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ISSN 1560-3547, Regular and Chaotic Dynamics, 2007, Vol. 12, No. 2, pp. 153­159. c Pleiades Publishing, Ltd., 2007.

RESEARCH ARTICLES

Rolling of a Non-Homogeneous Ball over a Sphere Without Slipping and Twisting
A. V. Borisov* and I. S. Mamaev
**

Institute of Computer Science, Udmurt State University, Universitetskaya ul. 1, Izhevsk 426034, Russia
Received December 9, 2006; accepted February 28, 2007

Abstract--Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found. MSC2000 numbers: 37N05, 76M23 DOI: 10.1134/S1560354707020037 Key words: nonholonomic mechanics, reducing multiplier, hamiltonization, isomorphism

1. CONSTRAINT REALIZATION AND THE EQUATIONS OF MOTION Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over the surface of a sphere. Suppose that during such motion the contact point has zero velocity and the projection of the angular velocity vector to the normal vector of the sphere equals zero. This model was introduced in the paper [1] and was presented in details in [2, 3]. It differs from the classical nonholonomic model of rolling without slipping, in which only the contact point velocity is assumed to be zero. We start with the classical nonholonomic model. S. A. Chaplygin [4] studied the rolling of a ball over a plane. He found the invariant measure and the additional first integral needed for integration by the theory of Jacobi's multiplier. He fulfilled the explicit integration using sphero-conical coordinates. For wider list of publications and the Chaplygin ball investigation see [5]. A. V. Borisov and Yu. N. Fedorov [6] considered the rolling of a ball over an stationary sphere. For the general case, this system is not integrable; the integrable case, for which the ratio of radii of the (moving) ball and the (stationary) sphere is two, was found in [7]. (In this paper this integrable system is generalized to the case of motion of the ball on a stationary sphere with additional constraint described above.) The corresponding system has not been integrated by quadratures and its dynamics have not been studied. The hierarchy of the dynamics in nonholonomic systems with a rigid body rolling without slipping over a plane or a sphere is discussed in [7]. We now consider the new model of rolling that can be physically interpreted as the motion of the ball under the no-slip and no-twist constraints. We construct the equations (not integrable in the general case) of the ball's motion under such constraints. For this purpose, we use the equations of nonholonomic mechanics in quasi-coordinates with undetermined multipliers. This form of the equations was probably first presented by G. K. Suslov in his famous paper [8]. Such equations are ´ called the Euler­Poincare­Suslov equations. For the quasi-coordinates and the quasi-velocities take, respectively, the unit normal vector n of the surface and the vectors v , s of the center of mass velocity and the angular velocity. All vectors are referred to the principal central coordinate system strictly attached to the ball.
* **

E-mail: borisov@ics.org.ru E-mail: mamaev@ics.org.ru

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The kinetic energy of the free system has the form 1 1 T = mv 2 + ( , I ). (1) 2 2 Here m is the mass center of the ball, I = diag(I 1 , I2 , I3 ) is the central inertia tensor. Two nonholonomic constraints mentioned above are f = v + â r = 0, (2) g = ( , n) = 0, (3) where r = -bn is the radius vector of the contact point with respect to the ball center, b is the ball radius. Note that the constraint (2) expressing the no-slip condition has the vector form f = (f 1 , f2 , f3 ). The ´ Euler­Poincare­Suslov equations have the form fi T d T = â+ Ni + F , dt v v v (4) fi d T T g = â+ Ni + + MF dt or, explicitly, mv = mv â + N + F , I = I â + r â N + n + MF . (5)

Here N = (N1 , N2 , N3 ) and are the undetermined multipliers (constraints reactions), F and M F stand for the external force and its torque with respect to the mass center. Eliminating N , we come to a system that is separated from the equations for v : Here MQ = MF + F â r is the torque of the external forces with respect to the contact point, and r = -bn. I + mr â ( â r ) = I + mr â ( â r ) â - mr â ( â r ) + n + MQ . (6)

Fig. 1.

In order to obtain a closed system, we construct the kinematic equation describing the evolution of n. First note that the velocities of the points representing the contact point on the movable and the stationary surfaces are equal. This condition can be written as follows: Here y is the vector drawn from the center of the stationary sphere to the contact point (Fig. 1). We have y = an, where a is the radius of the stationary sphere. From (7) one easily obtains a n = kn â , k = . (8) a+b
REGULAR AND CHAOTIC DYNAMICS Vol. 12 No. 2 2007

r = y + â y.

