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ISSN 1028 3358, Doklady Physics, 2015, Vol. 60, No. 6, pp. 269­271. © Pleiades Publishing, Ltd., 2015. Original Russian Text © A.V. Borisov, I.S. Mamaev, 2015, published in Doklady Akademii Nauk, 2015, Vol. 462, No.6, pp. 657­659.

MECHANICS

A New Integrable System of Nonholonomic Mechanics
A. V. Borisov
a, b

* and I. S. Mamaevc

Presented by Academician V.V. Kozlov October 20, 2014 Received November 11, 2014

DOI: 10.1134/S1028335815060087

S.A. Chaplygin [10] integrated in quadratures the problem of rolling without slipping for a dynamically asymmetrical balanced sphere (Chaplygin sphere) on a plane. For this purpose, he explicitly found an invari ant measure and first integrals. He also gave a geomet rical interpretation of the motion. The motion of the Chaplygin sphere with an additional constraint forc ing the point of contact to remain on a straight line was investigated by A.P. Veselov and L.E. Veselova in [6]. They gave an invariant measure and first integrals making it possible to integrate the system by the Euler­Jacobi theorem. However, it should be noted that no explicit quadratures for this system have been obtained until now. In this study, we consider an inte grable generalization of the system [6] and propose a new mechanical implementation of the arising non holonomic constraint. We consider the problem of a sphere (Chaplygin sphere) rolling on a plane and subject to a nonholo nomic constraint (the Veselova constraint) [7] (, E) = 0, where E is the unit vector fixed in space. If E , where is the normal to the contact plane, the sphere moves along a straight line and, as is shown in [6], the system is integrable. The motion of the sphere along the straight line can be implemented with the help of absolutely smooth walls [6] (see Fig. 1). If E || , we obtain a model of rolling without slip ping and spinning. As is well known [4], the arising system proves to be equivalent to the Veselova problem and is integrable.

Consider a more general situation assuming that E is an arbitrary unit vector fixed in space. To describe the realization of this constraint, we first introduce the concept of spherical suspension and the nonholo nomic joint. The spherical suspension (introduced in [9]) describes the motion of a rigid body with a fixed point O enclosed in a spherical shell to which an arbi trary number of massive dynamically symmetric balls adjoin. It is assumed that there is no slippage at the points of contact of the spheres with the shell, and that the centers of the spheres are fixed in space. As is shown in [9], this system is integrable at an arbitrary number of spheres; in the case where there is only one peripheral sphere, we obtain a problem equivalent to the problem of the Chaplygin sphere rolling on a plane. The initial design of the nonholonomic joint was proposed by V. Wagner in [5] for implementation of the constraint in the Suslov problem (, a) = 0, where a is the body fixed vector (see Fig. 2). In this case, flat rollers (disks) are attached to the body with a fixed point, which roll without slipping on the internal sur face of a fixed spherical shell; it is assumed that the roller is so sharp that its velocity in the direction per pendicular to its plane is zero. Similarly, it is possible to consider the motion of the body with a fixed point enclosed in a spherical shell which the rollers (disks) touch with the axes fixed in space (Fig. 3). It is clear

E

a

National Research Nuclear University MEPhI, Moscow, 115409 Russia b Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow oblast, 141700 Russia c Udmurt State University, Izhevsk, 426034 Russia *e mail: borisov@rcd.ru 269

Fig. 1. Rolling of a sphere along a straight line (A.P. Ve selov, L.E. Veselova).


270

BORISOV, MAMAEV

OP

Fig. 2. Suslov constraint in the Wagner implementation.

body with a fixed point is enclosed in a spherical shell which one ball and one disk adjoin (Fig. 4). In the coordinate system attached to the principal axes of the body, the constraint equations are repre sented in the form R â + R11 â = 0, (, E) = 0, (1) where is the angular velocity of the body, R is the radius of the spherical shell, 1 and R1 are the angular velocity and the radius of the adjoining ball, is the unit vector of the axis connecting the centers of the balls, and E is the vector of the normal to the plane containing the center of the ball and the axis of the disk. The equations of motion with undefined multipli ers have the form · I = I â + R â N + E + Q, · D1 1 = D11 â + R1 â N, (2) · · = â , E = E â , where I = (diagI1 , I2 , I3) is the tensor of inertia of the body, D1 is the tensor of inertia of the adjoining ball, N =(N1 , N2 , N3), are the undefined multipliers cor responding to constraint reactions (1), and Q is the moment of external forces. Using the second equa tion, we find that (1, )· = 0. Thus, · · ( 1 , ) = ­(1, ) = ­(1, â ). With the help of this in (2), we eliminate tions. As a result, we · · I + D â ( â relation and the second equation â N from the remaining equa obtain ) = I â + E + MQ , (3)



Fig. 3. Implementation of the Veselova constraint.

