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ABSTRACTS
The pseudo-rigid body model is viewed in the context of continuum mechanics and elasticity theory. A Lagrangian reduction, based on variational principles, is developed for both anisotropic and isotropic pseudo-rigid bodies. For isotropic Lagrangians the reduced equations of motion for the pseudo-rigid body are a system of two (coupled) Lax equations on $so(3)\times so(3)$ and a second order differential equation on the set of diagonal matrices with positive determinant. Several examples of pseudo-rigid bodies such as stretching bodies, spinning gas could and Riemann ellipsoids are presented.
Relative equilibria in linear elasticity:
Relative equilibria for Hamiltonian dynamical systems modelling the motionBifurcations from relative equilibria of Hamiltonian systems
of hyperelastic, homogeneous, frame indifferent and isotropic bodies, are studied. We
find conditions on the potential energy function $V$ for the existence of relative equilibria
with certain prescribed symmetries, namely for those of type $S$.
A symplectic version of the slice theorem for compact group actions is used to prove a bifurcation theorem for relative equilibria of symmetric Hamiltonian systems. The bifurcation theorem is applied to two examples, the classical dynamics of an $XY_2$ molecule near a symmetric linear equilibrium, and the dynamics of a system of two coupled identical axisymmetric rigid bodies near an equilibrium for which the two bodies are aligned on the top of each other. In addition reduction of these systems to appropriate "slices" is used to describe other aspects of their dynamics. The results suggest that this might be a useful general technique for describing the dynamics of systems near relative equilibria which are singular points of the associated momentum map.Symmetries of Riemann ellipsoids
The results of Dirichlet, Dedeking and Riemann on "ellipsoidal figures of equilibrium" of rotating self-gravitating fluids are reviewed in the context of the geometric theory of Hamiltonian systems with symmetry. In particular Riemann's classification is derived using only the existence of physically natural rotational symmetries, and so is shown to be applicable to models of liquid drops, atomic nuclei and elastic bodies as well as self-gravitating fluids. Similarly Dedekind's transposition symmetry is obtained as a simple consequence of the rotational symmetries. The symmetry groups of the different types of ellipsoidal figures of equilibrium are also computed, with particular attention being paid to the role of the transposition operator.
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