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A. V. BORISOV
Department of Theoretical Mechanics Moscow State University, Vorob'ievy Gory 119899, Moscow, Russia E-mail: borisov@rcd.ru

I. S. MAMAEV
Laboratory of Dynamical Chaos and Nonlinearity Udmurt State University, Universitetskaya, 1 426034, Izhevsk, Russia E-mail: mamaev@rcd.ru

arXiv:nlin.SI/0303024 v1 12 Mar 2003

A. A. KILIN
Laboratory of Dynamical Chaos and Nonlinearity Udmurt State University, Universitetskaya, 1 426034, Izhevsk, Russia E-mail: aka@rcd.ru

THE ROLLING MOTION OF A BALL ON A SURFACE. NEW INTEGRALS AND HIERARCHY OF DYNAMICS
Received February 10, 2002

DOI: 10.1070/RD2002v007n02ABEH000205

The pap er is concerned with the problem on rolling of a homogeneous ball on an arbitrary surface. New cases when the problem is solved by quadratures are presented. The pap er also indicates a sp ecial case when an additional integral and invariant measure exist. Using this case, we obtain a nonholonomic generalization of the Jacobi problem for the inertial motion of a p oint on an ellipsoid. For a ball rolling, it is also shown that on an arbitrary cylinder in the gravity field the ball's motion is b ounded and, on the average, it does not move downwards. All the results of the pap er considerably expand the results obtained by E. Routh in XIX century.

Contents
1. Intro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 2. The equations for a ball moving on The integrals of motion. . . . . . . . . . . Rolling on a surface of the second order. A ball on a rotating surface. . . . . . . . 3. Ball's motion on a surface of A parab oloid of revolution. . . . . An axisymmetric ellipsoid. . . . . Historical comments. . . . . . . . A circular cone. . . . . . . . . . . A circular cylinder. . . . . . . . . A ball on a rotating sphere. . . . . The rolling of a ball on a rotating a . . . surf ... ... ... . . . . . . . . ace . . ..... ..... ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... .... .... .... . . . . . . . .. . . . . . . . . . . . . . . .. . . . .. . . . . . . . 201 202 203 203 203 206 207 208 208 208 209 210

revolution ....... ....... ....... ....... ....... ....... sphere. . . .

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Mathematics Sub ject Classification 37J60, 37J55

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4. Rolling of a ball on surfaces of Comment I. . . . . . . . . . . . . . An elliptic (hyp erb olic) parab oloid. Comment I I. . . . . . . . . . . . . . Motion of a ball on an elliptic cone.

t . . . .

he . . . .

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se . . . .

cond ... ... ... ...

order .... .... .... ....

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210 211 211 213 213

5. Motion of a ball on a cylindrical surface . . . . . . . . . . . . . . . . . . . . . . . . . 214 An elliptic (hyp erb olic) cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

1. Intro duction
In this pap er we consider the problem of sliding-free rolling of a dynamically symmetrical ball (the central tensor of inertia is spherical I = µE) on an arbitrary surface. As it was indicated by E. Routh in his famous treatise [10], if the surface is a surface of revolution the problem is integrable even in the presence of axisymmetric p otential fields. Here we give a more complete analysis of the Routh solution for the b ody of revolution, and present new integrals for the case of ball rolling on non-symmetrical surfaces of the second order.

2. The equations for a ball moving on a surface
In rigid b ody dynamics it is customary to introduce a b ody-fixed frame of reference. However, while studying motion of a homogeneous ball, it is more convenient to write the equations of motion with resp ect to a certain fixed in space frame of reference. In such a frame a balance of linear momentum and of angular momentum with resp ect to the ball's center of mass, involving the reaction and the external forces, may b e written as mv = N + F , (I ). = a â N + MF . (2.1)

The sliding-free condition reads (the contact p oint velocity vanishes) v + â a = 0. (2.2)

Here m is the mass of the ball, v isthe velocity of its center of mass, is the angular velocity, I = µE is the (spherical) central tensor of inertia, a is the vector from the center of mass to the p oint of contact, R is the ball's radius, N is the reaction at the contact p oint (see Fig. 1), F and MF are the external force and the moment of forces with resp ect to the p oint of contact resp ectively. Fig. 1. The rolling of a ball on Eliminating from these equations the reaction N and making use of a surface (G is the center of the fact that the contact p oint velocities on the surface and on the ball mass, Q is a point of contact coincide, we obtain the system of six equations: with the surface) M = D â ( â ) + MF , r + R = â R . (2.3)

Here D = mR2 . These equations govern the b ehaviour of the vector of the angular momentum (with resp ect to the contact p oint M ), and of the vector = -R-1 a normal to the surface (Fig. 1). The vectors and r (the p osition vector of the contact p oint) are obtained from the relations M = µ + D â ( â ),
202

=

F (r ) , |F (r )|

(2.4)

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F (r ) = 0 is the equation of the surface on which the ball rolls. The last equation in (2.4) defines the Gaussian map. Further on, following Routh, we will use the explicit form of the surface on which the bal l's center of mass is moving. This surface, defined by the p osition vector r = r + R , is equidistant relative to the surface on which the contact p oint is moving. In the case of p otential forces, the moment MF is expressed in terms of the p otential U (r ) = = U (r + R ). This p otential dep ends on the p osition of the ball's center of mass as follows MF = = R â U .
r Remark 1. In his treatise [10] Routh obtained the equations of motion of the ball in semifixed axes and presented the cases when these equations can be analytically solved. Actually, in the ma jority of the subsequent publications [1, 9] Routh's results were just restated without any essential expansion. It should be noted that Routh was especially interested in the stability of particular solutions (e. g. a ball rotating around the vertical axis at the top of a surface of revolution). Here we are not giving the original form of Routh's equations. Equations (2.3) are, in many respects, similar to the equations describing an arbitrary body motion on a plane and a sphere [5]. This allows us to consider many problems (e. g. those of integrability) from the single point of view.

