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I. S. MAMAEV
Institute of computer science 426034, Russia, Izhevsk Universitetskaya St., 1 E-mail: mamaev@rcd.ru

NEW CASES WHEN THE INVARIANT MEASURE AND FIRST INTEGRALS EXIST IN THE PROBLEM OF A BODY ROLLING ON A SURFACE
Received September 16, 2003

DOI: 10.1070/RD2003v008n03ABEH000249

Some new cases when the invariant measure and an additional first integral exist in the problem of a rigid body rolling on a sphere and on an ellipsoid are discussed in the paper. These cases generalize the results obtained previously by V. A. Yaroshchuk and A. V. Borisov, I. S. Mamaev, A. A. Kilin.

This article concerns two classical problems of nonholonomic mechanics. The first one was indicated by E. Routh [1] and involves rolling of a homogeneous ball on an arbitrary surface (in particular, on a three-axial ellipsoid). The second problem was considered by P. V. Voronets [2], who studied the equations of rolling of an arbitrary rigid body on a sphere and reduced to quadratures the problem for the cases with axial symmetry. Hereinafter, by rolling we will always mean the classical nonholonomic model of the interaction between a body and a surface, where a body rolls without slipping, i. e. the velocity of the contact point is equal to zero. In this case, the energy is conserved and so we have the simplest model to describe interaction between a rigid body and a surface with friction (dry friction, for example, or viscous friction). For the specified problems, the equations of motion can be written in the form of a system of six equations, similar to the Euler­Poisson equations. In the general case, however, these equations no longer have the invariant measure and the integral of areas. The problem of specifying the conditions of their separate or simultaneous existence was systematically studied in papers [3, 4, 5, 6]. In paper [6], for example, the most complete list of different cases when the measure and the integral exist was compiled, showing a sort of integrability hierarchy in the equations of nonholonomic mechanics. The extreme cases in this hierarchy are dynamics of rattlebacks (that, due to absence of integrals and invariant measure, allows appearance of strange attractors [7]) and completely integrable behavior (eg., the behavior of the Chaplygin ball with the three-axial ellipsoid of inertia). In the latter case, the system becomes Hamiltonian under a certain change of time and can be integrated according to the Liouville theorem [8].

1. Rolling of a dynamically symmetric ball on a surface
The equations of motion for such a system can be written [3, 5] as (m + me ) = m( , n )n + 1 n â F , R x = R â n ,
Mathematics Sub ject Classification 37J60, 70F25, 70G45

(1.1)

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I. S. MAMAEV

where m is the ball's mass, R is its radius, me is the effective mass of the ball, and I = me R2 is the ball's moment of inertia relative to any axis, passing through its center. We assume that the vectors , n , and F are pro jected on the axes of the fixed frame of reference (absolute space) and represent, respectively, the ball's angular velocity, the normal to the surface, and the external force. Radius-vector x joins a point in the fixed space with the ball's center, while the ball is moving across a fixed surface, defined by equation f (x ) = f (x1 , x2 , x3 ) = 0. We have the following geometric identities: n= ni = f , | f| ni = 1 f , | f | xi f f 2f xi xj xj x
k

2f 1 - 12 | f | xi xk | f|

(1.2) xk .

j

If the external forces are potential (i. e. F = - U ), the energy is the first integral of system (1.1) H = 1 (I + mR2 ) 2 - 1 mR2 ( , n )2 + U (x ) 2 2 (1.3)

and there exists an obvious geometric integral f (x ) = 0, or (n , n ) = 1. For the system to be solvable by quadratures (according to the Euler­Jacobi theorem) one needs two additional first integrals and the invariant measure. V. A. Yaroshchuk proved [3] that when F = - U , system (1.1) always has the invariant measure, and the measure's density is = | f |.
Remark 1. The existence of the invariant measure was also shown in paper [9], where the author uses the following interesting form of equations of motion (found by J. Hermans [6]): v = -(v , n )n + u = (n â n , v ), F - (F , n )n me un â n + , me + m me + m x = v,

(1.4)

where v is the velocity vector of the ball's center, and u = R( , n ). In [9], equations (1.4) were used to analyze the limit case as the ball's radius tends to zero.

