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ISSN 1064 5624, Doklady Mathematics, 2014, Vol. 90, No. 2, pp. 631­634. © Pleiades Publishing, Ltd., 2014. Original Russian Text © I.A. Bizyaev, 2014, published in Doklady Akademii Nauk, 2014, Vol. 458, No. 4, pp. 398­401.

MATHEMATICAL PHYSICS

Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top
I. A. Bizyaev
Presented by Academician V.V. Kozlov February 24, 2014 Received April 4, 2014

DOI: 10.1134/S1064562414060192

EQUATIONS OF MOTION AND FIRST INTEGRALS We consider the problem about the rolling of an unbalanced dynamically asymmetric ball on a plane surface [1­3, 15]. Following [10­12], we refer to this system as a nonholonomic Chaplygin top. In [10­13], a nonholonomic model in which a Chaplygin top rolls without slipping was considered. In this paper, we con sider another rolling model, in which both slipping and spinning are absent at the contact point of the ball with the plane. A possible realization of such a cou pling was proposed in [14], where it was assumed that the body is covered by a sufficiently soft rubber ensur ing an appropriate contact with the plane. In the mov ing coordinate system Cxyz attached to the ball, the equations of motion describing the evolution of the angular velocity of the ball and the vector normal to the plane at the point of contact have the form [2] 1· U( ) ~· ~ J = (J) â ­ + â + , 2 · = â , 2 2 ~ J = J + E , J = I + m ( b + c ) ­ mc c , = 2 mb ( c, ) , · ~ ~ ( J ) â ­ 1 , J ­1 2 , 0 = ­ ~ ­1 ( , J ) where m is the mass of the ball, I = diag(I1, I2, I3) is the inertia tensor of the ball, b is the radius of the ball, c is the displacement vector of the center of mass with respect to the geometric center (see Fig. 1), and U() is the potential of external forces. System (1) admits the two general first integrals

F0 = ,

2

F 1 = ( , ) ,

whose physical values equal [1] F0 = 1 and F1 = 0. At the level F1 = 0, there also exists an energy integral, namely, ~ E = 1 ( , J ) + U ( ) . 2 Let us restrict Eqs. (1) to the manifold
4

= { ( , ) = 1, ( , ) = 0 } ,

2

(2)

(1)

which is diffeomorphic to the tangent bundle of the two dimensional sphere TS 2; as a result, we obtain a four dimensional system with energy integral E. Its properties depend essentially on the presence or absence of additional (tensor) invariants, such as an invariant measure or an additional integral. In this connection, we briefly mention previous results on the additional invariants of system (1). In [1], a necessary condition for the existence of an invariant measure depending on the positional vari ables was obtained. This condition is satisfied only in the cases of (i) axial symmetry (I1 = I2, c1 = c2 = 0), (ii) full dynamical symmetry (I1 = I2 = I3), and (iii) the balancedness of the ball (c = 0).

z C O c x

y

b

Udmurt State University, Universitetskaya ul. 1, Izhevsk, 426034 Russia e mail: bizaev_90@mail.ru 631

Fig. 1. The ball on the plane.


632 1.25 1.20 1.15 1.10 1.05 0 1.1668 1.1667 1.1666 1.1665 1.1664 0.5385 0.5390 0.5395 0.5400 0.5405 0.5410 0.2 (b) 0.4 0.6 0.8

BIZYAEV (6) 0.002 0.001 0 1.0 -0.001

(a)



2

Fig. 3. The deviation of the sixth iteration of the PoincarÈ mapping from the identity mapping on the circle for the 5 torus with rotation number = at = 0.54 (J1 = 15, J3 = 3 20.5, a = 6).

however, the necessary condition for the existence of an invariant measure remains unsatisfied. DECOMPOSITION INTO TWO CONFORMALLY HAMILTONIAN FIELDS It turns out that, under conditions (3), system (1) can be represented as the sum of two conformal Hamiltonian vector fields (this representation is called a conformal Hamiltonian decomposition [9]). To explicitly specify such a representation, we introduce new variables M1, M2, and M3 so that M 2 ( M1 2 ­ 1 M2 ) ­ 1 23 32 , 1 = 2 2 J1 + 2 a 1 ( 1 + 2 ) g1 ( 1 + 2 ) 2 = 1 ( M2 1 ­ 2 M1 ) J1 + 2 a 1 ( + ) 3 =
2 1 2 2

