Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

The problem of a disc rolling in a plane has been treated in classical works of P.Apple, D.J.Korteweg, E.J.Routh and S.A.Chaplygin. It is described by a dynamical system of 4 equations, has 3 integrals of motion and is integrable. When sliding is allowed there are 2 more dynamical variables, equations are dissipative, non-integrable and have energy as a monotonously decreasing function of time. The key for understanding the dynamics are asymptotic solutions, their stability properties and a La´Salle type theorem on asymptotic behavior of solutions. These results, together with with computer simulations of solutions starting in vicinity of the asymptotic solutions provide a basis for global understanding of what happens for different initial conditions. I shall explain formulation of the problem, present analytical results and tell how much we have learnt from numerical simulations.