Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ics.org.ru/upload/iblock/31e/120-the-dynamics-of-chaplygin-ball-the-qualitative-and-computer-analysis_ru.pdf Äàòà èçìåíåíèÿ: Wed Oct 28 19:30:52 2015 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 00:21:36 2016 Êîäèðîâêà: ISO8859-5 Ïîèñêîâûå ñëîâà: m 8

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

THE DYNAMICS OF CHAPLYGIN BALL: THE QUALITATIVE AND COMPUTER ANALYSIS
Received June 3, 2001

DOI: 10.1070/RD2001v006n03ABEH000178

The motion of Chaplygin ball with and without gyroscope in the absolute space is analyzed. In particular, the tra jectories of the point of contact are studied in detail. We discuss the motions in the absolute space, that correspond to the different types of motion in the moving frame of reference related to the body. The existence of the bounded tra jectories of the ball's motion is shown by means of numerical methods in the case when the problem is reduced to a certain Hamiltonian system.

1. Intro duction
The dynamics of nonholonomic systems formed an indep endent branch of the theoretical mechanics when it b ecame clear that the standard Lagrangian formalism is inapplicable to the systems with nonintegrable constraints. The development of the nonholonomic systems dynamics is asso ciated with the names of S. A. Chaplygin, H. Herz, P. App el, D. K. Bobylev, N. E. Zhukovski and others. A great numb er of studies in this field are connected with the extension of the develop ed analytical metho ds for holonomic system on the systems with nonholonomic constraints. One can find a sufficiently complete description of problems and metho ds of nonholonomic mechanics, for example, in [8]. At present the dynamics of nonholonomic system has many applications in the problems of mo dern technology, such as the motion of automobiles, landing devices of airplanes, railway wheel, etc. Besides, its metho ds are widely used in the theory of electrical machines. A great contribution to the development of the nonholonomic dynamics has b een made by the famous research by S. A. Chaplygin which concerns the problem of rolling of a heavy solid b o dy of rotation on the horizontal plane [10]. In this pap er he analyzed A. Lindelef 's error and obtained the correct equations of motion; he also carried out the complete investigation of the problem for a numb er of particular cases. In another work [11] S. A. Chaplygin gave the complete investigation of the problem of rolling of a dynamically nonsymmetrical ball on the plane under the only assumption ab out the coincidence of the center of mass and the geometric center of the ball. In this work S. A. Chaplygin presented the integrals of motion of the system, found the integrable multiplier, and obtained the solution of equations of the motion in quadratures. Despite the presented geometrical interpretation, the motion of Chaplygin ball in the absolute space was practically unstudied. Recently in [3] the connection b etween the systems with nonholonomic constraints and Hamiltonian systems with a nonlinear Poisson bracket was shown using the problem of Chaplygin ball as an example. The extension of the Chaplygin ball problem on the cases with the gyroscop e is of great interest. For the first time (b efore Chaplygin research) the particular case of the given problem when the ball is dynamically symmetric, was considered by D. K. Bobylev in [1]. D. K. Bobylev shown that despite the simplicity of the b o dy geometry the tra jectories of the ball's motion may have an interesting nontrivial form. N. E. Zhukowski considered an even more particular case of the given problem when its analysis can b e considerably simplified in [4]. Later in [7] A. P. Markeev shown the integrability of
Mathematics Sub ject Classification 70E18



REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001

291


A. A. KILIN

the problem in the general formulation when the ball is dynamically nonsymmetrical, and the moment of the gyroscop e has an arbitrary direction. However, despite the proven integrability, nob o dy has succeeded yet to integrate the given system in quadratures (or to obtain the solution in any class of sp ecial functions). The top ological analysis of the problem also hasn't b een made yet. Thus far, there is nothing known ab out the motion in the absolute space, b esides the particular cases. This article presents the analysis of Chaplygin ball's motion with and without gyroscop e in the absolute space. In particular, the sp ecial emphasis is made on studying of the tra jectories of the p oint of contact that can b e repro duced in the natural exp eriments making the ball rolling on the surface p owdered with, for example, likop o dium. We study the motions in the absolute space corresp onding to the different typ es of motion in the frame of reference related to the b o dy. The existence of b ounded tra jectories of the ball's motion in the case when the problem is reduced to a certain Hamiltonian system is shown with the help of numerical metho ds.

