r and write the equations of motion: r n â ( - ). M = M â , K = K â , n = - R-r K = I + (2.8) (2.7)

(2.9)

In this case the vector M is the angular momentum of the system with resp ect to the p oint at which the ball is fixed. From insp ection of Eqs. (2.3)­(2.4), (2.8)­(2.9) and (1.14) it is clear that the following prop osition holds Prop osition 2. The systems of equations that govern 1. rol ling of a homogeneous bal l of mass m over a dynamical ly asymmetric spherical shel l of mass M in zero-gravity, 2. rol ling of a homogeneous bal l of mass µ = shel l with a fixed center,
Mm M +m

over a dynamical ly asymmetric spherical
Mm M +m

3. rol ling of a dynamical ly spherical shel l of mass µ = fixed center

over a homogeneous bal l with a

are equivalent to each other up to additional quadratures and are special cases of the reduced system of equations (1.14).
REGULAR AND CHAOTIC DYNAMICS Vol. 16 No. 5 2011


ROLLING OF A HOMOGENEOUS BALL

473

3. PARTICULAR CASES OF MOTION Now consider particular cases of motion (i. e. we assume several additional restrictions on the initial conditions to b e satisfied) for which the system of nine equations (1.14) reduces to the system of six equations. 3.1. The case K = M Using the integrals (1.15), one can reduce Eqs. (1.14) by two degrees of freedom, thus obtaining a system of five differential equations with an energy integral and invariant measure. In the general case the system (1.14) does not admit any first integrals of motion except for (1.15). However, there exist a series of particular cases in which the equations can b e reduced even further and their integrability can b e proved. Consider one of the most interesting particular cases of Eqs. (1.14) where the vectors M and K are parallel, that is, aM + bK = 0. (3.1)

This equation determines an invariant submanifold of the system (1.14). Its invariance follows from the fact that the vectors M and K are constant in the fixed coordinate system. Assume that a = 0 and b = 0 (the cases a = 0 or b = 0 are examined in detail b elow) and represent the manifold as K = M. The equations of motion restricted to it read M = M â , where k= ( - )i - d( - 1)2 , (1 - )(( - )i + d( - 1)) c= -1 , ( - )i - d( - 1)2 -1 I â n . - n = kn â ( - cI), (3.2)

and the moment M is related to the angular velocity as follows: M= d( - 1) -1 I + nâ - ( - )i + d( - 1) i - (3.3)

The first integrals of the system (3.2) are n2 = 1, M2 = const, 1 -1 E= M2 - (M, I) + ( - )(M, ) , 2(1 - ) i and the density of the invariant measure dM dn is given by the relation (1.16). These equations simplify further in the following case. Let = , then the equations of motion and the relation b etween the angular velocities and the momenta b ecome M = M â , n= 1 n â , 1- (3.4)

d (M, (1 - D A)n) = A M- n, i+d (n, (1 - D A)n) where A and D are given by (1.9).

(3.5)

Numerically obtained Poincar´ cross-sections for the systems (3.2) and (3.5) indicate their e nonintegrability.
REGULAR AND CHAOTIC DYNAMICS Vol. 16 No. 5 2011


474

BORISOV et al.

3.2. The Case K = 0 If at the initial moment we set K = 0, then, since the comp onents of K in the fixed frame are constants of motion, this equality will b e satisfied at all moments of time. In this case the equations of motion b ecome 1 n â ( - ), (3.6) M = M â , n = 1- and the relation b etween the momentum and the angular velocity is given by =- 1 I , i M= -1 d I + D n â ( â n) + n â (I â n) . i (3.7)

Eqs. (3.6) admit three first integrals n = 1, M = const, 1 E= M, (i-1 I + 2 E) 2( - 1) and the measure dM dn with density (1.16). For these equations to b e integrable one additional first integral is needed. However, the Poincar´ e cross-section indicates that the system (3.6) is not generally integrable. 3.3. The Case M = 0 By analogy with the previous section we set M = 0. In this case the equations of motion and the relation b etween the momentum and the angular velocity are K = K â , 1 n â ( - ), 1- K d K = + nâ - ( - 1)i i + d ( - 1)i (An, P ) D An , = AP + 1 - D (n, An) n= d n â (K â n) , i+d

