| I investigate the stability of the relative equilibria in a class of mechanical systems which have rotational symmetry and a stable invariant equilibrium. Such systems generally have one or more families of relative equilibria, called whirling modes, that bifurcate from the stable equilibrium at different angular velocities. The main results are: (1) In each family, the relative equilibria sufficiently close to the stable equilibrium are linearly orbitally stable. (2) The relative equilibria in the first family are nonlinearly orbitally stable, at least until a pair of eigenvalues of the linearization collide at zero. (3) The standard method for proving nonlinear orbital stability of a relative equilibrium fails in all of the whirling modes after the first. (4) In a special class of singular systems, the relative equilibria in the first whirling mode are nonlinearly orbitally stable, and the relative equilibria in all higher modes are unstable.;The linear stability result applies to the whirling modes of a hanging chain. Kolodner (Comm. Pure Appl. Math. 8, 395-408) proved the existence of these whirling modes in 1955, but their stability has remained an open problem. |