The integrals of motion. In the case of p otential field with p otential U (r + R ), equations (2.3) p ossess the integral of energy and the geometric integral H = 1 (M , ) + U (r + R ), 2 F1 = 2 = 1. (2.5)

In the case of an arbitrary surface F (r ) = 0, apart from these integrals, the system (2.3) has neither measure, nor two additional integrals, which are necessary for the integrability according to the last multiplier theory (the Euler ­ Jacobi theory). The system's b ehavior is chaotic. As it is shown later, in some cases there exists a measure and only one additional integral. In such a case chaos b ecomes "weaker". As it was noted by Routh, for a surface of revolution there exist two additional integrals, the system is integrable, and its b ehavior is regular. The reduced system b ecomes a Hamiltonian one after an appropriate change of time. Rolling on a surface of the second order. We are going now to revisit equations(2.3) for the case when the ball's center of mass is moving on a surface of the second order. The surface is defined by the equation r + R , B-1 (r + R ) = 1, B = diag(b1 , b2 , b3 ) (2.6)

(for an ellipsoid, bi > 0 and are equal to the squares of the main semi-axes). Solving (2.6) for the p osition vector r gives B . (2.7) r + R = ( , B ) We obtain the equations of motion in terms of the variables M , : M =- D (M , ) , µ+D = R ( , B ) â â B-1 ( â M ) µ+D (2.8)

A ball on a rotating surface. Let us also consider the motion a ball on a surface rotating with constant angular velocity . One particular case of this problem (a plane and a sphere) was also investigated by Routh [10]. By analogy with the previous case, replacing the nonholonomic relation (2.2) by v + â a = â r, (2.9) we get M = m a â ( â a ) + a â ( â r ) + M r + R = â R + â r .
F

(2.10)

Here a = -R .

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It should b e noted that in the case of an arbitrary surface the system of six equations (2.10) is not closed b ecause the p osition vector of the p oint on the surface, r , is not expressed in terms of only; one should also introduce the equation for the angle of rotation of the surface around the fixed axis. Nevertheless, if the surface is axisymmetric, and the symmetry axis coincides with the axis of rotation, equations (2.10) get closed. We assume this to b e fulfilled in further text. Equations (2.2), (2.10) are in many resp ects similar to the equations, defining the rigid b ody rolling on a plane or a sphere, which we thoroughly investigated in the pap er [5]. We shall use this fact while transferring the corresp onding results onto systems (2.2), (2.10).

3. Ball's motion on a surface of revolution
First of all, let us consider the cases of integrability of equations (2.3), (2.10), associated with the rotational symmetry of the surface on which the ball rolls. Here we are using the technique introduced in [5] and concerned with the analysis of a certain reduced system in terms of new variables K1 , K2 , K3 , 3 . We shall also assume the surface to b e rotating around the axis of symmetry with constant angular velocity = (0, 0, ), = 0. The particular case, = 0, was discussed by Routh, who obtained the ma jority of the following results (although he missed some cases of equal interest). The equation of a surface of revolution with resp ect to an immovable reference frame may b e written as r1 = (f (3 ) - R)1 , r2 = (f (3 ) - R)2 , (3.1) 1 - 2 f (3 ) - 3 f (3 ) d3 - R3 , r3 = 3 where f (3 ) is a certain function sp ecifying the surface parametrization. The parametrization (3.1) is to b e so chosen as to make the form of the reduced system as simple as p ossible. In the case b eing considered equations (2.3), (2.10) allow an invariant measure with density = f (3 ) g(3 ),
3

where g(3 ) = f (3 ) -

2 1 - 3 3 f (3 ).

(3.2)

Apart from the invariant measure, the equations also have a simple symmetry field ^ v = M1 - M2 + 1 - 2 , M2 M1 2 1 (3.3)

which is caused by the rotational symmetry. In our b ook [3] we make a frequent use of the fact that to obtain the simplest form of the reduced system in the presence of symmetries, one needs to choose the most relevant integrals of the field of symmetries which, in this case, may b e written as K1 = (M , )f (3 ), K2 = µ3 = µM3 + D (M , )3 , µ+D K3 = M1 2 - M2 1 , 1 - 3 3 . (3.4)

In terms of the chosen variables, the reduced system takes the form 3 = kK3 , f f K1 = kK3 K2 + 1 - g µ , 3 R 1 K - 1 - g µ , K2 = kK3 D 3 µ+D f 1 R (µ + D )g 3 K1 - f K2 2 (µ + D 3 )K1 - 3 f (µ + D )K2 + K3 = -k 2 2 )f 2 2 µ (1 - 3 f (1 - 3 ) g 2 + 1- (µ + 2D 3 )K1 - 3 f (µ + 2D )K2 µ, R
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where k =