ân ,

(3.8)

where the following notation is used P =- 1 -1 K- A = (I + D E)-1 , D= id 2 . i+d (3.9)

The integrals of motion have the form n2 = 1, K2 = const, 1 E= (K, - ), 2( - 1) and there is an invariant measure dK dn with density (1.16). Numerical exp eriments show that, as in the previous case, Eqs. (3.8) are not integrable. 4. INTEGRABLE PARTIAL CASES Below we present the most interesting (from an integrability standp oint) partial cases of the system (3.2), i. e. on the invariant manifold M = K.
REGULAR AND CHAOTIC DYNAMICS Vol. 16 No. 5 2011


ROLLING OF A HOMOGENEOUS BALL

475

4.1. The Axisymmetric Case Consider the axisymmetric case of Eqs. (3.2) where I2 = I1 . Write the time derivatives of the pro jections of the momentum M3 , Mn = (M, n) and the comp onent of the normal vector n3 a1 M3 - 2kµ1 n3 Mn a2 Mn - 2kµ2 n3 M3 ÔM , Mn = ÔM , M3 = 2 1 + k n3 1 + k n2 3 n3 = ÔM , (4.1)

where M = (M, n â e3 ), e3 = (0, 0, 1), µ1 =

The coefficients a1 , a2 , µ1 , µ2 and k are related by a1 a2 = -4kµ1 µ2 ,

( - )i - ( - 1)I1 1 (( - )i + ( - 1)d)( - 1) 1 , µ2 = , 2 d - ( - )i + ( - 1)I 2 ( - 1) 2 ( - 1)2 d - ( - )i + ( - 1)I1 1 -(( - )i + ( - 1)d)kI1 -(( - )i - ( - 1)I1 )( - 1)d a1 = , a2 = , (4.2) 2 d - ( - )i + ( - 1)I )d (( - 1) (( - 1)2 d - ( - )i + ( - 1)I1 )I1 1 (I3 - I1 )id2 ( - 1)2 d - ( - )i + ( - 1)I1 k= , Ô= , I1 (id2 + (i + d)I3 ) ( - 1)2 (id2 + (i + d)I1 ) 1 µ1 + µ2 = - . 2 (4.3)

Dividing the first two equations of the system (4.1) by the third, we obtain a closed system of two first-order differential equations with n3 as an indep endent variable: M = n a1 M3 - 2kµ1 n3 Mn , 1 + k n2 3 M = 3 a2 Mn - 2kµ2 n3 M3 ; 1 + k n3 (4.4)

here the prime denotes the derivative with resp ect to n3 . The system (4.4) can b e written as a single second-order equation, e.g., in Mn : Equation (4.5) is an equation of the hyp ergeometric typ e. In the present case the solution to the equation can b e expressed in terms of the elementary functions: 2µ1 -2µ1 C k n + k 1 + k n2 1 + C2 kn3 + k 1 + kn2 , k(1 + kn2 ) > 0, 3 3 3 3 Mn = C1 sin 2µ1 arcsin -kn3 + C2 cos 2µ1 arcsin -kn3 , k(1 + kn2 ) < 0. 3 (4.6) Here the constants C1 and C2 are the first integrals of motion for the original system. Differentiating (4.6) with resp ect to n3 yields the dep endence of C1 and C2 on Mn and M : n
2µ1

M (1 + kn2 ) + kn3 M - 4kµ2 Mn = 0. n 3 n 1

(4.5)

C1 = (2kµ1 Mn - C2 = (2kµ1 Mn + for k(1 + kn2 ) > 0 and 3

k(1 + kn2 )M ) kn3 + n 3 k(1 + kn
2 )M n 3

k(1 + kn2 ) 3 k(1 + kn2 ) 3

,
-2µ1

(4.7)

) k n3 +

C1 = Mn sin +

M n 2µ1 -k M n C2 = Mn cos - 2µ1 -k = 2µ1 arcsin -k n 3
Vol. 16 No. 5

1 + kn2 cos , 3 1 + kn2 sin , 3 (4.8)