Let us recall that the problem of the motion of a free point moving across an ellipsoid (Jacobi's problem) is integrable only due to the fact that it has the quadratic integral of motion that was found by Joachimstahl. Jacobi solved this problem with the method of separation of variables, introducing the remarkable elliptic coordinates [10]. It was shown in [5, 11] that if F = 0 and the ball's center moves across a three-axial ellipsoid, the first integral of motion exists that is analogous to Joachimstahl's integral: K= (n â , B (n ,
-1

(n â )) n)

B-1

,

(1.5)

where (x , B-1 x ) = 1 is the equation of the ellipsoid, while B = diag(b1 , b2 , b3 ). Here we generalize the integral (1.5) for the case of forces with the potential of the form: U = kx2 + 1 2 2 ci , x2 i k , ci = const. (1.6)

i

It is known that addition of similar forces to the classical Jacobi system does not affect its
2 integrability [12]. Note also that quadratic potential kx was added by Jacobi himself; besides, in this

case, the system can also be solved using the elliptic coordinates and separation of variables [10].
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NEW CASES WHEN THE INVARIANT MEASURE AND FIRST INTEGRALS EXIST

Generalization of the first integral (1.5) for the system (1.1) with potential (1.6) is K= (n â , B-1 (n â )) (n , B
-1

n)

- 4(x , B

-2

x) k -

1 (x , B-1 x )

i

ci . bi x2 i

(1.7)

In paper [8], the nonholonomic Chaplygin system is discussed, which describes the motion of a dynamically nonsymmetric ball on a plane in force fields. In this case, the system admits the invariant measure and an additional integral, and, after a suitable change of time, becomes Hamiltonian (though, it is not integrable in the general case). The question whether the system (1.1) with potential (1.6) can be similarly reduced to a Hamiltonian form is still open.
Remark 2. For the classical Jacobi system, there exists a limit as one of the ellipsoid's axes tends to zero -- the integrable Birkhoff billiard. The existence of a similar limit in the case of a nonholonomic system has not yet been shown. The nonholonomic elliptic billiard that we offer does not seem to be integrable, but it can admit an additional integral and the measure. A ball inside this billiard also moves along a straight line, however, it is not clear if the known law of reflection holds when the ball hits the wall of the billiard. This system still needs further studying.

2. Rolling of a rigid b o dy on a sphere
The hierarchy of [11, 13] lacks two cases, when the equations of rolling of a rigid body on a sphere admit the invariant measure. One of these cases was found by V. A. Yaroshchuk [3] (here we give this result in a slightly modified form), while the other case is new. Let us discuss the two cases in question, one after the other. Rolling of E = ij ) on pro jected on the vector that joins an arbitrary b o dy with the scalar inertia tensor (I = µE, µ = const, a sphere. The equations for this problem are written via vector variables , r , axes bound to the body. Here is the angular velocity vector and r is the radiusthe body's center of mass G with contact point Q (Fig. 1).

Fig. 1

Without external forces, the equations of motion have the form [13] kr + n = n â , (µ + mr 2 ) = m( , r )(r + â r ) - m(r , r ) ;
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(2.1)

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I. S. MAMAEV f holds, where f (r ) = 0 is the equation of the body's | f| surface, n is the normal to the surface, m is the body's mass, while the sign of curvature k = 1 R0

besides, obvious geometrical equality n = -

(R0 is the radius of a fixed sphere) is selected from Fig. 1. The first of equations (2.1) can be written as follows [3]: q (k + V)r = n â , V = -q
-1

(1 + n n ),

=

2f xi x

j

,

q = | f |.

