ISSN 1560-3547, Regular and Chaotic Dynamics, 2013, Vol. 18, No. 3, pp. 277–328. c  Pleiades Publishing, Ltd., 2013. The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere Alexey V. Borisov 1, 2, 3* , Ivan S. Mamaev 1, 2, 3** , and Ivan A. Bizyaev 1*** 1 Institute of Computer Science; Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia 2 A.A. Blagonravov Mechanical Engineering Research Institute of RAS Bardina str. 4, Moscow, 117334, Russia 3 Institute of Mathematics and Mechanics of the Ural Branch of RAS S. Kovalevskaja str. 16, Ekaterinburg, 620990, Russia Received March 12, 2013; accepted May 8, 2013 Abstract—In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior. MSC2010 numbers: 37J60, 37J35 DOI: 10.1134/S1560354713030064 Keywords: nonholonomic constraint, tensor invariant, first integral, invariant measure, integra- bility, conformally Hamiltonian system, rubber rolling, reversible, involution Contents INTRODUCTION 278 1 ROLLING OF A RIGID BODY WITHOUT SLIPPING AND SPINNING: EQUATIONS OF MOTION, FIRST INTEGRALS AND DYNAMICAL BEHAVIOR 280 1.1 Rolling of a Body on a Plane 280 1.2 Rolling of a Body on a Sphere 282 1.3 The Poincar´ e Map and Hierarchy of Dynamics 284 2 CONDITIONS FOR THE EXISTENCE OF AN INVARIANT MEASURE AND CONFORMALLY HAMILTONIAN REPRESENTATION 287 2.1 Conditions for the Existence of an Invariant Measure for the Case of a Plane 290 2.2 Conformally Hamiltonian Representation of the System in the Case of Rolling on a Plane 292 2.3 Conditions for the Existence of an Invariant Measure for the Case of a Sphere 295 2.4 Conformally Hamiltonian Representation of the System in the Case of Rolling on a Sphere 295 * E-mail: borisov@rcd.ru ** E-mail: mamaev@rcd.ru *** E-mail: bizaev 90@mail.ru 277