Chaplygin's sphere
  Chaplygin proved the integrability by quadratures of a round sphere, rolling without slipping on a horizontal plane, with center of mass at the center of the sphere, but with arbitrary moments of inertia. Although the system is integrable in every sense of the word, it neither is a hamiltonian system, nor is the integrability an immediate consequence of the symmetries. On the other hand the constants of motion are obtained as a consequence of a Nother's principle and the system can be related to the geodesic flow on the Euclidean motion group for a left invariant metric.   In this paper we analyse the global dynamics and in the process we will explain almost all of Chaplygin's results. At the end of each section we describe in a subsection ``Chaplygin'' the relation between our text and Chaplygin's.   We also obtain several new results, such as the proof that the level sets of the constants of motion in the phase space for the rotational motion are two-dimensional tori on which, after a suitable time reparametrization, the rotational motion is quasiperiodic. After suitable completion of the level surfaces this is also true for the complexified system, which is algebraically integrable in the sense of Adler and van Moerbeke. This also follows, in a quite different way, from Chaplygin's integration in terms of hyperelliptic integrals. 
J.J. Duistermaat