REGULAR AND CHAOTIC DYNAMICS

2011


476

BORISOV et al.

for k(1 + kn2 ) 3 coordinates. It (for k(1 + kn2 ) 3 a factor can b e

< 0. These integrals are linear in velocities but are not algebraic functions of is interesting that their product (for k(1 + kn2 ) > 0) or the sum of squares 3 < 0) is an algebraic integral of degree 2 in velocities, which with an accuracy up to written as
2 F = Mn - 2 1 + k n3 2M 4kµ1 2 n

(4.9)

or, by expressing M in terms of M3 , n F = M2 - n

(a1 M3 - 2kµ1 n3 Mn )2 . 4kµ2 (1 + kn2 ) 1 3

(4.10)

Note that the integral (4.10) is a linear combination of the energy integral and the square of the momentum M2 . Thus, in the case I2 = I1 Eqs. (3.2) admit four indep endent integrals E , n2 , C1 , C2 and an invariant measure and therefore are integrable by the Euler­Jacobi [21, 22] theorem. It is clear that theoretically, one can explicitly evaluate the solution in terms of quadratures given the linearity of the integrals. The study of the system reduces to the analysis of a one-degree system with gyroscopic function (in much the same manner as for the Lagrange top or a circular disk [5]). It would b e interesting to make a qualitative analysis of the motion of this integrable system. 4.2. The Case K = M ( = 1) In the case K = M the equations of motion coincide with the equations governing the rolling of a dynamically asymmetric ball over the surface of a fixed sphere M = M â , where k= Equations (4.11) determine a well-kn a spherical surface [5, 6]; the difference only the ratio of the curvatures of the dynamical characteristics of the b odies system plays an imp ortant role in the holonomic mechanics [10, 22, 27]. 1 1- 1- d . i+d M = I - D n â ( â n), n = k n â , (4.11)

own system describing the rolling of the Chaplygin ball over is that in the problem of the Chaplygin ball k determines adjoining surfaces, whereas in this case k dep ends on the (mass and moment of inertia i) as well. Observe that this analysis of p ossible dynamical effects which occur in non-

For k = 1 the system (4.11) defines the well-known integrable Chaplygin problem of a dynamically asymmetric balanced ball rolling over a horizontal plane. Moreover, as shown in [5] (see also [6]), Equations (4.11) admit another case of integrability for k = -1. Prop osition 3. The equations of motion (1.14) admit a particular case of integrability on the invariant manifold K = M for k= 1 1- 1- d i+d = ±1.

For k = 1 this leads to the equation d µr 2 = = 1, i+d i + µr 2 which holds true only if r , i. e., as stated ab ove, the dynamically asymmetric balanced ball of radius R is rolling over the plane.
REGULAR AND CHAOTIC DYNAMICS Vol. 16 No. 5 2011


ROLLING OF A HOMOGENEOUS BALL

477

For k = -1 we find = d R =2 1+ r i+d

-1

,

i. e. 1 < < 2, which corresp onds to the rolling scenario depicted in Fig.1. It is interesting that, in contract to the case considered in [5], in this case the condition k = -1 imp oses constraints not only on the relative dimensions of the balls but also on their dynamic characteristics. 5. LIMITING AND RESTRICTED PROBLEMS Consider particular cases of Eqs. (1.14). All the particular cases considered b elow allow three different physical interpretations (a zero-gravity problem and two problems where the center of one of the b odies is fixed). However, for brevity we will present for each case only one interpretation which suits b est. 5.1. The Case = 0 Putting = 0 in Eqs. (1.14) and using the notation: M-K ; i+d give a homogeneous quadratic system of the following form: m = K, = m=mâI n2 = 1,
-1

m,

= âI

-1

m,

n = n â (I

-1

m - ).