Fig. 2. (a) The rotation number of system (6) as a function of the parameter for J1 = 15, J3 = 20.5, a = 6 (I1 = 6, I2 =2, I3 = 7.5, R = 3, m = 1, and c1 = 2) and (b) its 7 zoomed fragment near the resonance = . 6

Nevertheless, in the absence of external forces (U() = 0), the system has the additional quadratic integral [3] 1 ~ ~ F 2 = ( â J , â J ) ­ E . 2 In [2], it was shown that the integrals F2 and H deter mine a Liouville foliation of the manifold 4 into invariant tori, which is equivalent to the Liouville foli ation in the Euler case in the dynamics of rigid bodies with a fixed point. In the same paper, it was shown that when the cen ter of mass is displaced along one of the axes and U() = 0, there arise limit cycles on tori with rational rotation numbers in system (1), which prevents the existence of an invariant measure. In this paper, we consider the special case in which c = ( c 1, 0, 0 ) , as we see, we have J = diag ( J 1, J 1, J 3 ) ,
2 2

­

2 3 M g1 ( +
2 1

3 2 2

)

,

I = diag ( I 1, I 1 ­ mc 1, I 3 ) , U = u ( 3 ) , J 1 = I 1 + mb ,
2

2

(3)

M3 . g1 Then E and F2 take the form 2 2 2 E = 1 ( M1 + M2 + M3 ) + u ( 3 ) , F2 = M3 . 2 Note that, in the new variables, the energy E coincides with the Hamiltonian of the ball top in the potential field u(3). As a result, the equations of motion (1) are represented in the form F · x = N 1 J p E + N 2 J p 2 , x = ( M, ) , x x 1 , N1 = J1 + 2 a 1 N2 =
2

J3 = I3 + m ( b + c1 ) , = 2 a 1 , a = m b c1 . It turns out that, in this case, there arises an additional linear integral in system (1), namely, F2 = g1 3 , g1 = 2 a 1 + J1 3 + J2 ( 1 ­ 3 ) ,
2 2

(4)

1 1 ­ , 2 2 2 2 2 g1 ( 1 + 2 ) 2 J1 + 2 a 1 ( 1 + 2 ) where Jp denotes the Lie­Poisson bracket corre sponding to the algebra e(3):
DOKLADY MATHEMATICS Vol. 90 No. 2 2014


NONINTEGRABILITY AND OBSTRUCTIONS TO THE HAMILTONIANIZATION

633

{ M i, M j } = ­ ijk M k ,

{ M i, j } = ­ ijk k ,
2

{ i, j } = 0 . The fact that we have managed to represent the vector field (1) in the form (4) says that the foliation into invariant tori of the initial system is the same as in the case of a Hamiltonian system. In other words, there are no topological obstructions to the Hamilto nianization of the foliation. However, as shown in what follows, such obstructions arise in studying the flow on invariant tori. THE ABSENCE OF INVARIANT MEASURES AND THE EXISTENCE OF LIMIT CYCLES Let us show that, in case (iii) with u(3) = 0, there exist tori with limit cycles. For this purpose, we intro duce the Euler angles 1 = sin sin , 2 = cos sin , 3 = cos , M 1 = cos p ­ sin cot p , M 2 = ­ sin p ­ cos cot p , M 3 = p , with respect to which the equations of motion take the form · = g1 = p
2

d d =

2

( ) ( 2 a ( ) sin + ( 1 ­ ) ( J 1 ­ J 3 ) sin + J 3 ) , ( 2 a ( ) sin + J 1 ) (6)
2 2