(7)


ROLLING OF A NON-HOMOGENEOUS BALL OVER A SPHERE

155

Fig. 2.

Fig. 3.

a| If a, b < 0, we have the outer rolling as shown in Fig. 1. If a < 0, b > 0, |a| > b and k = |a||-b > 0, we a have the inner rolling of the ball over the sphere (Fig. 2), and if a > 0, b < 0, a < |b|, k = - |b|-a < 0, we have the outer rolling of the sphere over the ball (Fig. 3). Put a = , k = 1 for the case of the ball rolling over the plane and b = , k = 0 for a planar rolling over the sphere. Differentiate the constraint (3), by virtue of (8) we obtain ( , n) = 0. Hence, using the constraint itself we get the relations

Therefore,

I + mr â ( â r ) = I + mb2 ,

I + mr â ( â r ) = I + mb2 . n = kn â ,
ij

J = J â + n + MQ , J = I + mb2 E, where =-

E=

,

(9)

(J â , J-1 n) + (MQ , J (n, J-1 n)

-1

n)

.

Prior to the analysis of the system (9), note that the additional constraint g = (n, ) = 0 has a clear physical interpretation. In this case we are dealing with a rolling model where the torque of the twisting friction with respect to the contact point is extremely large. This is true, for example, in the case of viscous friction when the dissipation coefficient generated by twisting tends to infinity. One possible realization of such rolling is the motion of a rubber coated ball over a rough surface (this realization is proposed in [1]). In this case, due to the elasticity of the rubber, the contact takes place in some domain rather than at one point. It leads to a large torque due to twisting friction. Another more classical realization of the constraints (2) and (3) was considered by the authors in [9]. Nevertheless, it is important to understand that the possibility of applying either model to the description of the real motion must be established experimentally. All of these models are of the approximate type only. More exact models use various laws of friction and, unfortunately, do not admit any theoretical analysis. Note also that one of the nonholonomic models was proposed in [10], where the rolling of a round disk with a sharp edge over horizontal ice was considered. The nonholonomic constraint on the system supposes that the velocity of the point of contact is parallel to the horizontal diameter of the disk. This rolling model differs from classical models (Appell, Chaplygin, Korteweg), and the question of its applicability was solved experimentally. 2. INTEGRABLE CASES In the sequel, we suppose no external forces: M Q = 0. In this case the system (9) preserves the measure (n, J
-1

n) 2k d dn.

1

(10)

Note that this measure is also preserved in the case when M Q depends only on n rather than on . The measure (10) was found in the paper [2]. Equations (9) have the energy integral and the geometrical one: 1 H = (J , ), (n, n) = 1. 2
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The constraint ( , n) = 0 can be also considered as an additional partial integral. For complete integrability by the theory of the last multiplier (the Euler­Jacobi theorem) it is necessary to have one more integral we will call F . It can be found in two cases. 1. Let k = 1 (a = ). This corresponds to the ball rolling over the horizontal plane. Then F = (J â n, J â n). It is easily drawn from equations (9) that the resulting system is equivalent to Veselova's system [11] (it was shown by us in [9]) and it can be integrated in sphero-conical coordinates. 2. Let k = -1 (b = -2a). This is the case where the inner surface of the dynamically asymmetric sphere is rolling over the stationary ball. The new integral is F= (J , J )n2 + det J( , J )(J (n, J-1 n)
-1

n, J

-1

n)

.

(11)

We find this integral establishing the analogy of the system (9) with some Hamiltonian system describing the motion of a material point on a two-dimensional sphere. We now present this analogy in detail. 3. HAMILTONIAN STRUCTURE AND ALGEBRAIC APPROACH Introduce the sphero-conical coordinates ( , ) on the sphere |n| = 1. These are the roots of the equation f (z ) = n2 n2 n2 (z - )(z - ) 1 2 3 + + = , J1 - z J2 - z J3 - z A(z ) (12)

where A(z ) = (J1 - z )(J2 - z )(J3 - z ). It can be shown that 0 < J1 < < J2 < < J3 . From (2) we easily find that n2 = 1 (J1 - )(J1 - ) , (J1 - J2 )(J1 - J3 ) n2 = 2 (J2 - )(J2 - ) , (J2 - J1 )(J2 - J3 ) n2 = 3 (J3 - )(J3 - ) . (J3 - J1 )(J3 - J2 ) (13)

From the constraint equation we obtain an expression for in terms of n and n, namely, = k -1 n â n. The angular velocity is found in terms of , , and from (13). The kinetic energy T = 1 ( , J ) 2 is also expressed in terms of , , and : T= - 8k 2 - 2 2 + A( ) A( ) . (14)

It can be shown, after S. A. Chaplygin [12], that the equations of motion are d T T - = S, dt d dt T - T = - S, .