2 R 1 - 3 . It is easy to show that this system of equations p ossesses an invariant measure (µ + D)g (3 )

with density = k-1 . The system (3.5) can b e explicitly integrated in the following way. Let us divide the second and the third equation of system (3.5) by 3 . Then we obtain the nonautonomous system of two linear equations with the indep endent variable 3 f f dK1 g µ , = K2 + 1 - 3 R d3 dK2 1 K - 1 - g µ . =D 3 µ+D f 1 R d3 (3.6)

This system of linear equations always p ossesses two integrals which are linear in K1 , K2 . The coefficients in the integrals are functions of 3 , and in the general case cannot b e obtained in the explicit (algebraic) form. Having divided the last equation of the system (3.5) by 3 and substituting into this equation the known solution of the system (3.6), we obtain the explicit quadrature for K3 (3 ). Using the first equation from (3.5) we can obtain the expression for 3 (t). In the case = 0, the system (3.5) has the integral of energy
2 2 K1 K3 (µ + D )(3 K1 - f K2 )2 H = (M , ) = 1 . + + 2 2 f 2 µ+D µ2 f 2 (1 - 3 )

(3.7)

The quadrature for 3 (t) may b e obtained up on substitution of K3 = k For the reduced system (3.5) the following theorem is valid:

-1

3 into (3.7).

Theorem 1. For = 0 by the change of time k dt = d , the system (3.5) can be represented in the Hamiltonian form dxi = {xi , H }, x = (3 , K1 , K2 , K3 ). (3.8) d The bracket is degenerate and specified by the relations {3 , K3 } = µ + D , f {K1 , K3 } = (µ + D ) K2 , 3 {K2 , K3 } = D K1 f (3.9)

(for the other pairs of variables the brackets are zero ). Proof is a straightforward exercise consisting in derivation of (3.5) ( = 0) from (3.8) and verification of the Jacobi identity for the bracket (3.9). One can assert that the equations for the linear integrals of the system (3.6) exactly coincide with the equations for the Casimir function of the bracket (3.9). It should also b e noted that for = 0 the system (3.5) has "skew-symmetrical notation", similar to the Hamiltonian form (3.8), (3.9) dxi = Jij (x) H ; d xj Jij = -Jj i . (3.10)

The matrix J is of a somewhat more general form than that of the corresp onding bracket (3.9) 0 0 0 µ+D f 0 0 (µ + D ) 3 K2 + u D K + v J = , 0 - 0 1 f (3.11) f D K + v -(µ + D ) -(µ + D ) 3 K2 - u 0 1
f

u=

(µ + D )2 (3 K1 - f K2 )
2 µ2 f (1 - 3 )K3

,

v=

2 (µ + D ) (µ + D 3 )K1 - f 3 (µ + D )K2 2 µ2 f 2 (1 - 3 )K3

.

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Here is an arbitrary function of 3 , Ki . For = 0, one again gets bracket (3.9), and the tensor J satisfies the Jacobi identity. However, in the general case, J does not satisfy the Jacobi identity (it is not of the Poisson typ e). Nevertheless, if we assume = w(3 )K3 , (3.12)

then the tensor -1 J is a Poisson one, and the quantity is a reducing multiplier (according to Chaplygin). Thus, we obtain the following Hamiltonian vector field v = (-1 J )H, for which div v = 0 (div v = 0). The examples of Hamiltonian fields with nonzero divergency were almost not discussed earlier. One particular case of the Poisson tensor -1 J was found by J. Hermans in [12]. Hermans used his own system of reduced variables which slightly differs from ours. To clarify the b ehavior of a ball on a surface of revolution, we will discuss the cases, when this surface is a parab oloid, a sphere, a cone, and a cylinder. These problems were investigated by Routh in [10] for = 0. Here we will expand the results for = 0. A parab oloid of revolution. Supp ose the ball's center of mass is moving on the parab oloid of revolution z = c(x2 + y 2 ). In equations (3.1) we, therefore, put f (3 ) = - 1 . 2c3 The density of the invariant measure (3.2) (up to an unessential factor) can b e written as = 16 . 3 In this case, the two-dimensional system (3.6) reads 1 + 2cR3 dK1 µ , = 1 3 K2 - 2cR3 d3 2c3
3 (1 + 2cR3 ) dK2 µ . 2c3 K1 + =- D 2 µ+D d3 2cR3

(3.13)

(3.14)

(3.15)

For the variable K2 we obtain a homogeneous second-order linear equation whose coefficients are homogeneous functions of 3 (at = 0) D (2 + cR3 ) 1 K2 - K2 + D 12 K2 = µ . 3 3 µ + D 3 (µ + D )cR3
Its general solution may b e represented as 3 , = const 1 1 K2 = c1 3 - - c2 3 + + µ 1 +

1, 2D cR(µ + 4D ) 3

2 =

µ . µ+D

(3.16)

For the variable K1 from (3.15) we, similarly, obtain K1 = µ+D - -(1 - )3 2cD
1- - c1 + (1 + )3 1+

c2 -

µ µ 1. 1- 3 2c 2cR(3µ + 4D ) 3

(3.17)

Solving for the constants c1 , c2 from (3.16), (3.17), we obtain the integrals of the system (3.6). These integrals are linear in to K1 , K2 and have form F2 = F3 = 2 2 (µ + D )(1 + ) µ+D 2+ D K2 + µ 3 - - 3 2c3 K1 + 2 D 3 D 3 2cR(2 - )3 µ(µ + D ) D - 3 µ(µ + D )


(µ + D )(1 - ) µ+D 2- 2c3 K1 + K2 + µ 3 - - 2 D 3 D 3 2cR(2 + )3

(3.18)