(5.1)

The functions (1.15) remain integrals of motion for Eqs. (5.1) and can b e represented as m2 = const, 2 = cons 1 E = (m, I-1 m) + 2 t, (m, ) = const, 12 . 2

These equations govern the rolling of a p oint with dynamical prop erties (e.g. it p ossesses rotary inertia) on the surface of a homogeneous ball that rotates ab out its center. In the context of the rolling models describ ed ab ove, the system (5.1) is quite formal as an infinitely small ball with finite moments of inertia is physically unfeasible. For the analysis of integrability of the general equations (1.14), the system (5.1) is imp ortant as it admits an additional integral ( , n) = const and can b e represented in Hamiltonian form where the Poisson brackets are determined by the relations {mi , mj } = -ij k mk , {i , j } = -ij k k , mk = {mk , E }, k = {k , E }, nk = {nk , E }, k = 1, 2, 3,

The Poisson bracket (5.2) p ossesses three Casimir's functions G1 = n2 , G2 = (m + )2 ,

{mi , j } = -eij k k , {i , nj } = -ij k nk ,

{mi , nj } = -ij k nk , {ni , nj } = 0.

(5.2)

G3 = ( , n).

(5.3)

On a symplectic sheet we obtain a Hamiltonian system with three degrees of freedom and three involutive integrals of motion: F1 = E , F2 = m2 , F3 = 2 . (5.4) Thus, according to the Liouville theorem [28­31], Eqs. (5.1) are integrable noting that generic solutions are the cables of three-dimensional tori since each common level of the first integrals is compact.
REGULAR AND CHAOTIC DYNAMICS Vol. 16 No. 5 2011


478

BORISOV et al.

th th D

th

5.2. The Case r = 0 ( , d2 = D = const) Consider the limit case of the problem when the radius of the homogeneous ball vanishes. As in e previous section, such a passage to the limit is formal, however, it provides a further insight into e integrability of (1.14). In this case we let go to infinity in Eqs. (1.14) such that the quantity = d2 , the angular velocities and remain finite. Assume that M - K K , (5.5) = , m= -1 en after the passage to the limit we obtain the system of equations m = m â J-1 m, = âJ
-1

,

n = 0,

(5.6)

where J = I + D (E - n n). As can b e seen from Eqs. (5.6), the vector n is constant in the coordinate system attributed to the sphere. The equations in m and are the classic Euler­Poinsot equations (here they are written in pro jections not onto the principal axes of inertia, since the matrix J is not diagonal). We note that the case under study is closely related to the problem that was considered in Section 5.1. Furthermore, these two problems must coincide when the tensor of inertia I is spherical. However, it is evident that Eqs. (5.6) and (5.1) are not identical if I = E. This is due to the fact that these equations are written in different coordinate systems. Eqs. (5.1) are written in the system rotating with the material p oint while Eqs. (5.6) are written in the system fixed to the rotating sphere. The identity of these equations can b e shown, e.g., by writing the third equation of the system (5.1) in the coordinate frame attributed to the rotating sphere. To do so note that the derivatives of the vector n pro jected onto different coordinate axes are related as follows: n
abs

where nabs is the vector n written in pro jections onto the fixed coordinate system, n - onto the axes related to a dynamically asymmetric material p oint and n - onto the axes related to the rotating sphere. Writing the third equation of the system (5.1) in the form and comparing it to (5.7), we obtain n = n â ( - ) n = 0, This coincides with the equation for the vector n in the system (5.6). Remark 3. The fact that the "material p oint" on the spherical shell is at rest does not contradict the earlier results concerning the "non-holonomic rolling of the material p oint" on various surfaces [32]. Indeed, in the passage to the limit, when the radius vanishes, we leave the angular velocity of the ball's rotation finite. As a result, the ball's velocity relative to the shell b ecomes zero. On the other hand, in [32], while the radius is vanishing, the angular velocity of the ball's rotation increases infinitely such that the linear velocity of the material p oint relative to the surface remains finite. 5.3. Rolling of a Homogeneous Ball Over an Indep endently Rotating Dynamically Asymmetric Shell Let all the three moments of inertia I1 , I2 and I3 in (1.14) tend to infinity, while their ratios I1 I2 I3 I3 remain finite and, in addition, the angular velo cities , and the parameters i and d also remain finite. Putting 1 M-K I1 = µI1 , I2 = µI2 , I3 = µI3 , m = K, = ; µ (1 - )(i + d(1 - )) m=mâI
-1

