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A. V. BORISOV, I. S. MAMAEV, A. A. KILIN
Institute of Computer Science Universitetskaya, 1 426034, Izhevsk, Russia E-mail: b orisov@rcd.ru E-mail: mamaev@rcd.ru E-mail: aka@rcd.ru

DYNAMICS OF ROLLING DISK
Received August 22, 2002

DOI: 10.1070/RD2003v008n02ABEH000237

In the paper we present the qualitative analysis of rolling motion without slipping of a homogeneous round disk on a horisontal plane. The problem was studied by S. A. Chaplygin, P. Appel and D. Korteweg who showed its integrability. The behavior of the point of contact on a plane is investigated and conditions under which its tra jectory is finit are obtained. The bifurcation diagrams are constructed.

1. Intro duction
For the first time the motion of a heavy dynamically symmetrical round disk on a horizontal absolutely rough plane was investigated by G Slesser (1861) [28], N. Ferrers (1872) [9], K. Neumann (1886), and A. Firkandt (1892). These studies eventually (after unsuccessful attempts by Neumann and LindelЁ ) of lead to the correct form of equations of motion. This form differs from the usual (Lagrangian or Hamiltonian) equations of mechanics b ecause of the nonholonomic constrain showing that the velo city of the p oint of contact of a disk with a plane is zero. We shall not discuss in detail the general forms of the equations of the nonholonomic mechanics (they are presented, for example, in [23], [26]). Instead, we concentrate on the pretty obvious form of these equations obtained from the general principle of dynamics -- the conservation law of the moment of momentum written in the disk-fixed axes. S. A. Chaplygin (1897) was the first to show the integrability of the problem on rolling motion of a disk. He presented the reduction of the problem to the analysis of hyp ergeometric quadratures in pap er [6], where he showed also the integrability of the problem on rolling motion of an arbitrary heavy dynamically symmetric b o dy of rotation on a horizontal plane -- in the latter case the problem is reduced to the integration of the linear differential second-order equation. The integration of equations of motion of a disk in hyp erelliptic functions was also p erformed in 1900 indep endently from each other and from Chaplygin by P. App el [2] and D. Korteweg [14]. Sometimes the problem on rolling motion of a disk is referred to as App elKorteweg problem (or simply App el problem), but this is, probably, not quite correct. In 1903 the same result has b een rediscovered by E. Gellop [10], however he used the Legendre functions. Despite of the explicit hyp ergeometric quadratures the various qualitative prop erties of disk motion were not studied for the long time. There were mainly studies of stationary motions and of their stability (the corresp onding bibliography is presented in b o ok [23]). Some qualitative prop erties of the disk motion have b een discussed only in pap ers S. N. Kolesnikov [13] and Yu. N. Fedorov [8]. The first pap er shows that for almost all initial conditions the disk never falls onto a plane and the second one present the pro cedure of investigation of the reduced system. Analogous results for the dynamically asymmetrical disk and disk moving on an inclined plane (nonintegrable problems) were obtained in [1], [18]. Among the mo dern works analyzing the rolling motion of the disk we shall note pap ers O. M. O'Reily [27], R. Cushman, J. Hermans, D. Kemppainen [7], and
Mathematics Sub ject Classification 37J60, 37J35, 70G45

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A. S. Kuleshov [21] devoted to the study of bifurcations and stability of stationary motions of the disk. General results of a qualitative analysis for the rolling motion of a heavy b o dy of rotation were obtained in pap er N. K. Moshuk [24]. The pap er include the frequency analysis, application of the KAM-theory, and basic qualitative prop erties for the motion of the p oint of contact. It app ears that the p oint of contact p erforms the comp osite b ounded motion: it p erio dically traces some closed curve which rotates as a rigid b o dy with some constant angular velo city ab out the fixed p oint. Thus the realization of some resonance relation b etween frequencies makes p ossible the drift of the b o dy of rotation to the infinity. In this pap er we develop these qualitative considerations and complement them with the computer analysis. We also present various typ es of tra jectories which are traced by the p oint of contact in the b o dy-fixed and relative frames of references since they have curious forms which are difficult to predict. Using the computer mo delling we explicitly investigate the hyp othesis ab out the drift to the infinity under the resonance conditions. We present the most general three-dimensional bifurcation diagram in the space of the first integrals and the complete atlas of its sections by various planes, constructed with the help of computer mo delling. In this pap er we also present a new metho d of reduction of the problem to an one-degree integrable Hamiltonian system and explicitly consider the existence of Hamiltonian formulation for different variants of equations of motion of the problem.