206

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The product F2 F3 gives an algebraic quadratic integral. For = 0 the equations also have the integral of energy H=
2 2c2 3 2 1 µ + D 2 µ K1 + 2 µ2 (1 - 2 ) 2c3 K1 + K2 3 2 2 K3 . +1 2µ+D

(3.19)

Remark 2. Some forms of surfaces (on which a ball is rolling) are investigated in the paper [6] (details are given below). In this paper it is shown that if a ball is rolling on a paraboloid of revolution, then the system (2.3) is reduced to a particular class of Fuchsian equations. Routh himself considered the case when not the contact point, but the center of mass is moving on a paraboloid. We should note again that in this case equations (2.3) have algebraic integrals. The motion of a homogeneous ball on a surface of revolution was also studied by F. Noether [11].

An axisymmetric ellipsoid. Consider the motion of a dynamically symmetrical ball when its center of mass is moving on a fixed ellipsoid ( = 0). In this case f (3 ) = b b1 (1 -
1 2 3

2 ) + b3 3

,

(3.20)

where b1 , b2 are the squares of the ellipsoid principal semi-axes. The density of the invariant measure (3.2) (up to a constant factor) is
2 2 = b1 (1 - 3 ) + b3 3 -3

.

(3.21)

In this case the variables N1 , N2 , K3 are more convenient than the variables (3.4). Here N1 = (M , ), N2 = µ f (3 ) 3 (M , ) - M3 , µ+D (3.22)

which satisfy the system, similar to (3.5) for = 0 f N1 = -kK3 N2 , 3 f 2 N2 = kK3 µ f N1 , µ+D

(µ + D )g 2 K3 = -k 2 N µf (1 - 3 )N1 + (µ + D )3 N2 , 2 )2 f 3 2 µ 3 (1 - 3 where k =
2 R 1 - 3 . (µ + D)g

(3.23)

Using (3.20), we obtain two linear equations with indep endent variable 3 dN1 =- d3 b (b1 - b3 )
2 3

1

b1 (1 -

)+b

2 3 3

N2 ,

µb1 dN2 N1 . = 2 2 d3 (µ + D ) b1 (1 - 3 ) + b3 3

(3.24)

It is easy to show that system (3.24) has a quadratic integral with constant coefficients F2 = b
2 1

µ 2 N 2 + (b1 - b3 )N2 . µ+D 1

(3.25)

Below we will show that this integral may b e generalized to the case of a three-axial ellipsoid. The system (3.24) can b e solved in terms of elementary functions. Dep ending on the sign of the difference b1 - b3 , the solution may b e written as:
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A. V. BORISOV, I. S. MAMAEV, A. A. KILIN b1 > 1. b1 - b3

1. b1 > b3 , a2 =

N1 = c1 sin (3 ) + c2 cos (3 ), (3 ) = arctg

N2 = a 3
2 a2 - 3

µb1 -c1 cos (3 ) + c2 sin (3 ) , µ+D , = µ . µ+D

(3.26)

2. b1 < b3 , a2 =

b1 > 0. b3 - b1

N1 = c1

-

+ c2 ,

N2 = a a2 +
2 3

µb1 -c1 µ+D , =

-

+ c2



, (3.27)

(3 ) = 3 +

µ . µ+D

Here c1 , c2 = const. Solving for these constants, we can obtain linear integrals of motion.
Historical comments. It is an interesting fact that neither Routh, nor his followers succeeded in obtaining the simplest reduced equations (like (3.24)) and solving the problem of a ball rolling on an ellipsoid of revolution in terms of elementary functions. To integrate the equations one must appropriately choose the reduced variables such as (3.22).

A circular cone. In this case, due to the fact that the Gaussian map = F is degenerate, one should use the comp onents of the vector r (the p osition vector of the ball center) as the p ositional variables in equations (2.10). For the cone (Fig. 2) we have 3 = cos = const,
Fig. 2
2 r1 , 1 = k 2 r3 1+k 2 2 r1 + r2 , 2 r2 2 = k , 2 r3 1+k (3.28)

|F |

r3 = k

k = tg .

The measure of equations (2.3), (2.10) in which is expressed in terms of r in accordance with (3.28), can b e written in explicit form:
k2 R + r 0 3 1 - k2
2 r3 3

=

,

(3.29)

For the reduced system let us choose the variables r3 D 1 = 3 + , 2 µ+D R 1+k
2 R 2 = r3 + k 1 + k2

k 2 µ2 M- , . µ+D

(3.30)

In terms of these variables we obtain the equations d1 = 0, dr3
208

d2 = dr3

1 + k 2 1 +

µ r3 -k µ+D R

2

.