= n + â n = n + â n ;

(5.7)

(5.8) (5.9)

and passing to the limit µ , after some simplification, we obtain the system m, where k = i + d(1 - ) . (i + d)(1 - ) =âI
-1

m,

n = kn â (I

-1

m - ),

(5.10)

REGULAR AND CHAOTIC DYNAMICS

Vol. 16

No. 5

2011


ROLLING OF A HOMOGENEOUS BALL

479

Physically, the system (5.10) governs a restricted problem for which a dynamically asymmetric shell rotates ab out a fixed p oint as a free rigid b ody (i. e. the Euler­Poisson case from the rigid b ody dynamics [33]), and a dynamically symmetric ball rolling over it has no influence on its dynamics. Equations (5.10) have five first integrals n2 = 1, m2 = const, 2 = const,
-1

(m, ) = const,

E = (m, I

m)

and an invariant measure I dm d dn with constant density. For k = 1 we obtain an integrable Hamiltonian system with three degrees of freedom, describ ed in Section 5.1. For 0 the system (5.10) takes the form and is a limit case of the Chaplygin ball rolling over a sphere, describ ed in detail in [33]. As shown in [33], for integer odd k the system has an algebraic integral of degree |k| in variables m. For m 0 the system (5.10) also admits a trivial case of integrability. 5.4. Rolling of a Dynamically Asymmetric Shell over a Constantly Rotating Ball If we let i in Eqs. (1.14) keeping the other parameters I, and d and the angular velocities and finite, then, denoting m= we obtain the following equations m = m â , M - K , -1 = â , = = lim
i

m = mâI

-1

m,

n = kn â I

-1

m

(5.11)

K , i

1 n â ( - ), 1- (5.12) (m + dn â ( â n), J-1 n) n, J = m + dn â ( â n) + 1 - (n, J-1 n) d2 where J = I + d2 E. Physically, this case can b e regarded as a restricted problem for the system describ ed in Section 2.2, where the mass (and, accordingly, the moment of inertia i) of a dynamically symmetric ball is so great that its motion is not influenced by the shell rolling over it. The system (5.12) also has five integrals of motion n= m2 = const, 2 = const, (m, ) = const, 1 E = (m, ) + d(n â (( - ) â n), ) 2 and an invariant measure i dm d dn with density i = (n, IJ
-1

n2 = 1,

n).

Since the vector is constant in the fixed coordinate system, we can put = 0, thus obtaining the equations of the Fedorov system (see [6]) dealt with in Section 4.2: m = m â , 1 n â , 1- (m, J-1 n) J = m + n. 1 - (n, J-1 n) d2 n=

(5.13)

For = 2 the system remains integrable in the limit case 0, d2 D = const. This case corresp onds to the well-known Chaplygin problem of a ball rolling on a plane.
REGULAR AND CHAOTIC DYNAMICS Vol. 16 No. 5 2011


480

BORISOV et al.

Remark coincides equations in the sys

4. Note that if the tensor of inertia is spherical, that is, I = E this particular case with the case of Section 5.3 with I = E. The only difference is that in Section 5.3 the are written in a fixed coordinate system, and in the case considered here they are written tem attributed to the rolling ball.

5.5. Passage to a Ball Susp ension Now we consider the passage to the limit M for the problem of Section 2.2. In this case the equation in V of the system (2.6) b ecomes V = V â , (5.14)

whence it follows that the velocity V is constant in the fixed coordinate system. The only value of V that satisfies the constraint equation (2.7) for non-zero angular velocities is V 0. Thus, the equation of motion in this case can b e written as I + â I = -Rc â N , i( + â ) = rc â N , n = n â . (5.15)

These equations admit the first integral of motion C = ( , n) = const and can b e written as K = K â , n = n â , R R K = I + i , = n â ( â n) + C n r r or, after the change of M = K -
R r

(5.16)

(5.17)

iC n, as the equations for the Chaplygin ball [3]: n = n â , (5.18)

M = M â , M = I +

R in â ( â n). r2

2

In conclusion, we mention some of the most interesting unresolved problems. First of all, it would b e interesting to find stationary solutions to the general system and to explore their stability (p erhaps it would clear up the question of representability of the system in a Hamiltonian form). One could also look for integrable cases of the system considered. But what is most interesting, esp ecially from the p ersp ective of p ossible applications, is the problem of motion in the case of dynamic symmetry (in particular, a generalization of the results obtained for the submanifold M = K to the general system).