2. The motion equations and their integration
2.1. The motion equations and the integrals
Let us consider the problem of rolling of a balanced dynamically nonsymmetrical ball on the horizontal rough surface [11]. The motion of the ball in pro jections on the principal axes related to the ball is describ ed by the system M =M ç =ç (2.1) M = I + D ç ( ç ), D = ma2 , where is a vector of the angular velo city, is a vertical unit vector, I is a tensor of inertia of the ball relatively to its center, m is the ball's mass, a is its radius. The vector M is the kinetic moment of the ball relatively to the p oint of contact. As S. A. Chaplygin has proved in [11] the equations (2.1) have the integrating multiplier Å=
1 1 - D( , J
-1

)

,

(2.2)

J = I + D E, and four indep endent integrals h = 1 (M , ), 2 C = (M , ),

E=

ij

( , ) = 1,

n = (M , M ),

(2.3)

thus, according to the Euler - Jacobi theorem it is p ossible to integrate the system. S. A. Chaplygin also integrated in this study the system (2.1) in terms of hyp erelliptic functions. In order to determine the ball's motion in the absolute space, one should add the equations for the unit vectors of the fixed system to (2.1) = ç , = ç . (2.4)

The tra jectory of the p oint of contact on the surface is esp ecially imp ortant for understanding the ball's motion in the absolute space. This tra jectory coincides obviously with the tra jectory of the center of mass. The equation of motion of the p oint of contact can b e obtained using the condition that its velo city is equal to zero: v = r ç ,


(2.5)
3, 2001

292

REGULAR AND CHAOTIC DYNAMICS, V. 6,


THE DYNAMICS OF CHAPLYGIN BALL: THE QUALITATIVE AND COMPUTER ANALYSIS

where v is the velo city of the center of mass, r = with the p oint of contact. Let us write the equality co ordinates x = (v , ) = y = (v , ) =

-a is a vector, connecting the center of mass (2.5) in terms of pro jections on the fixed axes of a( , ), -a( , ). (2.6)

The obtained equations (2.4) and (2.6) determine the tra jectory of the p oint of contact (and of the center of mass) on the surface.

2.2. Integration of the motion equations in case of zero value of area integral
First let us consider the reduction of the problem to quadratures in the case of zero value of area integral C = (M , ) = 0. We p erform the change of variables in (2.1) discussed in [3] by A. V. Borisov and I. C. Mamaev d t Åd , 1 M ÅM , (2.7)

Then the motion equations can b e represented in the Hamiltonian form with the Hamiltonian H = 1 (1 - D ( , J 2 and with the nonlinear bracket {Mi , Mj } = The motion integrals take the form 1 C = Å (M , ), n = 12 M 2 , Å 2 = 1. (2.10)
ij k -1

))(M , J

-1

M ) + 1 D ( , J 2

-1

M )2 ,

(2.8)

{Mi , j } =

(Mk - D (M , )Å2 J
ij k k

-1 k ,k k

),

,

{i , j } = 0.

(2.9)

As it follows from (2.9) the Poisson bracket at zero constant of the area integral (M , ) = 0 corresp onds to the algebra e(3), and the corresp onding symplectic leaf is the cotangent bundle to the sphere. As it was mentioned ab ove, the explicit reduction of (2.1) to the quadratures (Ab elian - Jacobi equations) was made by S. A. Chaplygin who, however, by no means asso ciated it with the Hamiltonian - Jacobi metho d b ecause the Hamiltonian structure (2.9) was unknown to him. Let us show that integration of the system is p ossible with the help of the usual separation of variables, giving the geometrical meaning to those virtuoso but non-evident Chaplygin's transformations. Let us pro ceed to sphere-conical co ordinates q 1 , q2 which are defined as the ro ots of the quadratic equation 2 2 2 1 2 3 f (q ) = + + = 0, (2.11) a1 - q a2 - q a3 - q where ai = 1 , and a3 < q2 < a2 < q1 < a1 . The expressions for in terms of the new co ordinates
I
i

have the form
2 1 =

(a1 - q1 )(a1 - q2 ) , (a1 - a2 )(a1 - a3 )
2 3

2 2 =

(a3 - q1 )(a3 - q2 ) = . (a3 - a1 )(a3 - a2 )

(a2 - q1 )(a2 - q2 ) , (a2 - a3 )(a2 - a1 )

(2.12)



REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001

293


A. A. KILIN

The expression for the moment M using the new co ordinates q 1 , q2 and their conjugate impulses p1 , p2 have the form q1 q2 M = p ç = p1 ç + p2 ç , (2.13) where
q1 q2 , and are expressed via q1 and q2 by formulas (2.12). Finally, we obtain