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From these equations the following integrals can b e easily obtained: F2 = 1 , F3 = 1 + k2 r3 1 - 2 +
2 r3 µ - k2 r3 . µ + D 2R

(3.31)

A circular cylinder. The motion of a bal l within a cylinder is a well-known problem which is usually used to illustrate some unrealistic conclusions, derived by means of nonholonomic mechanics. It can b e shown that a homogeneous ball, moving within a vertical cylinder, on the average, is not rolling downwards due to the gravity force. Nevertheless, this physical fact may b e observed while playing basketball, when the ball has almost hit the basket, but then rapidly jumps out of it, suddenly lifting upwards. The addition of a viscous friction to this nonholonomic system, which leads to a vertical drift, is analyzed in [8], where the explicit solution of this problem has also b een obtained. For a cylinder, = -
r r1 , - 2 , 0 , where Rc is the cylinder radius (Fig. 3), in Rc Rc

terms of the variables (M , r ) or ( , r ), the invariant measure's density is constant. The kinetic energy looks like 1 H = 1 (M , ) = 2 2(µ + D )
2 2 2 (M1 + M2 ) + D (M , )2 + M3 µ

=

2 = 1 (µ( , )2 + (µ + D )(3 + (1 2 - 2 1 )2 )). 2

The reduced system in terms of the variables 1 = 3 , 2 = ( , ) looks like
1 = 3 = 0, 2 =

Fig. 3

R 3 - Rc . (Rc - R)R

(3.32)

Thus, we have two integrals 3 = const, ( , ) - R3 - Rc r3 = const. Rc - R R (3.33)

On writing = (1 , 2 ,

R3 - Rc ), the second integral takes the form Rc - R

( , r ) = const, and the kinetic energy b ecomes r 2H = µ ( , ) + 3 3 R
2 2 + (µ + mR2 )3 + (µ + mR2 ) 2 3 R2

r

.

Hence, the variable r3 (resp onsible for the vertical displacement of the ball) can b e easily found to b e r3 = - ( , ) ± 3
2 2H - (µ + mR2 )3 sin 2 µ 3

3

µ (t - t0 ) , µ + mR 2

where t0 is a constant dep ending on the initial conditions. It is clear that the average displacement of the ball equals zero even in the presence of the gravity field (see formula (5.6)).
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Let us now consider two different variants of problem concerning the rolling of a ball on a sphere. These problems were integrated and analyzed by Routh [10]. A ball on a rotating sphere. Consider a sphere of radius Rs rotating around a certain axis with a constant angular velocity . Let R b e the ball radius, a = -R , r = Rs (Fig. 4). The equations of motion in the p otential field with p otential V ( ) may b e represented as D1 =
Fig. 4

D R ( , ) â + Rs R (, )(R + R ) â + â V , s Rs + R Rs + R = (R + Rs ) â Rs + R

(3.34) where D = mR2 , D1 = µ + D , and µ is the moment of inertia of the ball. Whether the ball rolls outside or inside the sphere is determined by the sign of R. Using the angular momentum vector M = D1 - D ( , ), the first equation of (3.34) may b e written as Rs R R(( â )(, ) + (, â )) - Rs (, )( â ) + â V . M= Rs + R Using = +
Rs in (3.34), we get R

(3.35)

This coincides with the original system (3.34) for = 0. Thus, it will b e enough to consider only the case when = 0. In terms of the variables (M , ) equations (3.34) for = 0 are written as M = â V (3.37) R â . = Rs + R They p ossess the integrals H = 1 (M , ) + V ( ), 2 The change of time t - F1 = (M , ) = c, F2 = 2 = 1. (3.38)

D = DR 1 Rs + R = Rs +

R R

( , )( â ) + â V â .

(3.36)

R transforms the second equation of (3.37) into an ordinary Poisson Rs + R 2

equation = â , and the p otential is multiplied by some nonessential factor. Thus we set a system (3.34) for which we know the cases when it is integrable. For example, if V = 1 ( , C ), C = diag(c1 , c2 , c3 ), one gets famous Neumann problem of the motion of a p oint on a sphere in the quadratic p otential (equations (3.34) incorp orate even more general situation, when (M , ) = c = 0, which corresp onds to the Clebsch case [4]). The equations (3.36), (3.37)for the motion of a ball on a sphere also manifests an analogy (discussed in [9, 10]) of problems, concerning the rolling of a homogeneous ball on a sphere in the gravity field, and the Lagrange case in the Euler ­ Poisson equations (the motion of a heavy dynamically symmetrical top). Indeed, for a p otential of the typ e V = V (3 ), the systems (3.36), (3.37), b ecause of the axial symmetry, have "Lagrangian integrals" M3 = const or 3 = const.
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The rolling of a ball on a rotating sphere. Now supp ose that the sphere on which a ball rolls rotates freely ab out its center. The dynamical equations can b e written as: mv = N , µ = a â N , µs = - r â N ,

v + â a = â r.

Here , , µ, µs are the angular velocities and the moments of inertia of the ball and the sphere, resp ectively. Using the relations r = Rs , a = -R , for the quantities = write (1 + D ) = D â ( â ), = â ,
R + Rs , , we can R + Rs

2 R 2 mR D = mµ + µ s . s

(3.39)

By the change of time dt dt, = const, this system can b e reduced to equations (3.36) and, for this reason, is integrable.

