The heavy top: a geometric treatment

Abstract

The authors consider the steady group motions of a rigid body with a fixed point moving in a gravitational field. For an asymmetric top, rotation about the axis of gravity is the only permissible group motion; for a Lagrange top, simultaneous rotation about the axis of gravity and spin about the axis of symmetry of the top is permissible. The analysis of the heavy top follows the reduced energy momentum method of Simo et al. (1991), which is applicable to a wide range of conservative systems with symmetry. Steady group motions are characterized as solutions of a variational problem on the configuration space; local minima of the amended potential correspond to nonlinearly orbitally stable steady motions. The combination of a low-dimensional configuration space and a relatively large number of parameters that produce substantial qualitative changes in the dynamics makes possible a thorough, detailed analysis, which not only reproduces the classical results for this well known system, but leads to some results which we believe are new. They determine general equilibrium and nonlinear stability conditions for steady group motions of a heavy top with a fixed point. They rederive the classical equilibrium and stability conditions for sleeping tops and precessing Lagrange tops, analyse in detail the stability of a family of steady rotations of tilted tops which bifurcate from the branch of sleeping tops parametrized by angular velocity, and classify the possible stability transitions of an arbitrary top as its angular velocity is increased. They obtain a simple, general expression for the characteristic polynomial of the linearized equations of motion and analyse the linear stability of both sleeping tops and the family of tilted top motions previously mentioned. Finally, they demonstrate the coexistence of stable branches of steadily precessing tops that bifurcate from the branch of sleeping Lagrange tops throughout the range of angular velocities for which the sleeping top is stable.