2. The rolling motion of a rigid b o dy on a plane
2.1. Equations of motion and their integrals
Let the rigid b o dy in an exterior field of force p erform a rolling motion on a plane without sliding. In this case the equations of motion have the most convenient form in the b o dy-fixed frame of references which axes are directed along the principal axes of inertia of the b o dy and the origin is situated at the center of mass. In the following text all vectors are assumed to b e pro jected on these axes. The condition of absence of slipping thus b ecomes v + в r = 0, (2.1)

where v , are the velo city of the center of mass and the angular velo city of the b o dy and r is the vector directed from the center of mass to the p oint of contact (see fig. 1). Let's denote the pro jections of the fixed basis vectors to the moving axes by , , (the vector is p erp endicular to the plane) and by (x, y ) we shall denote the co ordinates of the pro jection of the center of mass onto the plane in the fixed frame of references. We Fig. 1 assume that the field of force is p otential with a p otential dep ending only on the orientation of the b o dy U = U (, , ). The complete set of the equations of motion defining the given system can b e represented in the form M = M в + mr в ( в r ) + в U + в U + в U , = в , = в , = в . (2.2) (2.3)

The equation (2.2) describ es the evolution of the vector of moment of momentum for the b o dy with resp ect to the p oint of contact M and (2.3) concerns the evolution of the fixed basis vectors in the b o dy-fixed frame of references.

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The motion of the center of mass can b e obtained in quadratures from solutions of the equations (2.2), (2.3) as follows (2.4) x = (r в , ), y = (r в , ). The expression of the vector of moment of momentum with resp ect to the p oint of contact M can b e written in the following form M = I + mr в ( в r ), (2.5)

where I = diag(I1 , I2 , I3 ) is the tensor of inertia of the b o dy. In turn r can b e uniquely expressed (for a convex b o dy) through the normal to the plane from the equation =- F (r ) , | F (r )| (2.6)

Here F (r ) = 0 is the equation of the b o dy's surface. Let's consider a motion of the p oint of contact on a plane. If we denote the p osition of the p oint of contact on the plane in the fixed frame of references as (X, Y ), then the equation of motion for the p oint of contact can b e presented in the form X = (r , ), Y = (r , ) . (2.7)

where r is determined from equations (2.2) (2.6). Actually X and Y are pro jections of the velo city of the p oint of contact in the relative frame of reference onto the fixed axes. The equations of motion in the form similar to (2.2) (2.3) are presented, for example, in b o ok [11]. They can b e obtained also by means of Poincarґ Chetaev formalism [3] with undetermined Lagrangian e co efficients; these co efficients shall b e eliminated with the help of the constrains' equations (2.1). The system (2.2) (2.3) generally has seven indep endent integrals of motion, six of them are trivial geometrical integrals: 2 = 1, (, ) = 0, The seventh is the integral of energy 1 (M , ) + U (, , ) = h = const. 2 (2.9) 2 = 1, ( , ) = 0, 2 = 1, ( , ) = 0. (2.8)

Generally the given system has no other additional integrals and the p ossibility of its integrability in concrete cases dep ends on the presence of additional tensor invariants (measure, fields of symmetry, integrals).

2.2. The rolling motion of a heavy disk
Let's consider the case of rolling motion for an axially symmetric disk of radius R in the field of gravity. The field is, obviously, also axially symmetric with the p otential dep ending only on . Moreover, we supp ose that the disk is dynamically symmetric, i. e. I 1 = I2 . The p otential energy in this case has the following form 2 U = -mg (r , ) = mg R 1 - 3 . (2.10)
2 2 The equation of surface for the disk is F (r ) = r 1 + r2 - R2 . Substituting it in the equation (2.6) and solving with resp ect to r we obtain

r1 = -

R

1 2 3

1-

,

r2 = -

R

2 2 3

1-

,

r3 = 0.