4. Rolling of a ball on surfaces of the second order
Consider the dynamics of a ball in greater detail for the case when its center of mass is moving on a surface of the second order r + R , B-1 (r + R ) = 1, B = diag(b1 , b2 , b3 ). (4.1)

In this case the equations of motion are identical in form to (2.8). It can b e shown that these equations p ossess an invariant measure and a quadratic integral of the form = ( , B )-2 , F2 = ( â M , B-1 ( â M )) . ( , B ) (4.2)

Here B is an arbitrary (nondegenerate) matrix.
Comment I. The invariant measure's density was found rather easily, after the authors had obtained equations (2.8) for rolling of a homogeneous ball on an ellipsoid. These equations describe the Jacobi nonholonomic problem. The problem's name stems from the following fact. When the ball's radius is tending to zero it seems that we get the holonomic classical problem concerning the geodesic lines on an ellipsoid (solved by Jacobi in terms of elliptic functions). Apparently, one should get convinced in the correctness of such a limiting transition, which, however, does not prevent us from using the given terminology. One can treat the integral (4.2) as a generalization of the Ioachimstal quadratic integral in the Jacobi problem. Originally, the authors found this integral by numerical experiments, using the Poincar´ three-dimensional map in terms of the e Andoyae ­ Deprit variables (L, G, H, l, g , h). These variables for nonholonomic mechanics were introduced in [5] (see also earlier paper [2]). Fig. 5 illustrates three-dimensional sections of a phase flow in the phase space (l, L/G, H, g ) at a level of energy E = const. The secant plane is g = /2. It is seen that the levels F2 = const make the three-dimensional chaos "foliated" into two-dimensional chaotic surfaces. The fact that the motion on the two-dimensional surfaces F2 = const ensures that no additional integrals (necessary for integrability of the problem) exist.

Let us consider various particular (may b e degenerate) surfaces of the second order for which an integral of the typ e (4.2) exists. An elliptic (hyp erb olic) parab oloid. Let the ball's center of mass move on an elliptic parab oloid defined by the equation 2 x2 + y = 2z . (4.3) b1 b2
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Fig. 5. Examples of three-dimensional mappings (on the left) and the corresponding level surfaces of the integral F2 (on the right). All the mappings are constructed at a level of energy E = 1 for B = diag(1, 4, 9). To the frames on the left correspond the values F2 = 1.7, 2, 4, 8, 10 (from top to bottom). 212 REGULAR AND CHAOTIC DYNAMICS, V. 7,

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Although in this case all the results may b e obtained from the previous ones by means of a passage to the limit, we will derive them "of scratch". The Gaussian map (2.7) looks like: 1 r1 + R1 = -b1 , 3 2 r2 + R2 = -b2 , 3 r3 + R3 =
2 2 b1 1 + b2 2 2 23

,

(4.4)

and the equations motion (2.8) take the form M =- D (M , ) , µ+D = R3 â â Bi ( â M ) , µ+D

where Bi = diag(b-1 , b-1 , 0) is a degenerate matrix. 1 2 The invariant measure density dep ends on 3 only = 14 , 3 and the quadratic integral (4.2) can b e written as F2 = â M , Bi ( â M )
2 3

(4.5)

.

(4.6)

Comment I I. The integrals (4.2), (4.6), being quadratic with respect to the velocities (M or ), depend on the positional variables in a rather complex way. Maybe for this reason the classics (in particular, Routh and F. Noether who obtained only particular results) did not find these integrals. As it has been already noted, the integrals (4.2), (4.6) were originally found through numerical experiments. Their analytic form was obtained by means of the following considerations. As we have shown above, the problem concerning the rolling of a ball on a paraboloid of revolution such that b1 = b2 is integrable and has two additional linear integrals. This integrals (in the absence of rotation = 0) may be written as follows I1 = 3 1-k-1 (1 - 2 3 + 2 3 1 - k ), (4.7) - I2 = 3 1-k-1 (1 - 2 3 - 2 3 1 - k ), k = D µ+D where 1 = 3 , 2 = ( , ). The product of these integrals is also an integral which is quadratic with respect to i , and a rational function of 1 , 2 and 3 J= (2 3 - 1 )2 2 2 2 - 2 (1 - k ) = 1 - 22 1 + k 2 . 2 2 3 3 3 (4.8)

2 Let us eliminate, using the expression for energy (3.7), the term k 2 from the integral (4.8), and consider the integral 2 3 F2 = J + 2E = 2 - 22 3 + 2 . (4.9) 3 3

Substituting the expressions for 2 into (4.9) and isolating perfect squares, we obtain F2 = (2 3 - 3 2 )2 + (3 1 - 1 3 )2 . 2 3 (4.10)

The integral F2 , written in such a form, is easily generalized to the case of an arbitrary paraboloid (b1 = b2 ) 1 ( - )2 + 1 ( - )2 23 32 31 13 b1 b2
2 3

F2 =

=

( â , Bi ( â )) , 2 3

Bi = diag(b

-1 1

,b

-1 2

, 0)

(4.11)

and any surface of the second order (4.2). The integrals (4.2), (4.6) may be used for stability analysis of stationary motions of a ball near the points of intersection of the surface with the principal axes. The symmetrical case of this problem was considered by Routh [10]. REGULAR AND CHAOTIC DYNAMICS, V. 7,

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Motion of a ball on an elliptic cone. Supp ose that the ball's center of mass is moving on the surface of an elliptic cone, defined by the equation (rc , B-1 rc ) = 0, B = diag(b1 , b2 , -1), (4.12)

where rc = r + R are the coordinates of the center of mass, and b1 , b2 are p ositive quantities such that b1 and b2 determine the slop e of the generatrices with resp ect to the coordinate axes. Given the coordinates of the center of mass, we can calculate the normal to the surface at this p oint as follows: B-1 rc . = (4.13) (B-1 rc , B-1 rc ) In our case r (or rc ) is not uniquely defined by (b ecause is constant on a generatrix). Therefore, as phase variables, we will use not (M , ) (as we did earlier), but (M , rc ). Up on substitution of (4.13) into the equations of motion (2.3), we get the equations of motion in terms of the variables M , rc : R r c = (M â B-1 rc ), -1 r , B-1 r ) (µ + D) (B c c (4.14) DR M = (B-1 (M â B-1 rc ), B-1 rc â (B-1 rc â M ))B-1 rc . 2 -1 -1 5/2
(µ + D) (B rc , B rc )