ACKNOWLEDGMENTS The authors are grateful to Yu. N. Fedorov, M. Przybylska and A. Yu. Moskvin for useful comments and discussions. This research was supp orted by the Grant of the Government of the Russian Federation for state supp ort of scientific research conducted under sup ervision of leading scientists in Russian educational institutions of higher professional education (contract no. 11.G34.31.0039) and the federal target programme "Scientific and Scientific-Pedagogical Personnel of Innovative Russia" (pro ject code 02.740.11.0195). A. A. Kilins's research was supp orted by the grant of the President of the Russian Federation for the Supp ort of Young Russian Scientists­Candidates of Science (MK-8428.2010.1).
REGULAR AND CHAOTIC DYNAMICS Vol. 16 No. 5 2011


REGULAR AND CHAOTIC DYNAMICS Vol. 16 No. 5 2011 Fig. 5. Hierarchy of partial and limiting cases in the problem of rolling motion of a dynamically asymmetric ball over a homogeneous sphere in zero-gravity.

ROLLING OF A HOMOGENEOUS BALL 481


482

BORISOV et al.

REFERENCES
1. Chaplygin, S. A., On the Ball Rolling on a Horizontal Plane, in Col lected works, vol. 1, pp. 76­101, Moscow­Leningrad: Gostekhizdat, 1948 (Russian). 2. Chaplygin, S. A., On some generalization of the area theorem with applications to the problem of rolling balls, in Col lected works, vol. 1, pp. 26­56, Moscow­Leningrad: Gostekhizdat, 1948 (Russian). 3. Fedorov, Yu. N., The Motion of a Rigid Body in a Spherical Support, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1988, no. 5, pp. 91­93 (Russian). 4. Markeev, A. P., Integrability of the Problem of Rolling of a Sphere with a Multiply Connected Cavity Filled with an Ideal Fluid, Izv. Akad. Nauk SSSR. Mekhanika tverdogo tela, 1986, vol. 21, no 1. pp. 64­65 (Russian). 5. Borisov, A. V. and Fedorov, Y.N., On Two Modified Integrable Problems of Dynamics, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1995, no. 6, pp. 102­105 (Russian); English transl.: http://ics.org.ru/doc?pdf=384&dir=e 6. Borisov, A. V., Fedorov, Yu. N., and Mamaev, I. S., Chaplygin Ball over a Fixed Sphere: An Explicit Integration, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 557­571. 7. Borisov, A. V., Mamaev, I. S., and Marikhin, V.G., Explicit Integration of one Problem in Nonholonomic Mechanics, Dokl. Akad. Nauk, 2008, vol. 422, no. 4, pp. 475­478 [Dokl. Phys., 2008, vol. 53, no. 10, pp. 525­528]. 8. Veselov, A. P. and Veselova, L. E., Integrable Nonholonomic Systems on Lie Groups, Mat. Zametki, 1988, vol. 44, no. 5, pp. 604­619 [Math. Notes, 1988, vol. 44, no. 5, pp. 810­819]. 9. Borisov, A. V., Mamaev, I. S., and Kilin, A. A., The Rolling Motion of a Ball on a Surface: New Integrals and Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 201­219. 0. Borisov, A. V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443­490. 1. Fedorov Yu. N., Multidimensional Integrable Generalizations of the Nonholonomic Chaplygin Sphere Problem, Report, Nov. 29, 2006, 13 p. See also: http://www.cirm.univmrs.fr/videos/2006/exposes/32/Fedorov.pdf 2. Jovanovi´, B., Hamiltonization and Integrability of the Chaplygin Sphere in Rn , J. Nonlinear Sci., 2010, c vol. 20, no. 5, pp. 569­593. 3. Kilin, A. A., The Dynamics of Chaplygin Ball: The Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291­306. 