A New Analysis of the Tippe Top: Asymptotic States and Liapunov Stability

Abstract We study the asymptotic behaviour of a spinning top whose shape is spherical, while its mass distribution has axial symmetry only and which is subject to sliding friction on the plane of support (so-called tippe top). By a suitable choice of variables in the equations of motion we construct explicitly all solutions of constant energy. The latter are the possible asymptotic states of the solutions with arbitrary initial conditions. Their stability or instability in the sense of Liapunov is determined for all possible choices of the moments of inertia. The stability analysis makes essential use of the conservation law L3 − αL3 = const. (Jelett′s integral), where L3 and L3 are the projections of the angular momentum onto the vertical and onto the symmetry axis, respectively, α being the ratio of the distance of the center-of-mass from the center to the radius of the sphere. The article concludes with some numerical examples which illustrate our general analysis of Liapunov stability.