CONTINUUM MECHANICS AND HAMILTONIAN PDES

4.1      Preliminary analytical work on H² well-posedness, long time existence, uniqueness and Lyapunov stability for the $\alpha$-Euler equations will follow the methods of classical fluid dynamics, but using the fact that it is not enstrophy that is preserved by the flow, but the L² norm of $1 - \alpha^2 \Delta$ applied to the vorticity. Analogous equations will be derived from the equations of elasticity theory and analysed by techniques similar to those used for fluids. REs and RPOs will be studied using standard methods combined with the new techniques developed under objective 1.1. The effects of viscosity will be studied using the methods of objective 1.3, with particular emphasis on `shadowing curves'. The application of geometric integrator techniques for numerical simulating these models will also be considered.

4.2      REs and RPOs of point vortex systems will be studied using the methods developed in objective 1.1. Heteroclinic cycles will be sought in systems with non-trivial finite symmetry groups resulting from permutations of identical vortices using ideas from non-Hamiltonian theory. The methods of objective 1.4 will be applied to these to obtain results on symmetric chaos. Reduction and reconstruction of the dynamics of point vortex systems on the plane will be studied in detail, with particular emphasis on exploring the effects of the non-compact Euclidean symmetry group. The effects of viscosity on point vortex models will be investigated using the methods of objective 1.3, particularly for systems derived from $\alpha$-Euler equations for which finite viscosity perturbations are better behaved than for the standard Euler equations.

4.3      Convergence properties of Hamiltonian structures will be used to derive geometrically exact shell and rod models as limits of three-dimensional nonlinear elasticity models. This will be done under various constitutive assumptions for the underlying materials, e.g. for Saint Venant Kirchhoff or Euler Kirchhoff materials. Within the context of such theories, REs and RPOs will be studied using the methods of objective 1.1. The influence of adding small viscosity to the model equations will also be studied.

4.4      REs and RPOs of affine rigid bodies will be studied using the methods developed in objective 1.1 combined with numerical computation of global bifurcation diagrams using methods similar to those applied to molecules in objective 3.2. Heteroclinic cycles and symmetric chaos will be studied by restricting to low dimensional subsystems in symmetry fixed point spaces and by using integrable approximations. Methods developed in objective 1.4 will be applied to these. The methods of objective 1.5 will be applied to non-holonomic affine rigid bodies.

4.5      The stability and bifurcation studies of pulses in optical fibres will use the methods developed under objective 1.1 supplemented by studies of multi-symplectic Evans functions. For non-Hamiltonian perturbations the methods of objective 1.3 will be used and a dissipative theory of multi-symplectic Evans matrices developed.