Equations (4.14) p ossess an energy integral H=

D (M , B-1 rc )2 1 (M 2 + ) 2(µ + D ) µ(B-1 rc , B-1 rc ) = (B-1 rc , B-1 rc ). R d t. (µ + D ) (B-1 rc , B-1 rc )

and an invariant measure Let us now make one more change of variables and time y = B-1 rc , This results in d =

y = B-1 (M â y ), M =

D (B-1 (M â y ), y â (y â M ))y (µ + D)(y , y )2

(4.15)

which p ossesses two "natural" integrals: the energy integral H= and the geometrical integral (y , B y ) = 0. The latter defines the surface in terms of the new variables along which the ball's center of mass is moving. Moreover, equations (4.15) p ossess invariant measure with constant density. The generalization of the nontrivial integral (4.2) to the case of equations (4.15) looks like F2 = ((M â y ), B-1 (M â y )), or, in terms of the original physical variables, we set F2 = ((M â B-1 rc ), B-1 (M â B-1 rc )). The question of existence of one more additional integral for equations (4.15) remains op en. Apparently, in the general case, when b1 = b2 , it does not exist.
214 REGULAR AND CHAOTIC DYNAMICS, V. 7,

(M , y )2 1 (M 2 + D µ (y , y ) ), 2(µ + D )

(4.16)

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5. Motion of a ball on a cylindrical surface
Let us consider the rolling of a ball whose center of mass is moving on a cylindrical surface. It can b e shown that, in the absence of external fields, this system may b e integrated by quadratures; but when an external force is directed along the cylinder generatrix, the equations are reduced to a Hamiltonian system with one and a half degree of freedom . Let us choose a fixed frame of reference with one axis (Oz ) directed along the cylinder generatrix (see Fig. 6). In this case, a normal vector is expressed as = (1 , 2 , 0),
2 2 1 + 2 = 1.

(5.1)

Denote the pro jections of the normal to the center of mass and the p osition vector of the center of mass of the ball onto the normal cross-section by r = (r1 + R1 , r2 + + R2 ), = (1 , 2 ). For these pro jections we have evident geometrical relations (r , ) = ( , ) = 0. Hence, we conclude that is parallel to r , = ( )r . The factor ( ) is completely determined by the geometry of the cylinder cross-section and does not dep end on angular velocity. Using equations (2.3), we get the equations of motion for the ball on the cylindrical surface with the assumption that the cylinder is sub ject to a force (with p otential U (z )) directed along a cylinder's r generatrix (z = 3 ):
R Fig. 6

M= =

M3 ( ) D (M â , ez ) + U ez â , µ+D µ+D z M3 ( )ez â , µ+D z= 1 (M â , e ). z µ+D

(5.2)

Here not only the energy is conserved but also the pro jection of the (angular velocity) moment on the cylinder axis: H = 1 (M , ) + U (z ), 2 (5.3) F2 = M3 = (µ + D )3 = const. Moreover, the system (5.3) p ossesses an invariant measure with density ( ) =
-1

( ).

(5.4)

It follows from (5.2) that the equations for the vector get uncoupled. Let us use parametrization 1 = cos , For the angle (t) we obtain the equation = M3 (cos , sin ) = Q µ+D
-1

2 = sin .

(),

(5.5)

where Q() is, in the general case, a 2 -p eriodic function of , determined by the shap e of a cylinder cross-section.
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In the remaining equations of the system (5.2) we put K1 = M1 1 + M2 1 , K2 = M1 2 - M2 1 ,

and replace the time (as an indep endent variable) by the angle , and thereby obtain the nonautonomous system with 2 -p eriodic coefficients µ dK1 K, =- µ+D 2 d dK2 = K1 - Q()U (z ), d

dz = Q() K . µ+D 2 d This system has an integral of energy
2 K2 K2 + U (z ). H = 1 µ1 + 2 µ+D

(5.6)

(5.7)

In the case of the gravity field, U (z ) = mgz and equation (5.6) are integrated by quadratures:


K1 () = - mg K2 () = -mg

0

sin ( - )Q( )d + A cos + B sin , (5.8)



cos ( - )Q( )d + A sin - B cos ,
0

where A, B are constants and 2 =

µ . µ+D

Let us show that the integrals in (5.8) are b ounded functions. We expand the function Q( ) in a Fourier series Q( ) = Qn ein . (5.9)
nZ

The integrals in the expressions for K1 () and K2 () in (5.8) may b e considered as the real and imaginary parts of the integral ei Q( ) d =
n

Qn ei(n

+ )

d .

(5.10)

Using the well known theorems of the Fourier analysis and the fact that n + = 0 (as far as 0 < < 1) we put the integral under the sum sign and integrate the series term-by-term: Qn ei(n
n + )

d =
n

Qn ei(n i(n + )

+ )

.