(2.2)

Using equations (2.1) and (2.2), one can obtain the following relation for the density of the invariant measure: = (µ + mr 2 )3/2 q det(k + V). This density was also given in [3], though in somewhat less symmetric form. An ellipsoid with a sp ecial mass distribution on a sphere. The result given below generalizes the similar result from paper [4] that concerns rolling of a special ellipsoid on a plane. Indeed, the equations of rolling of the ellipsoid defined by equation (r , B-1 r ) = 1 on a sphere look as follows (see, for example, [13]): kr + n = n â , IQ = (IQ ) â - mr â â( â r ), (2.3)

where IQ = I + mr 2 E - mr r , and I is the ellipsoid's inertia tensor relative to its geometric center. In the case of a special mass distribution, such that the central inertia tensor can be presented in the form I = µ + mB, (2.4) the equations (2.3) have the invariant measure = (µ + mr 2 )-
1/2

det IQ q det(k + V).

(2.5)

In conclusion, it should be mentioned that almost all the results of this paper have required a large amount of computer-aided analytical calculations, and it is hard to imagine what it has cost for V. A. Yaroshchuk, who, as we know, has managed without a computer. The author is grateful to A. V. Borisov for useful discussions. This work was supported in part by Leading Scientific School of Russia Support grant no. ìü-136.2003.1.

References
[1] E. J. Routh. Dynamics of a System of Rigid Bodies. Dover Publications, N.-Y.. 1960. [2] P. V. Voronets. On the problem of motion of a rigid body, rolling without slipping on a given surface under the action of some given forces. Univers. Izvestiya, St. Vladimir University. 1909. P. 1­11. (In Russian) [3] V. A. Yaroshchuk. New cases when an integral invariant exists in the problem of a rigid body's rolling without slipping on a fixed surface. Vestn. MGU, Ser. Math. Mech. 1992. 6. P. 26­30. (In Russian) [4] V. A. Yaroshchuk. An integral invariant in the problem of rolling of an ellipsoid with a special mass distribution on a fixed surface without slipping. Izv. RAN, Mekh. tv. tela. 1995. 2. P. 54­57. (In Russian) [5] A. V. Borisov, I. S. Mamaev, A. A. Kilin. A new integral in the problem of a ball rolling on an arbi-

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[6] [7]

[8]

[9]

[10]

trary ellipsoid. Doklady RAN. 2002. V. 385. 3. P. 338­341. (In Russian) J. Hermans. Asymmetric sphere rolling on a surface. Nonlinearity. 1995. V. 8. 4. P. 493­515. A. V. Borisov, I. S. Mamaev. Strange attractors in the dynamics of rattlebacks. Uspekhi fiz. nauk. 2003. V. 173. 4. P. 407­418. (In Russian) A. V. Borisov, I. S. Mamaev. Hamiltonianicity of Chaplygin's Ball Rolling Problem. Matem. zametki. 2001. V. 70. 5. P. 793­795. (In Russian) M. V. Deryabin. On the invariant measure in the problem of a symmetric ball rolling on a surface. Prikl. mat. i mekh. 2003. V. 67. 3. P. 384­389. (In Russian) C. G. J. Jacobi. Vorlesungen uber Dynamik. Berlin, ¨ G. Reimer. 1884. P. 300.

[11] A. V. Borisov, I. S. Mamaev, A. A. Kilin. Rolling of a ball on a surface. New integrals and hierarchy of dynamics. Reg. & Chaot. Dyn. 2001. V. 7. 2. P. 201­220. [12] V. V. Kozlov. Some integrable generalizations of Jacobi's problem of geodesics on an ellipsoid. Prikl. mat. i mekh. 1995. V. 59. 1. P. 3­9. (In Russian) [13] A. V. Borisov, I. S. Mamaev. Rolling of a rigid body on plane and sphere. Hierarchy of dynamics. Reg. & Chaot. Dyn. 2002. V. 7. 2. P. 177­200. [14] V. V. Kozlov. On the theory of integration of equations of nonholonomic mechanics. Uspekhi mekhaniki. 1985. V. 8. 3. P. 85­107. (In Russian) [15] A. V. Borisov, I. S. Mamaev. Dynamics of rigid body. Moscow­Izhevsk: RCD. 2001. (In Russian)

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