(2.11)

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As the p otential energy dep ends only on , in the equations of motion (2.2) (2.3) we get the separate system of six equations. M = M в + mr в ( в r ) + mg r в , = в . (2.12)

Expressing , r from relations (2.5), (2.11) we shall get the closed system for the variables M , similar in many asp ects is to the EulerPoisson system in the Lagrange case, however the obtained system is much more complicated than the last one. The equations (2.12) preserve the geometrical integral 2 and the energy (2.9), in addition they allows the standard invariant measure (with a constant density). For the integrability (by EulerJacobi [17]) of these equations we need two additional integrals. In the following we describ e the metho d of derivation of these integrals. The p ossibility of separation of the system (2.12) from the general system (2.2) (2.3) is connected to the symmetry with resp ect to the rotations ab out the vertical axis defined by the vector . The system (2.12) is invariant with resp ect to the field of symmetries commuting with the vector field of the problem. v = 1 - 1 + 2 - 2 + 3 - 3 , (2.13) 1 1 2 2 3 3 It is p ossible to show that the variables M , are the integrals of field (2.13) that is v (Mi ) = 0, v (i ) = 0, i = 1, 2, 3. According to the general Lie theory [19], variables M , define the reduced system. For the classical EulerPoisson equations the corresp onding reduction is the Raus reduction with resp ect to the cyclical angle of precession. In addition to the field of symmetries (2.13) the equations of motion (2.2) (2.3) for the axially symmetric b o dy allow one more field of symmetries corresp onding to the rotation ab out the axis of symmetry of the disk. v = M 1 - M 2 + 1 - 2 + M2 M1 2 1 (2.14) +1 - 2 + 1 - 2 . 2 1 2 1 It is p ossible to show that integrals of the field (2.14) are pro jections of the moment and normal to the plane of disk onto the fixed axes of co ordinates N = ((M , ), (M , ), (M , )), n = (3 , 3 , 3 ).

The equations of motion for these variables can b e presented in the following form N = mr в ( в r ) + mg r в n , n = в n, (2.15)
1

where symb ols , r denote the same vectors, but pro jected onto the fixed axes (that is = ( , ), . . . , r1 = (r , ), . . .). The explicit expression of the comp onents of the vector r is r= R 3
3 2 3

=

1-

,

R 3

3 2 3

1-

, -R

1-

2 3

.

(2.16)

The vector N is expressed through by the formula N = I1 + (I3 - I1 )( , n )n + mr в (r в r ).
Remark 1. Such reduction is also possible for an arbitrary body of rotation.

(2.17)

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2.3. A reduction to the integrable one-degree Hamiltonian system
Let's describ e the pro cess of reduction of order with resp ect to the b oth fields of symmetries (2.13) and (2.14). For that we shall cho ose the simultaneous integrals of these fields as variables of the reduced system. According to [5], the most convenient algebraic set of such variables is 3 , K2 = K1 = M1 1 + M2 2 = N3 - 3 (N , n ), I1 M3 = I3 + mR2 I1 (N , n ), I3 + mR2 (2.18)

K3 = 1 M2 - 2 M1 = N 1 n2 - N 2 n1 . The equations of motion in the new variables b ecome 3 = K1 = - K2 = - K3 , I1 + mR2 I (I1 + mR )
2 3

I1 (I3 + mR2 )

K3 K2 ,

K3 K1 mR2 , 2 ) I (I + mR2 ) 1 - 2 (I1 + mR 13 3 3 1-
2 3 2 2 K1 K2 + I1 I1 + mR 2

(2.19)

K3 = - +

+
2 1 - 3 .