M1 = F (q1 , q2 ) a2 - a M2 = F (q1 , q2 ) a1 - a M3 = F (q1 , q2 ) a1 - a F (q1 , q2 ) = 2 q1 - q

3

p

1

a1 - q 1 -p a1 - q 2 q1 - a 2 +p a2 - q 2
1

2

a1 - q a1 - q

2 1

,

3

p

1

2

a2 - q 2 q1 - a 2
2

, (2.14)
3 3

2

-p

q1 - a 3 +p q2 - a 3

q2 - a q1 - a

,

2

(a1 - q1 )(a1 - q2 )(q1 - a2 )(a2 - q2 )(q1 - a3 )(q2 - a3 ) . (a1 - a2 )(a2 - a3 )(a1 - a3 )

Substituting (2.12) and (2.13) in (2.8)-(2.10) we express the Hamiltonian as a function of the canonical variables -1 1 1 H=1 q -q (-p2 f (q1 ) + p2 f (q2 )), 1 2 2 1 2 (2.15) 1 (q - a )(q - a )(q - a )(1 + D q ), f (q ) = q 1 2 3 the additional integral can b e represented in the form n= - p2 f (q1 )q1 (1 + D q2 ) + p2 f (q2 )q2 (1 + D q1 ) 1 2 q1 - q 2 (2.16)

From (2.15) it is clear that in new variables the equations are separated. The solution of the Hamiltonian - Jacobi equation has the form S (q1 , q2 , 1 , 2 ) = Á 21 + q1 R(q1 )
2

d q1 Á

21 + q2 R(q2 )

2

d q2 ,

(2.17)

R(q ) = (21 + q 2 )(q - a1 )(q - a2 )(q - a3 )(1 + D q ), where 1 , 2 are separation constants related to the integral values (2.10) in the following way 1 = H , 2 = 2D H - n. (2.18)

Using S as a generating function one can pro ceed to new constants (named by Jacobi osculatory) , p = S, q = S. (2.19)

In these variables the motion equations have the form i = - H = 0, i and their solution is i = const,
294
0 i = 1i + i ,

i = H = 1i , i

i = 1, 2

(2.20)

i = 1, 2
3, 2001


(2.21)

REGULAR AND CHAOTIC DYNAMICS, V. 6,


THE DYNAMICS OF CHAPLYGIN BALL: THE QUALITATIVE AND COMPUTER ANALYSIS
0 where i are constants determining the initial p osition on the tra jectory. Thus, the solution (2.21) and the system (2.19) define the tra jectory (the first equation of the system) and the motion along it (the second equation) in the phase space (p , q ). The corresp onding Ab elian - Jacobi equations that determine the evolution of q1 and q2 have the form

dq

1

R(q1 ) q1 d q
1

Á Á

dq

2

R(q2 ) q2 d q
2

= d , (2.22) = 0.

R(q1 )

R(q2 )

The explicit dep endence of p ( ), q ( ) can b e obtained in terms of theta-functions. The obtained motion equations (2.22), (2.19) and (2.21) coincide with S. A. Chaplygin results [11] with the accuracy of the notation of the variables. The presence of the separating variables and the Poisson structure allows one to intro duce actionangle variables (for the system with new time (2.7)) in the usual way. Using the first equation of the system (2.19) we obtain for the action variables
a
1

a

2

I1 =

p1 d q 1 =
a
2

21 + q R(q )

2

d q,

I2 =

p2 d q 2 =
a
3

21 + q R(q )

2

d q.

(2.23)

One can get the expressions for the angle variables using the implicit dep endence (I ) i = Expressing the partial derivatives 1 = 1 1 F 2 2 = 1 1 F 2
a
2

S (q , (I )) . Ii

(2.24)

i from (2.23) we obtain finally the angle variables Ij
a
2



a

2

qd q
a

dq

1

R(q )
3

R(q1 ) q1 d q
1

+

dq

2

R(q2 ) q2 d q
a 2

-1 2 -1 2
a
1

dq
a

q1 d q

1

R(q )
3

R(q1 ) dq
1

+

q2 d q

2

R(q2 ) dq
2



a

1

a

1

dq
a

R(q )
2

R(q1 )
1

+
2

qd q
a

, ,

R(q2 ) dq 1 2 R(q )

R(q )
2

R(q1 )

+

R(q2 )

a

F= 1 2
a
3

qd q

1 2 R(q )

dq
a
2

-1 R(q ) 2

qd q
a

a

R(q )
2

. (2.25)

3

frequencies are determined by the relations
a
2

d i The motion equations in action-angle variables have the form I i = const, = i (I ) where the d
a
1

1 (I ) = 1 2 F
a
3

qd q R(q )

,

2 (I ) = - 1 2 F

qd q
a

R(q )
2

.