E

O

· · D = R2 D1 , = â , E = E â . R1

2

From the form of system (3), it is easy to conclude that these equations coincide with the equations for the rolling of the Chaplygin sphere with the additional Veselova constraint; in this case, the direction of the vector E can be arbitrary. The undefined multiplier is found from the condition (E, )· = 0: =­ ( I â + M Q, I E ) ( E, I Q E )
­1 ­1

Fig. 4. New implementation of constraints.

,

that in the simplest case we obtain constraint (, ) = 0 of the Veselova problem, where is the vector perpen dicular to the disk plane (this result is noted in [3]). If we fix the axis with the roller in space, in the sim plest case we obtain the Veselova constraint, and if we fix the spherical shell, we get the Suslov constraint. If both bodies are mobile, the problem is also integrable and considered in [1]. We consider now a combination of the spherical suspension and the nonholonomic joint, when the

IQ = J ­ D , J = I + D . It can be shown by direct verification that if MQ is independent of , Eqs. (3) have the invariant measure d3d3 with the density = ( ( E, I Q E ) det I Q ) . There are also explicit geometrical integrals 2 =1, E2 =1, (, E) = const.
DOKLADY PHYSICS Vol. 60 No. 6 2015
­1 1/2

(4)


A NEW INTEGRABLE SYSTEM OF NONHOLONOMIC MECHANICS

271

U U In the potential force field MQ = â +Eâ , E energy is also conserved H = 1 ( I Q , ) + U ( , E ) , 2 where U(, E) is the potential energy of external forces. If there are no external forces, U = 0 and E â 0, there are two more additional integrals (5) K = I Q ­ ( I Q , E ) E , and, hence, the system (3) is integrable (by the Euler­ Jacobi theorem). To prove the existence of the integrals F1 and F2, we write the evolution equations for the vector K: · K = K â . Hence, the vector K is fixed in space and all its projec tions to the moving axes are conserved, but since (K, E) 0, there are only two independent integrals. Thus, if E â 0, the system under consideration is almost completely identical to that considered in [6] with an additional restriction (E, ) 0. Until now, the explicit quadratures for (3) are unknown even for U = 0. In the particular case E = , the integrals (5) iden · tically vanish; however, we find (, )· = ( , )= 0 from the constraint equation (, ) = 0, and, hence, I Q = J ­ D ( , ) = J , · · · I Q = J ­ D ( , ) = J , I â = J â . Thus, the system under consideration is equivalent to the Veselova system after the replacement I J . This system is integrable and can be reduced to quadratures with the help of the sphero conical coordinates [7]. F 1 = ( K, E â ) , F 2 = ( K, E â ( E â ) ) ,

ACKNOWLEDGMENTS The work was performed as part of the base portion of the state universities jobs and supported by a grant of the President of the Russian Federation for support of leading scientific schools (NSh 2964.2014.1). REFERENCES
1. I. A. Bizyaev, A. V. Borisov, and I. S. Mamaev, Regul. Chaotic Dyn. 19 (2), 198 (2014). 2. A. V. Borisov, A. A. Kilin, and I. S. Mamaev, Regul. Chaotic Dyn. 16 (1/2), 104 (2011). 3. A. V. Borisov and I. S. Mamaev, in Nonholonomic Dynamical Systems. Integrability, Chaos, Strange Attractors (IKI, Moscow, Izhevsk, 2002) [in Russian]. 4. A. V. Borisov and I.S. Mamaev, Regul. Chaotic Dyn. 13 (5), 443 (2008). 5. V. V. Vagner, Tr. Seminara po Vektorn. i Tenzorn. Anal izu, No. 5, 301 (1941). 6. A. P. Veselov and L. E. Veselova, Mat. Zametki 44 (5), 604 (1988). 7. L. E. Veselova, in Geometry, Differential Equations and Mechanics (Izd vo MGU, Moscow, 1986), pp. 64­68 [in Russian]. 8. G. K. Suslov, Theoretical Mechanics (Gostekhizdat, Moscow, 1946) [in Russian]. 9. Yu. N. Fedorov, Vestn. MGU, Ser. 1. Matem. Mekh., No. 5, 91 (1988). 10. S. A. Chaplygin, in Collection of Works (OGIZ, Lenin grad, 1948) [in Russian]. 11. A. P. Kharlamov and M. P. Kharlamov, Mekh. Tverd. Tela, No. 27, 1 (1995). 12. F. B. Fuller, Proc. Nat. Acad. Sci. USA 68 (4), 815 (1971).

Translated by V. Bukhanov

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No. 6

2015