(5.11)

It is evident that the series obtained converges to a certain quasip eriodic function, hence, K1 () and K2 () are b ounded. This fact and also the conservation of energy in the reduced system (5.7) ensure that z () is b ounded. Thus, when the bal l is rol ling on an absolutely rough cylindrical surface of an arbitrary crosssection in the gravity field, the vertical secular drift is not observed. The time dep endence of the angle (and, hence, of all the other functions) is describ ed by (5.5).
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An elliptic (hyp erb olic) cylinder. Let us consider in greater detail a particular case, i. e., the ball's center of mass is moving on an elliptic cylinder with cross-section is defined by the equation
2 x2 + y = 1. b1 b2

(5.12)

We have r1 + R1 = b1 1 b
2 1 1

+

2 b2 2

,

r2 + R2 =

b2 2
2 2 b1 1 + b2 2

,

r3 = Rz ,

therefore, ( ) = Q
-1

R( , B )3/2 , B = diag(b1 , b2 , 0), b1 b2 M3 R () = (b1 cos2 + b2 sin2 )3/2 . (µ + D )b1 b2

We should mention an imp ortant distinction existing b etween an elliptic and a circular cylinder (see ab ove): in the case of an elliptic cylinder the dep endency of dynamical variables K1 , K2 , z is defined by two frequencies 1 = 1, 2 = , instead of a single frequency, as it happ ens in case of a circular cylinder. Thus, the integrals in (5.8) contain quasip eriodic functions; the integrals have very complicated nature, their analytical prop erties are thoroughly discussed in [7]. Graphs z () for various initial values of K1 and K2 are shown in Fig. 7. The main result is that whatever the ratio of the frequencies, the quantities K1 and K2 , and, therefore, the displacement z execute b ounded, quasip eriodic oscillations. It is the main result of this construction. The authors express are deeply thankful to V. V. Kozlov for useful remarks and discussions.

Fig. 7. The -dependence of the vertical coordinate of the point of contact z for various initial values K1 , K2 , z . For this figure the other parameters are: E = 1, µ = 1, D = 1( = 2-1/2 ), b1 = 1, b2 = 2, R = 1.

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References
[1] P. Appel l. Trait´ de m´canique rationnelle. Paris, e e Gauthier ­ Villars. [2] A. V. Borisov, K. V. Emel'yanov. Nonintegrability and stochasticity in rigid b ody dynamics.. Izhevsk, Izd. Udm. Univer. 1995. [3] A. V. Borisov, I. S. Mamayev. Poisson Structures and Lie Algebras in Hamiltonian Mechanics. Izhevsk: RCD. 1999. P. 464. [4] A. V. Borisov, I. S. Mamaev. Rigid Body Dynamics. Izhevsk: RCD. 2001. P. 384. [5] A. V. Borisov, I. S. Mamaev. Rigid b ody rolling on a plane and a sphere. Dynamical hierarchy. Reg & Chaot. Dyn. 2002 (in print). [6] Yu. P. Bychkov. On rolling rigid b ody on a stationary surface. PMM. 1965. V. 29. 3. P. 573­583 . [7] V. V. Kozlov. Qualitative Analysis Methods in Rigid Body Dynamics. Izhevsk: RCD. 2000. [8] S. N. Kolesnikov. Some Mechanics' Problems on Rolling of Rigid Bodies. Candidate Thesis. Moscow, Lomonosov's MSU. 1988. P. 88. [9] Yu. I. Neumark, N. A. Fufayev. Nonholonomic System Dynamics. M.: Nauka. 1967. P. 519. [10] E. Routh. Dynamics of a System of Rigid Bodies. Dover Publications, New York. ¨ [11] F. Noeter. Ub er rollende Bewegung einer Kugel auf Rotationsfl¨che. Leipzig, Teubner. 1909. 56 S. a [12] J. Hermans. A symmetric ball rolling on a surface. Nonlinearity. 1995. V. 8(4). P. 493­515.

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THE ROLLING MOTION OF A BALL ON A SURFACE

Table 1. The rolling of a ball on a surface
surface typ e measure additional integrals Hamiltonianity cylindrical surface measure exists system is integrable by quadratures surface of the second order ellipsoid, hyp erb oloid, cone of the second order parab oloid = ( , B )
-1
-2

surface of revolution arbitrary surface ellipsoid, hyp erb oloid = (f ( 3 ))
3

parab oloid, cone, cylinder f ( 3 ) -

sphere

=

(B

-1

rc , B

-1

rc )
-1

1

2 - 3 3 f

( 3 )

( âM , B ( âM )) ((M â B-1 rc ), B-1 (M â B = const = const ( , B )
(one integral) (one integral) nothing is known ab out the Hamiltonianity of these systems A.V.Borisov, I.S.Mamaev, A.A.Kilin (2001)

authors

generalizations and remarks

integrable addition of the gravity field along a cylinder generatrix is p ossible

two linear rc )) = integrals, defined there exist two linear integrals that can b e expressed in by the system of terms of elementary functions linear equations up on change of time (prescrib ed by the reducing multiplier) the reduced system b ecomes Hamiltonian A.V.Borisov, I.S.Mamaev, E.Routh (1884) E.Routh (1884) A.A.Kilin (2001) the problems of the A.V.Borisov, I.S.Mamaev, and A.A.Kilin have shown the rolling of a ball on integrability in terms of elementary functions of the case of an unconstrained the ball rolling on the ellipsoid of revolution; have found an and rotating sphere invariant measure for an arbitrary surface of revolution; for are also solved. a parab oloid, a cone, and a cylinder they have shown the System also allows integrability of the case, when the ball is rolling on a integrable additions surface rotating around its axis of symmetry of p otentials.

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Remark. The cases when the tensor invariants exist are indicated by gray color in the table. The partial filling corresp onds to the uncomplete set of integrals.

219