(2.26)

The rotation numb er for the tori that equals to the frequency ratio is expressed by the formula
a
2

qd q R(q ) qd q R(q )

dq . dq (2.27)

n(I1 , I2 ) = 1 = 2

a

3 1

a a

2



REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001

295


A. A. KILIN

V. V. Kozlov in [6] intro duced simpler angle variables on the tori for the system with the initial time t (2.7).
q1 d q
1

q2 d q

2

x=
1

R(q1 )
a a
1

qd q R(q )

,

y=
1

R(q2 )
a a
2

qd q R(q )

.

(2.28)

2

3

These variables are the linear combination of the (2.25) variables with the co efficients that dep end on the integral constants. Though the (2.28) variables are not the action-angle ones, they are of interest for studying the dynamics on the invariant tori.

2.3. Integration of the motion equations in case of nonzero value of area integral
In case of nonzero value of area integral C = (M , ) = 0 the equations of motion can b e integrated using the reduction to the case C = 0. For this purp ose let us intro duce new variables Mi = Å i + ÅMi , i = i + M
i

(2.29)

and select the constants , , Å, Å so that with an account of the motion integrals the following relations hold (M , M ) = n1 , ( , ) = 1, (M , ) = 0. Inserting (2.29) into the motion integrals, we obtain Å + Å2 n1 = n, + 2 n1 = 1, Å + Ån1 = C. Let us require the integrals of energy h 1 and of moment quantity n1 in new variables to have the form of the integrals on the level of the zero value of area integral: (J , ) - D ( , )2 = 2h,
2 2 2

(2.30)

(J , J ) - D ( , )2 = n,

(2.31)

where J = J = I + D E is a matrix of the moment of inertia relatively to the p oint of contact; it is invariant under the substitution (2.29). Using (2.31) we obtain the equations: ( Å - Å )2 h1 = h + Å2 , D Å2 ( Å - Å )2 1 = h2 + D D ÅÅ h + =0 D h ( Å - Å )2 1 = h , D D
2



296

REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001


THE DYNAMICS OF CHAPLYGIN BALL: THE QUALITATIVE AND COMPUTER ANALYSIS

The constants of relations of M , with M , are determined by the equations f2 - n - hD f - hD = 0 C Å = f
2 2

[n - C 2 + (C - f )2 ] = (C - f ) = C -f n - Cf Å

(2.32)

Å - Å = (C - f ), and the following equalities hold b etween the integral constants 2 h1 =
2

n - C2 n1 = . n - C 2 + (C - f )2

f [n - C 2 + (C - f )2 ]

(f - C )h

(2.33)

Remark 1. Let us note that the transformation (2.29) is determined within the accuracy of the expansion M M .

Since with variables M , , the integration (at the zero value of area integral) can b e made in theta-functions, it is also p ossible to p erform integration for C = 0. The case when the vector M is collinear to the vector should b e considered separately, since in this case the change of variables (2.29) turns out to b e inapplicable. In this case M = C , and the integrals have the form 2 2 2 1 + 2 + 3 = 1, C = (M , ), n = (M , M ) = C 2 , (2.34) h = 1 (M , ) = C ( , ), 2 2 The motion equation at M = C coincides with the Euler equation. Indeed, on the basis of (2.1) and (2.34) we obtain the expression for the angular velo city J = (C + 2Dh ) and the motion C equations of the form J = J ç . Thus, the solution is expressed in terms of elliptic functions in this case. S. A. Chaplygin gave in [11] an interesting geometric interpretation of this motion. Let us consider the ellipsoid x2 x2 x2 1 2 3 + + = K - a, (2.35) -a -a -a
I
1

I

2

I

3

where

and the ellipsoid defined by the formula rolls on another one. In this case the angular velo city is prop ortional to the length of the segment of the rotation axis which lies b etween the p oints of contact of the ball and the ellipsoid with the planes. However, despite such an interpretation, the qualitative nature of the motion in the absolute space and, in particular, the trace of the p oint of contact, remains undetermined.


