The heavy top: a geometric treatment
We 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. Our analysis of the heavy top follows the reduced energy momentum method of Simo et al, 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. 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 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. We 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, we 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.
Copyright © Institute of Physics and IOP Publishing Limited 1992. Received 6 June 1991. Accepted by J D Gibbon. Authors were: University of California Postdoctoral Fellow, partially supported by the Institute for Mathematics and its Applications and the Army Research Office; partially supported by NSF Grant DMS 9142613; partially supported by AFOSR Grants 2-DJA-544 and 2-DJA-771. partially supported by NSF Grant DMS-89227M and DOE Contract DE-FG03-88ER25064. We thank the Mathematical Sciences Institute at Cornell University, where this work was initiated, for its support during the program on Hamiltonian dynamics in the fall of 1989. DL thanks the Institute for Mathematics and its Applications, at the University of Minnesota, Minneapolis, for its support during much of this work and Wolfram Research Incorporated for generously providing Mathematica(TM). We thank J Maddocks for a valuable exchange of ideas which motivated us to analyse in detail relative equilibria other than sleeping tops.