2k - 1 ( - ) S= 8k 3 Introduce the generalized momenta P =
T

+ A( ) A( )
T -1 J

(15)

and P =

instead of the velocities , . In terms of these n)
-1+1/k

new variables, the preserved measure has the form (n, = (n, J
2 -1

dP dP d d . Let
-1+
1 2k

n) = , det J

N =

-2+

1 k

=

det J

.

(16)

Introduce the new time-variable such that d = N dt and denote by prime the derivative with respect T T to . Then = N , = N , p = = N T and p = = N T . The equations of motion become canonical T T T T , p = - , = , p = - , (17) = p p
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i.e., N is the reducing factor of Chaplygin. (See [9, 12] for details of the reducing factor method.) Thus we have shown that with respect to the new time the system is Hamiltonian (for k = 1 this was shown by us in [9] using Euler angles). Such systems are also called conformally Hamiltonian systems. Conformally Hamiltonian systems of type (9) and (15) was first found in [2] and [3] by means of slightly different techniques. If k = 1/2 (a = b), then the system (9), (15) is Hamiltonian in the usual sense. The coordinates and are local. To establish the complete form of the isomorphism of the system (9), (15) with the problem of motion of the point on the sphere we use a new algebraic coordinate system. Denote 1 det J N det J N= = (n, J-1 n) 2k . 2 k k Introduce the three-dimensional vectors M = NJ Obviously, ( , ) = 1, (M , ) = N ( , n) = 0. (19)
1/2

,

=

1 J

-1/2

n.

(18)

Using the expressions for and n in terms of , , p , p we easily calculate the Poisson brackets of M and to be and the expression for the Hamiltonian function 1 H=T = N 2
-2

{Mi , Mj } = -

ij k

Mk ,

{Mi , j } = -

ij k k

,

{i , j } = 0,

(20)

M2 =

1 k2 ( , J ) 2 det J

1/k

M2 = = 2k 2 - det J
(2k -1)/k

A( ) 2 A( ) p- p

2

. (21)

The bracket (20) is the Lie­Poisson bracket of the (co)algebra e(3). The functions F 1 = (M , ), F2 = ( , ) are its Casimir functions. According to (21), variables separate in these sphero-conical coordinates for k = 1, and do not separate for k = -1. Separation of variables for the case k = -1 will be introduced in Section 5. 4. TRAJECTORY ISOMORPHISM The Hamiltonian function (21) is the product of two functions depending on M and respectively: H = G( )F (M ). The equations of motion can be written in the form G-1 F M =G Mâ - F G â M , = G â F . M (22)

Introduce the time change ds = G( )dt and fix any level of the energy integral F G = h. At this level we obtain the system H H dM =Mâ +â , ds M d H =â , ds s

h H = F (M ) - . G( )

(23)

Thus, at any fixed level of the energy integral H = h the system (22) is trajectory equivalent to the system (23) with H = 0. The procedure described here is known in celestial mechanics as regularization and is due to Bolin and Levi­Civita.
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-1/k

1 For the Hamiltonian function (21) we have F (M ) = 2 M 2 , G( ) = ( , J )

, =

det J k2

h and (24)

1 H = M 2 - ( , J ) 2

-1/k

,

where h is the constant of the energy integral (21). It is well-known that the Hamiltonian function (24) describes the motion of a material point on the sphere surface (exactly under the condition (M , ) = 0) in the potential force field with the potential function V = h( , J ) -1/k . The integrable potentials correspond to k = 1 (the Braden system [13]) and k = -1 (the Neumann system). For the other cases the Hamiltonian function (24) seems to be not integrable. It can be shown, at least, that there exists no additional integral of the second degree with respect to M . Nevertheless, the existence of the integral of higher degree with respect to momenta is still possible.