C a . Then the ball rolls without sliding on one plane, and cho ose larger than the values I i a and K Cl

K = 2h ,

x = - a ,
l

l = C + 2Dh ,

REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001

297


A. A. KILIN

3. The bifurcational diagram, the p erio dic solution and the p oint of contact
3.1. The case of (M , ) = 0
Let us consider the case of zero value of area integral (M , ) = 0. The bifurcational curves at C = 0 are found using the condition of the multiplicity of the p olynomial ro ots R(q ) (2.17). Eventually we obtain three straight lines on the plane (h, n) pi : h= n, Ii + D i = 1, 2, 3. (3.1)

The region of p ossible motions is situated b etween the lines p 1 and p3 (Fig. 1). The bifurcational diagram in this case coincides with the bifurcational diagram of Euler - Poisson problem, the only difference is that one takes the moments of inertia I i + D relatively to the p oint of contact instead of the moments of inertia I i relatively to the center of mass.

Fig. 1. The bifurcational diagram at I1 = 1, I2 = 1.5, I3 = 3, D = 1, C = 0.

3.2. The case of (M , ) = 0
First let us note that the inequality n C 2 is always fulfilled for the real motions, so the b ounding line n = C 2 is always presented on the bifurcational diagram. Since, using substitution (2.29), this case can b e reduced to the case of zero value of area integral, the bifurcational lines have the form (3.1) for the transformed variables, i. e. n1 h1 = . I1 + D To obtain the bifurcational curves for the initial variables it is sufficient to p erform the inverse transformation of the co ordinates Mi , i Mi , i . The substitution (2.29) changes b oth the moments of inertia relatively to the center of mass I i and the parameter of nonholonomicity D , however, the moments of inertia relatively to the p oint of contact remain unchanged J i = Ii + D = Ii + D = Ji . Using (2.33) we find (f - C )h h1 1 i = 1, 2, 3. n1 = f (n - C ) = Ji ,


298

REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001


THE DYNAMICS OF CHAPLYGIN BALL: THE QUALITATIVE AND COMPUTER ANALYSIS

Then using the expression for f from (2.32) we finally obtain the bifurcational curves pi : hi = 1 Ji n + DC Ii
2

.

The region of p ossible motion is again situated b etween p 1 and p3 , and it is also b ounded by the line n = C 2 on the left.

Fig. 2. a) The bifurcational the level of energy h = 1; b) bifurcational diagram (O is trace of the point of contact

diagram at I1 = 1, I2 = 1.5, I3 = 3, D = 1, C = 1 and the PoincarÄ mapping on e The ball's motion in the absolute space which corresponds to the branch p 1 on the the point of contact, C is the trace of the point of contact on the plane, C is the on the ball).

Fig. 2 (a) shows the bifurcational diagram for C = 0 and the p erio dic solutions on the PoincarÄ e section using the Andoyer - Deprit variables, corresp onding to the p oints on the branches of the bifurcational diagram (see, for instance, [2]). These p erio dic solutions represent the rotations under which is parallel to one of the principal axes. Thus, the following p erio dic solutions corresp ond to the p oints on the branch p1 : = (1 , 0, 0), = (1 ,
2 1 - 1 cos(1 t), 2 1 - 1 sin(1 t)),

(3.2)

where 1 and 1 are expressed as functions of the integrals
2 1 = 2 2I1 h + D C 2

J1 I

2 1

, (3.3)

J C2 = 21 . 2I1 h + D C 2
2 1

Such motions represent the ball rotation ab out one of the principal axes in the absolute space. In this case the axis keeps the constant p osition in the space, and the trace of the p oint of contact on the surface is represented by the straight line (Fig. 2 (b)).


REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001

299


A. A. KILIN

The bifurcational diagram shows that to the stable rotation ab out the minor axis and the ma jor one corresp onds, resp ectively, to branches p 1 and p3 . The unstable rotations ab out the mean axis corresp onds to branch p2 . Besides, the separatrix motions that connect two p erio dical solutions with the opp osite directions of also corresp ond to this branch. Let us return to the case when C = (M , ) = 0. Cho ose unit vector of the axis x , which is parallel to the vector of the kinetic moment M . In this case, the second equation in the system (2.4) takes the form 2 M y = -( , ) = - h = const. n n

Fig. 3. The bifurcational diagram at I1 = 1, I2 = 1.5, I3 = 3, D = 1, C = 1 and the tra jectories of the points of contact which correspond to the different points of the diagram.