2

g 1 sin

,
2

p · = , g2
2

cos p · p = , 3 g 2 sin

2

( ) = cos + sin . Following [2], we calculate the rotation numbers of system (6) as V ( u ) ­ V0 = lim , u u ­ u0 for the initial point V0 = 0, u0 = 0. These numbers depend on the parameter (the other parameters being fixed). The graph of the dependence is shown in Fig. 2a. In the zoomed graph, one can see the horizon 7 tal fragment corresponding to the resonance = . 6 Let us show that the flow on the family of tori cor 7 responding to the resonance = has limit cycles. 6 For this purpose, on the separate torus corresponding to the given value of , consider the PoincarÈ section by the plane = 0, which determines a self mapping of the circle ( ) : S S . Figure 3 shows the deviation of the sixth iteration of this mapping from the identity mapping, i.e., the func (6) tion (). Note that the maximum deviation of the sixth iteration from the identity mapping does not exceed 0.002%. It is seen from Fig. 3 that, on the tori under consid eration, there are twelve limit cycles, six of which are stable and the other six are unstable. This means that the system admits no smooth invariant measure. Thus, although system (1) with parameters (3) exhibits a reg ular behavior and admits a conformal Hamiltonian decomposition, this system cannot be solved in quadratures. ACKNOWLEDGMENTS This work was supported by the program for sup port of young doctors of sciences, project no. MD 2324.2013.1. REFERENCES
1. A. V. Borisov, I. S. Mamaev, and I. A. Bizyaev, Regul. Chaotic Dyn. 18 (3), 277­328 (2013). 2. A. V. Bolsinov, A. V. Borisov, and I. S. Mamaev, Regul. Chaotic Dyn. 17 (6), 571­579 (2012). 3. A. V. Borisov and I. S. Mamaev, Regul. Chaotic Dyn 13 (5), 443­490 (2008). 4. I. A. Bizyaev and A. V. Tsiganov, J. Phys. A: Math. Theor. 46 (8), 1­11 (2013).
1 1

(5)

J 1 cos + J 3 sin + 2 a sin sin ,

g 2 = J 1 + 2 a 1 sin sin , and the integrals of motion take the form
2 1 p E = 1 p + , F2 = p . 2 sin2 2 We fix h and related to the values of the first inte grals as 2

p , 2h where 0 1. The evolution and are then described by the equations E = h, = . 2 2 sin g 2 g 1 sin Consider in more detail the family of tori for which 0 < < 1 , [ 0, 2 ) , [ 1, 2 ] , where 1 and 2 are the roots of the equation sin2 = in the interval (0, ); on these tori, we introduce the angular variables (, )mod2. The absolute value of is found from the relation [2]
2

2

· =

p



,

· =
2

p ( sin ­ )

2

2

sin ­ = ( 1 ­ ) cos . In the new variables, the trajectory on the torus is determined by the differential equation
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2

2


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BIZYAEV 11. J. Shen, D. A. Schneider, and A. M. Bloch, Intern. J. Robust Nonlinear Control 18 (9), 905­945 (2008). 12. P. Lynch and M. D. Bustamante, J. Phys. A: Math. Theor. 42, 425203 (2009). 13. A. V. Borisov and I. S. Mamaev, Regul. Chaotic Dyn. 7 (2), 177­200 (2002). 14. K. M. Ehlers and J. Koiler, in Proceedings of IUTAM Symposium on Hamiltonian Dynamics, Vortex Struc tures, Turbulence, Moscow, Russia, 2006 (Springer, Dordrecth, 2006). 15. A. O. Kazakov, Regul. Chaotic Dyn. 18 (5), 508­520 (2013).

5. S. A. Chaplygin, Regul. Chaotic Dyn 7 (2), 131­148 (2002). 6. V. V. Kozlov, Usp. Mekh. 8 (3), 85­107 (1985). 7. V. V. Kozlov, Proc. Steklov Inst. Math. 256, 188­205 (2007). 8. V. V. Kozlov, J. Appl. Math. Mech. 51 (4), 420­426 (1987). 9. A. V. Borisov, I. S. Mamaev, and A. V. Tsyganov, Math. Notes 95 (3), 308­315 (2014). 10. J. Shen, D. A. Schneider, and A. M. Bloch, in Proceed ings of the 42th IEEE Conference on Decision and Con trol, Maui, 2013 (Maui, 2013), Vol. 53, 4369­4374.

Translated by O. Sipacheva

DOKLADY MATHEMATICS

Vol. 90

No. 2

2014