I1 (I3 + mR2 ) I
2 1

K1 K2 + mg R

3

The equations (2.19) preserve the invariant measure with density =

1 . Dividing the second 2 1 - 3

and the third equations on the first and cho osing a new indep endent variable, the angle of nutation = arccos 3 , we shall get the system of linear equations dK1 = d I3 sin I1 (I3 + mR2 ) K2 , dK2 = d K1 mR2 . 2 ) sin I1 (I3 + mR (2.20)

The general solution of these equations can b e presented in the form [23] K1 = C -C
1

I3 sin2 2 I1 (I3 + mR ) I3 sin2 I1 (I3 + mR )
2 2

F (1 + , 1 + , 2,

1 - cos )- 2 (2.21)

2

2

F (1 + , 1 + , 2,

1 + cos ), 2

K2 = C1 F ( , , 1,

1 - cos 1 + cos ) + C2 F ( , , 1, ), 2 2
I3 mR2 = 0 and F ( , , n, z ) I1 (I3 + mR2 )

where and are the solutions of the quadratic equation x 2 - x + is the generalized hyp ergeometric function representable by series

F ( , , n, z ) =
k =0

( + k )( + k )(n) z k ( )( )(n + k ) k !

(2.22)

Thus, the relations (2.21) define (implicitly) the integrals of motion. In this case they are the "constants" C1 and C2 expressed through K1 , K2 , .

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The quadrature for the angle of nutation can b e obtained from the integral of energy written in the variables K1 , K2 , K3 , 2 = 2 sin2 (I1 + mR2 )P ( ), P ( ) = h - K2 - 1 2 - mg R sin . 2I1 sin2 2 I1 K
2 1

(2.23)

Here we assume that the variables K1 , K2 are expressed through the constants of integrals and angle according to the formulas (2.21). In this case the function P ( ) (dep ending on the constants of integrals) define the analog of gyroscopic function for the Lagrange top [3], [22].

Fig. 2. Phase portraits of the system (2.23) at various values C1 and C2 . Left: the case of existence of three periodic solutions (C1 = 0.05, C2 = 0.01). Right: the case of existence of one periodic solution (C1 = 0.08, C2 = -0.02).

Thus, the equation (2.23) at the fixed values C 1 and C2 define the one-degree Hamiltonian sys tem. The phase p ortraits of this system on the plane , are presented in fig. 2. All the variables 3 , K1 , K2 , K3 are p erio dic functions of time with the p erio d T and the corresp onding frequency .
Remark 2. According to [5], the system (2.19) is the Hamiltonian one with degenerated Poisson bracket which has two Casimir functions expressed through hypergeometric functions.

2.4. Quadratures for angles of prop er rotation and a precession
According to the general Lie theory [19], if the variables of reduced system (2.18) are the given functions of time, then all the variables of initial system (2.12) (and accordingly (2.15)) can b e obtained by one quadrature (if fields v (2.13) and v (2.14) are commuting). Indeed, using the equalities tg = 1 (and corresp ondingly tg = - n1 ) for angles and , we 2 2 obtain 3 K1 K2 K1 =- + , =- . (2.24) 2I 2 1 - 3 1 I1 (1 - 3 ) I1 (I3 + mR2 ) Thus, for each of the angles the dep endence on time is defined as an integral of a p erio dic function with the frequency , hence it can b e presented in the form (see, for example, [17], [24]) = t + (t), = t + (t), (2.25)
n

where (t), (t) are p erio dic function with frequency . Moreover, (2.24) and (2.25) imply also that al l the frequencies , , depend only on the constants of the first integrals.

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Fig. 3. The surface of regular precessions. Parameters of the system are I 1 = 0.25, I2 = 0.5, R = 1, m = 1, g = 1.

2.5. A motion of the p oint of contact
Following pap ers [20], [24] we present the equation for the velo city of the p oint of contact in the form Z=R 3 K1 - 2 1 - 3 I1 K
2 2

I1 (I3 + mR )

ei ,

(2.26)

where Z = X + iY and X, Y are the co ordinates of the p oint of contact in the fixed frame of references. Thus the co ordinates of the p oint of contact are determined by quadratures of quasip erio dic two-frequency (with the frequencies , ) functions of time.