Thus, the center of mass of the ball and the p oint of contact move uniformly along the direction which is p erp endicular to the vector of kinetic moment. Under this condition, in the general case, the ball's motion in the parallel direction is represented as a certain combination of translational motion and p erio dic oscillatory motion. S. A. Chaplygin first describ ed the ab ove feature of the ball's motion in [11]. If C = (M , ) = 0, using the substitution (2.29) one can show that the ball's motion in b oth directions is represented as a combination of translational motion and p erio dic oscillatory motion. S. A. Chaplygin in his work [11] obtained the explicit expression for the p oint of contact. However, due to complexity of the expressions that represent the integrals of the rational expressions of thetafunctions, analytical study of the motion of the p oint of contact on the plane is imp ossible to carry out. Fig. 3 shows the tra jectory of the p oint of contact for different p oints in the bifurcational diagram, which were obtained numerically. An issue of sp ecial interest is the separatrix motion which is shown in Fig. 3 (d), in this case the vector of the angular velo city turns around and takes the p osition that is opp osite to the initial one. Due to complexity of the exact hitting at the separatrix and computational errors in the numerical integration one can see a numb er of such "tumbles" in the figure. As it is seen from the figures and as it follows from the ab ove discussion, the unb ounded character of the motion of the p oint of contact is the usual situation for the ab ove-mentioned cases.


300

REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001


THE DYNAMICS OF CHAPLYGIN BALL: THE QUALITATIVE AND COMPUTER ANALYSIS

3.3. The case of M



Let us consider in detail the case of M , when the substitution (2.29) is invalid and the tra jectories of the p oint of contact are of the greatest interest.

Fig. 4. The tra jectory of the point of contact on the plane at M and h = h : a) I1 = 1, I2 = 1.5, 2 I3 = 2, C = 1, = 0.002858; b) I1 = 1, I2 = 3, I3 = 5, C = 1, = 0.411829.

The vertical rectilinear segment n = C 2 corresp onds to this case in the bifurcational diagram, this segment is b ounded by the branches p 1 and p2 of the diagram (Fig. 2). The p oints of intersection of the line n = C 2 with the branches p1 , p2 and p3 (h = h , h , h ) corresp ond to the rotations ab out 1 2 3 the principal axes, in this cases the p oint of contact do es not move. The exception is the p oint of intersection with the second branch, the separatrix motion also corresp onds to this p oint. Let us discuss this case in detail. Two p erio dical solutions corresp ond to the p oint (C 2 , h2 ) on the diagram, these solutions are the rotations ab out the mean axis in opp osite directions. These two solutions are connected by the family of the doubly p erio dical tra jectories (see, for example [5]); during the motion along these tra jectories the tumble of the ball takes place. Cho osing the solutions of equation (2.4) from this family so that they satisfy the orthogonality prop erty, we obtain a = a a = a a = a where aij = 1 - 1 , = C , = C J2 a a . Ji Jj I2 I2 32 21
32 31 32 31 32 31

1 , - th t, a a ch t a th t cos t + a a th t sin t - a

21 31

1 , ch t
21 31

21 31

a sin t, cos t , a ch t a cos t, sin t , a ch t

a th t cos t - a a th t sin t + a

32 31

sin t , cos t ,

(3.4)

21 31

21 31

32 31

Substituting (3.4) into (2.4) we obtain for the p oint of contact x = cos t ,
ch t

Bearing in mind that for t - the p oint of contact is situated in the origin of co ordinates, we


y = sin t .
ch t

(3.5)

REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001

301


A. A. KILIN

obtain

x(t) = y (t) =

t - t -

cos d , ch sin d . ch

(3.6)

The tra jectory of the contact p oint for such motion for different sets of moments of inertia is shown in Fig. 4. Letting t in (3.6) we will receive x() = cos d , - ch y () = 0. Thus, the tumbling of the rotation axis takes place together with the displacement of the center of mass of the ball by value


=
-

cos d . ch cos t


(3.7)

or, if = 0,


=
-

ch t

dt.

The magnitude of displacement dep ends only on the 1 combination = , i. e. the dynamic b ehavior of
J2 a21 a
32

the b o dy, and do esn't dep end on the integral level under examination. Fig. 5 shows the plot of ( ). As we will see later, the value of ( ) is very imp ortant for understanding the geometry of the p oint of contact near the p oint (h , C 2 ) on the diagram. 2
Remark 2. Under the motion of the point of contact along the sphere the angle of intersection of its tra jectory with meridians is constant along the whole tra jectory. This property can be easily obtained using the motion equations (2.1) and the reduced hyperbolic solution (3.4). Thus, the point of contact on the sphere is moving along the loxodrome, if h = h . 2

Fig. 5.

