The Lagrange–D'Alembert–Poincaré equations and integrability for the rolling disk

Classical nonholonomic systems are described by the Lagrange–d'Alembert principle. The presence of symmetry leads, upon the choice of an arbitrary principal connection, to a reduced variational principle and to the Lagrange–d'Alembert–Poincaré reduced equations. The case of rolling bodies has a long history and it has been the purpose of many works in recent times, in part because of its applications to robotics. In this paper we study the classical example of the rolling disk. We consider a natural abelian group of symmetry and a natural connection for this example and obtain the corresponding Lagrange–d'Alembert–Poincaré equations written in terms of natural reduced variables. One interesting feature of this reduced equations is that they can be easily transformed into a single ordinary equation of second order, which is a Heun's equation.