3. The qualitative analysis and results
Let's p erform the qualitative analysis of the dynamics of the disk motion. We will make a classification of all p ossible motions dep ending on the constants of the first integrals. Some features of the considered case essentially complicate this work in comparison with the case of the Lagrange top for EulerPoisson equations. For uniformity we recommend to study such analysis for the Lagrange case in b o ok [3]. The complexity of analysis is caused by the facts that the integrals of motion can not b e expressed in elementary functions (only in sp ecial one) and the system has no natural Hamiltonian presentation. Moreover, in addition to the motion of ap exes of the b o dy (disk) we shall classify tra jectories of the p oint of contact obtained by additional quadratures of quasip erio dic functions.

3.1. The bifurcation analysis of the reduced system
Possible typ es of motion for the axis of symmetry of the b o dy are completely determined by the form of the gyroscopic function P ( ) and by the energy level. Critical values of the integrals of motion C1 , C2 , h are determined by the equations P ( ) = 0, dP ( ) = 0. d (3.1)

In three-dimensional space with co ordinates C 1 , C2 , h equations (3.1) define a three-dimensional surface, so-called surface of regular precessions [3] (see fig. 3). This name is connected to the fact that at the given values of integrals the coin p erforms motion with the fixed angle = const, which is analogous to the precession for Lagrange top [22]. The full atlas of sections of the surface of regular precessions (bifurcation diagrams) by planes C 1 + C2 = const and C1 - C2 = const is presented in

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Fig. 4. Sections of the surface of regular precessions represented in fig. 3 by planes C 1 + C2 = const.

Fig. 5. Sections of the surface of regular precessions represented in fig. 3 by planes C 1 - C2 = const.

figs. 4 and 5 accordingly. In fig. 6 and 7 for two different sections we show the forms of the gyroscopic function P ( ) corresp onding to various values of integrals C 1 , C2 , h. Using these figures (and the rule of signs) we can easily study the stability of the corresp onding solutions lo cated on branches of the bifurcation diagram (branches corresp onding to unstable solutions are represented on the diagram by a dotted line). In figs. 6 and 7 vertical straight lines represent cases when C1 = 0 or C2 = 0. In these cases the disk motion corresp onds to the falling and planes determined by these equalities define in space of integrals C 1 , C2 , h the two-dimensional manifold of fallings. Thus for almost all initial conditions the disk do not fall p erforming the rolling motion on a plane. Other remarkable motions corresp ond to the cases C 1 = C2 , the rolling motion of the disk, and C1 = -C2 , the rotation of the disk ab out its axis passing through the diameter. During the latter motion the declination of the disk with resp ect to the vertical remains constant.
Remark 3. The bifurcation diagram (fig. 3, 4, 5) is different from one presented in papers [21], [27] since instead of the value of energy we use the value of angle of declination corresponding to the precession 0 and this function has no physical sense for other motions (when this angle is not preserved). Only the points on the surface of regular precessions have the physical sense. At the same time each value of constants C 1 , C2 , h in space of integrals in fig. 3 corresponds to some motion whether this point is situated on the surface of regular precessions or not and this is important for the qualitative analysis. Remark 4. One of sections of the three-dimensional diagram by a plane h = const and the corresponding gyroscopic functions are presented in paper [7].

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Fig. 6. Various types of the gyroscopic function for the section of the surface of regular precessions by plane C1 + C2 = 0.08.

Fig. 7. Various types of the gyroscopic function for the section of the surface of regular precessions by plane C1 - C2 = 0.08.

3.2. The qualitative analysis of motion of ap exes
The b ehavior of angles of prop er rotation and precessions that together with determined the motion of ap exes is defined by relations (2.25). The imp ortant feature in this problem is the twofrequency b ehavior of each of these angles. That is not usual for integrable systems. For example, for the Kovalevskaya top the angle (t) is defined by three frequencies [3]. In this case such phenomena is connected to the existence of two metho ds of reduction with resp ect to the symmetries of system (2.12), (2.15). From the geometrical p oint of view whole space of variables M , , , is foliated on threedimensional tori defined as the joint level surfaces of the integrals C 1 , C2 , h and the geometrical integrals. The motion represents a winding of the three-dimensional torus with frequencies , , [24]. (For the reduced systems (2.12) and (2.15) the corresp onding tori are two-dimensional.) Since the frequencies dep end only on constants of the first integrals, all motion on the torus have the identical frequency that not evident for nonholonomic systems. Even for the integrable