4. Duistermaat, J. J., Chaplygin's Sphere, in Chaplygin and the Geometry of Nonholonomical ly Constrained Systems, R. Cushman, J. J. Duistermaat, J. Sniatycki (Eds.), Singapore: World Sci. Publ., 2010. 5. Shen, J., Schneider, D. A., and Bloch, A. M., Controllability and Motion Planning of a Multibody Chaplygin's Sphere and Chaplygin's Top, Int. J. Robust Nonlinear Control, 2008, vol. 18, no. 9, pp. 905­ 945. 6. Alves, J. and Dias, J., Design and Control of a Spherical Mobile Robot, J. Systems Control Engrg., 2003, vol. 217, no. 6, pp. 457­467. 7. Bhattacharya, S. and Agrawal, S. K., Design, Experiments and Motion Planning of a Spherical Rolling Robot, Proc. IEEE Intern. Conference on Robotics & Automation (San Francisco, CA, April 2000), pp. 1208--1212. 8. Crossley, V. A., A Literature Review on the Design of Spherical Rol ling Robots, Pittsburgh, PA, 2006. 9. Behar, A., Matthews, J., Carsey, F., and Jones, J., NASA/JPL Tumbleweed Polar Rover, Proc. IEEE Aerospace Conference (Big Sky, MO, March 2004), pp. 1--8. 0. Routh, E. J., Dynamics of a System of Rigid Bodies: 2 vols, New York: Dover, 1905. 1. Kozlov, V. V., On the Theory of Integration of the Equations of Nonholonomic Mechanics, Adv. in Mech., 1985, vol. 8, no. 3, pp. 85­107 (Russian). 2. Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds, Regul. Chaotic Dyn., 2011, vol. 16, pp. ?­?. 3. Kozlov, V. V., Tensor Invariants of Quasihomogeneous Systems of Differential Equations, and the Asymptotic Kovalevskaya­Lyapunov Method, Mat. Zametki, 1992, vol. 51, no. 2, pp. 46­52 [Math. Notes, 1992, vol. 51, no. 2, pp. 138­142]. 4. Kozlov, V. V., Symmetries and Regular Behavior of Hamiltonian Systems, Chaos, 1996, vol. 6, no. 1, pp. 1­5. ´ 5. Kozlov, V. V., Integral Invariants after Poincar´ and Cartan, in Cartan, E., Integral Invariants, Moscow: e URSS, 1998, pp. 215­260 (Russian). 6. Kozlov, V. V. and Ramodanov, S. M., Motion of a Variable Body in an Ideal Liquid, J. Appl. Math. Mech., 2001, vol. 65, no. 4, pp. 579­587. 7. Borisov, A. V. and Mamaev, I. S., Rolling of a Rigid Body on a Plane and Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177­200.
REGULAR AND CHAOTIC DYNAMICS Vol. 16 No. 5 2011

1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2


ROLLING OF A HOMOGENEOUS BALL

483

28. Arnol'd, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1989. 29. Arnol'd, V. I., Kozlov, V. V., and Ne¨shtadt, A. I., Mathematical Aspects of Classical and Celestial i Mechanics, Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 1993. 30. Kozlov, V. V., Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergeb. Math. Grenzgeb. (3), vol. 31, Berlin: Springer, 1996. 31. Borisov, A. V. and Mamaev, I. S., Modern Methods of the Theory of Integrable Systems, Moscow­Izhevsk: Institute of Computer Science, 2003 (Russian). 32. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., On the Model of Non-Holonomic Billiard, Rus. J. Nonlin. Dyn., 2010, vol. 6, no. 2, pp. 373­385 (Russian). 33. Borisov, A. V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Moscow­Izhevsk: Institute of Computer Science, 2005 (Russian). 34. Borisov, A. V., Mamaev, I. S., and Kilin, A. A., Dynamics of Rolling Disk, Regul. Chaotic Dyn., 2003, vol. 8, no. 2, pp. 201­212.

REGULAR AND CHAOTIC DYNAMICS

Vol. 16

No. 5

2011