Let us consider again the whole line n = C 2 . As it was mentioned ab ove, the motion equations take the form of the Euler equations in this case. Under this condition the solutions of the motion equations for M and are expressed in elliptic functions. Thus, for the system that is rigidly asso ciated with the b o dy the p oint of contact moves along the closed tra jectories. The trace of the p oint of contact in the absolute space is no longer closed and is represented by the curves that are complicated enough. Fig. 6 shows these curves for different sets of the moments of inertia of the energy levels. As it can b e seen from the figure practically all the curves, except some critical ones, are b ounded. While moving along the line M 2 = C 2 from h = h to h = h the tra jectory of the p oint of contact undergo es a 1 3 numb er of bifurcations. Near the p oint h = h the tra jectories are represented by the quasi-p erio dical 1 curves, in which the radius of the envelop e curves tends to zero when approaching to a critical value h (Fig. 6 (a)). Near the h = h , dep ending on the parameters, the tra jectories b ehave in a similar way 3 or the radius of the envelop e curves go es up to infinity when approaching h = h . However, in this 3 case the difference b etween the radiuses of the envelop e curves and the velo city of the drift along the envelop e curves tends to zero (Fig. 6 b)). Thus, in the limit case the ball, as in the previous case,


302

REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001


THE DYNAMICS OF CHAPLYGIN BALL: THE QUALITATIVE AND COMPUTER ANALYSIS

Fig. 6. The tra jectory of the point of contact on the plane at M and C = 1 for different values of the moment of inertia and energy. I1 = 1, I2 = 3, I3 = 5: a) h = 0.4993; b) h = 0.1002; c) h = 0.1674; d) h = 0.16667; f ) h = 0.180278. I1 = 1, I2 = 1.5, I3 = 2: e) h = 0.33335.

rotates on the same place, and the tra jectory of the p oint of contact degenerates into the p oint. The neighb orho o d of the p oint h = h is of a greatest interest during the motion along the line under 2 consideration on the bifurcational plane. In the p oint itself the curve is represented by two segments of the spiral with the centers displaced by the value of (Fig. 4). Near the indicated value H the numb er of the wreaths of the spiral b ecomes finite, and the rest segment of the tra jectory b etween two tangency p oints of the external curve b egins to rotate in a quasi-p erio dical way ab out a certain center (Fig. 6 (c), (d)). In this case the radius along which the center of the spiral moves tends to 2 if h h . Note that dep ending on the value of and other parameters of the system, the center of 2 the envelop e curves may b e situated b oth inside and outside the spiral (Fig. 6 (d), (e)). As it moves away from the critical value of h, the numb er of the wreaths of the spiral decreases and b ecomes equal to 1. Let us also note that under the certain system parameters (in particular, under the large values of ) there exists some critical value of the energy, at which the motion of the p oint of contact b ecomes infinite (Fig. 6 (f )).

4. The ball rolling with gyroscop e
Let us discuss the problem of rolling of the dynamically nonsymmetrical ball with the gyroscop e within it on the surface. D. K. Bobylev studied the given problem in detail in [1] for the case of the


REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001

303


A. A. KILIN

dynamically symmetrical ball. Then N. E. Zhukowski in [4] describ ed the sp ecific case of the problem under consideration when the axis of the dynamical symmetry of the ball coincides with the axis of the gyroscop e, and the difference b etween the p olar and equatorial moment of the inertia for the ball equals to the equatorial moment of inertia for the gyroscop e. In this case, as N. E. Zhukowski has proved, the analysis of the ball's motion is simplified largely. The general formulation of the problem of rolling of the ball with the gyroscop e was considered by A. P. Markeev in [7], where he proved its integrability in case of arbitrary moments of inertia and direction of the gyroscop e moment.

Fig. 7. The tra jectories of the point of contact on the surface at I1 = 1, I2 = 3, I3 = 5, K = (0.1, 0.1 0.2), C = 1, M + K : a) h = 0.1366; b) h = 0.1415; c) h = 0.1533; M + K : d) h = 0.25, the tra jectory is near the separatrix; f ) h = 0.3, the tra jectory is near the elliptic fixed point.