5. SEPARATION OF VARIABLES FOR THE CASE k = -1 This isomorphism allows us to construct separating variables for the case k = -1. Let us choose the sphero-conical coordinates u and v as the roots of function
3

g (z ) =
i=1

ni (z - u)(z - v ) 1 = , Ji (Ji - z ) (det J)A(z )

where A(z ) = n2 = 1
2 J1

i

(Ji - z ) and 0 < J1 < u < J2 < v < J3 . Hence, we get n2 = 2
-1 2 J2

(J1 - u)(J1 - v ) , (J1 - J2 )(J1 - J3 )
-1

(J2 - u)(J2 - v ) , (J2 - J1 )(J2 - J3 )

n2 = 3

2 J3

(J3 - u)(J3 - v ) , (J3 - J1 )(J3 - J2 )

where 2 = (n, J

n) = (

i

Ji - u - v ) T=

. For kinetic energy (14) we have u2 v2 - A(u) A(v ) .

(det J)4 (u - v ) 8

For the reducing multiplier N we get from Eq. (15)
3/2

N=
i

Ji - u - v
T v

;

then for the variables u, v , pu = N T and pv = N u the canonical equations with the Hamiltonian H= det J(

after change of time N dt = d we obtain

i

2 A(u)p2 - A(v )p u Ji - u - v )(u - v )

2 v

.

Therefore, variables are separated.

ACKNOWLEDGMENTS We thank J. Koiller and K. Ehlers for formulating the problem and valuable discussions. This research was supported by the Russian Foundation for Basic Research (Grant 05-01-01058), the State Maintenance Programs for the Leading Scientific Schools of the Russian Federation (Grant NSh1312.2006.1) and INTAS (Grant 04-80-7297). We also acknowledge the partial support from the the Institute of Mathematics and Mechanics of the Ural Branch of Russian Academy of Sciences.
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REFERENCES
1. Ehlers, K., Koiller, J., Montgomery, R. and Rios, P.M., Nonholonomic Systems via Moving Frames: Cartan Equivalence and Chaplygin Hamiltonization, in Marsden, J. and Ratiu, T., Eds., The Breadth of Symplectic ¨ and Poisson Geometry, vol. 132 of Progr. Math, Boston: Birkhauser, 2005, pp. 75­120. 2. Ehlers, K. and Koiller, J., Rubber Rolling: Geometry and Dynamics of 2-3-5 Distributions, Proceedings of the IUTAM symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, Dordrecht: Springer, (to appear). 3. Koiller, J. and Ehlers, K., Rubber Rolling over a Sphere, Regul. Chaotic Dyn., 2007, vol. 12, pp. 127­152. 4. Chaplygin, S.A., On a Ball's Rolling on a Horizontal Plane, Math. Sbornik, 1903, vol. 24, pp. 139­ 168. Reprinted in: Sobranie sochinenii (Collected Papers), Moscow-Leningrad: Gostekhizdat, 1948, vol. 1, pp. 76­101. English translation in: Regul. Chaotic. Dyn., 2002, vol. 7, pp. 131­148. 5. Kilin, A.A., The Dynamics of Chaplygin Ball: the Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, pp. 291­306. 6. Borisov, A.V. and Fedorov, Yu.N., On Two Modified Integrable Problems of Dynamics, Vestnik Moskov. Univ. Ser. 1 Mat. Mekh., 1995, no. 6, pp. 102­105. 7. Borisov, A.V. and Mamaev, I.S., The Rolling of Rigid Body on a Plane and Sphere. Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, pp. 177­200. 8. Suslov, G.K., Teoreticheskaya Mekhanika (Theoretical Mechanics), Moscow: Gostechizdat, 1946. 9. Borisov, A.V. and Mamaev, I.S., Isomorphism and Hamilton Representation of Some Nonholonomic Systems, Sibirsk. Mat. Zh., 2007, vol. 48, pp. 33­45 [Siberian Math. J. (Engl. transl.), 2007, vol. 48, pp. 26­ 36]. 10. Kozlov, V.V. and Kolesnikov, N.N., On the Theorems of Dynamics, J. Appl. Math. Mech., 1978, vol. 42, pp. 28­33. 11. Veselova, L.E. New Cases of Integrability of the Equations of Rigid Body Motion with Nonholonomic Constraint, Geometry, Diff. Equations and Mechanics, Moscow: Moskov. Gos. Univ., Mekh.-Mat. Fak., 1986. 12. Chaplygin, S.A., On the Theory of the Motion of Nonholonomic Systems. Theorem on the Reducing Multiplier, Mat. Sbornik, 1911, vol. 28, no. 2, pp. 303­314. 13. Braden, H.W., A Completely Integrable Mechanical System, Lett. Math. Phys., 1982, vol. 6, pp. 449­452.

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