Nonlinear stability of rotating pseudo-rigid bodies

Abstract

A rigorous nonlinear stability analysis of rotating homogeneous elastic bodies is presented, which exploits the hamiltonian structure and symmetries inherent to homogeneous elasticity by means of the energy-momentum method. It is shown that stability of a relative equilibrium is implied by the definiteness of the second variation of a modified hamiltonian restricted to an appropriate subspace. The analysis makes crucial use of a special parametrization of the constrained space of admissible variations, which results in a nearly diagonal second variation. The stability conditions obtained by this method include the conditions for stability of the equilibrium configuration as a rigid body and satisfaction of the Baker-Ericksen inequalities. As an application of our results, we obtain complete, explicit stability conditions for a particular form of relative equilibria for three classes of materials: for two of these, Ciarlet-Geymonat and St Venant-Kirchhoff materials, these equilibria are always stable; for the third, a compressible Mooney-Rivlin material, both stable and unstable equilibria exist.