Lewis, D., T. S. Ratiu, J. C. Simo, and J. E. Marsden
We determine general equilibrium and nonlinear stability conditions for steady group motions of a heavy top with a fixed point. We rederive the classical equilibrium and stability conditions for sleeping tops and precessing Lagrange taps, analyze 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. We obtain a simple, general expression far the characteristic polynomial of the linearized equations of motion and analyze the linear stability of both sleeping tops and the family of tilted top motions previously mentioned. Finally, we demonstrate the coexistence of stable branches of steadily precessing tops that bifurcate from the branch of sleeping Lagrange lops throughout the range of angular velocities for which the sleeping top is stable.