ISSN 1560-3547, Regular and Chaotic Dynamics, 2012, Vol. 17, No. 6, pp. 571–579. c  Pleiades Publishing, Ltd., 2012. Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals Alexey V. Bolsinov 1, 2* , Alexey V. Borisov 2** , and Ivan S. Mamaev 2*** 1 School of Mathematics, Loughborough University United Kingdom, LE11 3TU, Loughborough, Leicestershire 2 Institute of Computer Science, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia Received August 4, 2012; accepted October 19, 2012 Abstract—In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles. MSC2010 numbers: 37J60, 37J35, 70H45 DOI: 10.1134/S1560354712060081 Keywords: non-holonomic constraint, Liouville foliation, invariant torus, invariant measure, integrability INTRODUCTION This work continues the cycle of recent investigations [2, 9–11] devoted to the study of the dynamics of a body rolling on a surface without slipping and spinning (twisting). For this model of rolling, the authors of [12, 15] proposed the name “rubber-rolling”. Roughly speaking, it means the rolling of bodies with rubber coating. Such a model differs from the usual classical nonholonomic problem of a body rolling on an absolutely rough plane, so that in addition to the condition for the point of contact to be equal to zero it is necessary that the angular velocity relative to the normal to the fixed surface also be equal to zero. The kinematics of such a rolling motion was explored already by Hadamard [13] and Beghin [1]; their results are treated in [9] from the modern point of view. For various questions of control theory and robotics the use of this model simplifies investigations. Nevertheless, the problems of the dynamics of such systems (both free and controllable) have only recently begun to be studied (see [10]). Here we give an example of a system with “rubber rolling”, which possesses a complete set of integrals necessary for integrability (we note that in this problem the Euler – Jacobi integrability requires finding another additional integral and an invariant measure), but nevertheless does not admit an analytical invariant measure. This is the problem of a dynamically asymmetric ball, whose center of mass is displaced from the geometric center, rolling on a plane. We let c denote the displacement vector. For c = 0 this system generalizes the well-known Chaplygin ball problem to the case of “rubber rolling”, its integrability is shown in [7, 12] (in this case the invariant measure exists and is found explicitly). In this paper we consider the c = 0 case, but in the absence of * E-mail: A.Bolsinov@lboro.ac.uk ** E-mail: borisov@rcd.ru *** E-mail: mamaev@rcd.ru 571