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Fig. 8. Tra jectories of the point of contact of the disk in the absolute space at various values of the integral of energy. Parameters of the system correspond to figure a). The closed tra jectories in the frame of references rotating with the angular velocity (see explanations in the text) are presented in the right upper corner of each figure (except the case of infinite motions). In figures a) and b) we present various types of motion of the disk at the energy h = 0.86. Figures c) and d) correspond to the energy h = 0.92217 when one of the motions (2) becomes resonant ( = ) and the secular drift (fig. d) is observed. The increase of the energy in figures e) and f ) to h = 0.961 makes both types of motion bounded again. In figure g) the motion of the disk is presented at h = 1.1 after merging of two domains of possible motions corresponding to various types of motion. The infinite motion in figure h) corresponds to the resonance = 2 at the energy h = 1.18169. In figure i) the motion of the point of contact of the disk is presented after the further increase of the energy up to h = 1.4.

nonholonomic systems on two-dimensional tori there is a non-uniform rectilinear motion and, generally sp eaking, the intermixing is p ossible (see pap er [4]). Practical ly these arguments prove that the given system is Hamiltonian one in the analytical sense (though the Hamiltonian function can be a different from the energy (2.9) [24]. Moreover, taking into account only the analytical point of view we can say that near the nonsingular torus the system becomes the Hamiltonian one by the infinite number of methods [16].
Remark 5. N. K. Moschuk in [25] observed a related phenomena studying the nonholonomic Chaplygin system possessing some number of the linear with respect to velocities first integrals.

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Fig. 9. Dependencies of frequencies , and from the energy at I1 = 1 , I3 = 1 , R = 1, m = 1, 4 2 g = 1, and various values of integrals C1 , C2 . In figures a) and c) the areas marked by the rectangles are separately presented in the increased scale. In the field of energies where two different types of motions 1 1 2 2 are possible in the absolute space we denote the frequencies by , and , . The resonance energies are marked on the graphs by thick dots. The orders of the resonances are indicated near the dots. The values of integrals for the dependencies presented here are: a) C1 = 0.04, C2 = -0.02; b) C1 = 0.09, C2 = -0.07; c)C1 = 0.065, C2 = 0.055; d) C1 = 0.09, C2 = 0.03.

At the same time the existence of a natural (algebraic) Poisson structure with a Hamiltonian defined by the energy (2.9) remains an op en problem. A. V. Borisov and I. S. Mamaev show that the reduced system (2.19) is the Hamiltonian one with some algebraic nonlinear bracket (see [5]), however the p ossibility of its lifting on the systems (2.12) and (2.15) is still not investigated.

3.3. The analysis of motion of the p oint of contact.
For the analysis of motion of the p oint of contact we decomp ose the velo city (2.26) in the Fourier series with resp ect to time. Then from (2.25) we get Z= Integrating with resp ect to time we obtain Z (t) = Z0 + e
i t

vn e
n

i( +n )t

.

n

vn e i( + n )

in t

.

Thus, if at + n = 0 we use the frame of references rotating ab out the p oint Z 0 with the angular velo city , then the p oint of contact traces some closed curve (see [24], [20]). Various typ es of such closed curves and tra jectories corresp onding to them in the fixed space are presented in figure 8. At the resonance + n = 0 we observe the secular drift of the p oint of contact. Graphs of frequencies (h), (h), (h) at the fixed values of integrals C 1 , C2 are presented in fig. 9. They show that the relation + n = 0 can b e fulfilled b oth in the case of existence of one and of three regular precessions. And at the same energy some initial conditions lead to a secular drift while the others are not (see fig. 8). Since all frequencies dep end only on the values of the first integrals the

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relation + n = 0 define in three-dimensional space of integrals some two-dimensional manifold corresp onding to the infinite tra jectories of the disk. Thus, for almost al l initial conditions (except the indicated manifold) al l trajectories of the disk are bounded. We can consider this result to b e opp osite to the one obtained from research of dynamics of the p oint of contact for the Chaplygin ball on a horizontal plane (see [12]) where the ma jority of tra jectories, on the contrary, were unb ounded.

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