When one adds the gyroscop e, the equations (2.1) are reduced to the form M = (M + K ) ç , = ç , (4.1)

M = I + D ç ( ç ),

D = ma2 ,

where K is a constant vector of the gyroscop e moment. As A. P. Markeev shown in [7], the given


304

REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001


THE DYNAMICS OF CHAPLYGIN BALL: THE QUALITATIVE AND COMPUTER ANALYSIS

system allows the integrals h = 1 (M , ), C = (M + K , ), 2 ( , ) = 1, n = (M + K , M + K ), (4.2)

that allow the reduction of the problem to quadratures. The measure of the given problem have the same form (2.2) as in the case of the absence of a gyroscop e. However, despite this, the explicit integration of the problem has not b een carried out so far. Let us consider the sp ecific case of the given problem [9] when the total moment of the ballgyroscop e system is vertical M + K = C . In this case the third equation of the system takes the form J + K = l , (4.3) where l = C + D
2h + (K , ) . Let us differentiate (4.3) with resp ect to time and make a substitution C

= lx,

dt = 1 d . l

After some transformations the equation in terms of new variables can b e written in the form ^ ^ J x = (J x + 1 K ) ç x, where ^ Jm n = J
mn

(4.4)

= C + 2Dh . C

- D Km Kn ,
C

The obtained equations coincide with Zhukovski - Volltera equations, which describ e the motion of the free gyroscop e. V. Volltera integrated these equations in [12]. However, due to the implicit nature and the necessity to solve the equations of p ower four, V. Volltera's solution do esn't allow one to get a full understanding of the dynamics. The motion of the p oint of contact in case under consideration is describ ed by the equations (2.4), (2.6), same as ab ove. At the same time on the discussed ball's motion the gyroscop e influence is imp osed. As a consequence the view of the tra jectory is yet more complicated, however, the tra jectory remains b ounded for the case under discussion. Fig. 7 shows some tra jectories that are most remarkable at M + K and M + K . The author is grateful to A. V. Borisov and I. S. Mamaev for the useful discussion of the problem.

References
[1] D. K. Bobylev. On a ball with a gyroscope inside rolling without sliding on the horizontal plane. Mat. Sbornik. 1892. V. XVI. (In Russian) [2] A. V. Borisov, I. S. Mamaev. Poisson structures and algebras Lie in hamiltonian mechanics. Izhevsk, NIC RCD. 1999. P. 464. (In Russian) [3] A. V. Borisov, I. S. Mamaev. Poisson structure of the problem of Chaplygin ball rolling. Mat. zametki (to be published). (In Russian) [4] N. E. Zhukovsky. On gyroscopic ball of D. K. Bobylev. Trudy otd. fizich. nauk Obsch. lyub. estestvoznaniya. 1893. V. VI. 1. P. 11-17. (In Russian) [5] S. L. Ziglin. Separatrix splitting, solution branching and non existence of the integral in rigid body dynamics. Trudy Mosk. Mat. Obschestva. 1980. V. 41. P. 287-303 . (In Russian) [6] V. V. Kozlov. On the theory of integration of nonholonomic mechanics equations. Advances in Mechanics. 1985. V. 8. 3. P. 85-107. (In Russian)




REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001

305


A. A. KILIN

[7] A. P. Markeev. On integrability of the problem of rolling of the ball with multilinked cavity filled with ideal fluid. Izv. Akad. Nauk SSSR, Ser. Mekh. tver1. P. 64-65. (In Russian) dogo tela. 1985. [8] Yu. I. Ne imark, N. A. Fufaev. Dynaics of nonholonomics systems. M.: Nauka. 1967. P. 520. (In Russian) [9] Yu. N. Fedorov. The explicit integration and isomorphisms of some problems of classical mechanics. An abstract of the dissertation for the academic degree of


candidate of physical and mathematical sciences. M.: MGU. 1989. (In Russian) [10] S. A. Chaplygin. On the motion of a heavy rigid body of rotation on a plane. Collection of papers. V. 1. M.-L.: GITTL. 1948. P. 57-75. (In Russian) [11] S. A. Chaplygin. On a ball rolling on the horizontal plane. Collection of papers. V. 1. M.-L.: GITTL. 1948. P. 76-101. (In Russian) [12] V. Volterra. Sur la theorie des variations des latitudes. Acta Math. 1899. V. 22. P. 201-358 .



306

REGULAR AND CHAOTIC DYNAMICS, V. 6,

3, 2001