MASIE
PROJECTS
 1 Geometry and Dynamics
2 Numerical methods    
3 Atoms and Molecules

Section 4

CONTINUUM MECHANICS
AND HAMILTONIAN PDES

Topics

4.1REs and RPOs of alpha-Euler and other reduced averaged equations
4.2REs, RPOs, HCs and chaos in systems of point vortices
4.3Geometrically exact elasticity models and limit behaviour
4.4Bifurcations and dynamics of affine rigid bodies
4.5Stability and bifurcations of optical pulses

Project objectives

This section will focus on REs, RPOs and other aspects of the dynamics of a number of model problems from continuum mechanics. Specific objectives include: Major breakthroughs are expected to include the development of new techniques which will greatly simplify the computation of stability properties and bifurcations in infinite dimensional continuum systems and the discovery of new dynamical phenomena in models of continuum systems.

Scientific originality

4.1      The application of techniques from bifurcation theory and geometry to the fundamental models of continuum mechanics has proved to be very successful in the stability analysis of REs and attendant bifurcation phenomena. These methods, combined with new techniques from section 1, will be applied to a number of continuum problems, including the $\alpha$-Euler equations. In the last two years it has shown that they are much better behaved than the usual Euler equations in terms of both the existence and uniqueness of solutions and non-zero viscosity perturbations. They therefore provide an excellent context for developing and testing rigorous stability, bifurcation and non-Hamiltonian perturbation theories for REs and RPOs of continuum systems. The same techniques will be used to derive and study analogous equations from nonlinear elasticity theory. Special emphasis will be put on new models for shape memory alloys for which the fundamental equations have not yet been studied mathematically.

4.2      There is a huge literature, stretching back over a century, on the derivation and dynamics of systems of point vortices. However it is only in the relatively recent past that geometric and symmetry techniques have been applied systematically, resulting in existence, stability and bifurcation theorems for REs, descriptions of dynamics for small numbers of vortices and computations of geometric phases. In this project we will extend the results for REs, develop analogous results for RPOs, investigate non-zero visocity perturbations and use systems of point vortices as test-beds for exploring and testing new ideas and techniques for symmetric chaos.

4.3      The derivation of equations for elastic rod and shell dynamics as limits of 3-D elastic body equations has been approached from several different points of view including asymptotic analysis and Galerkin methods. However only in the past few years has a start been made on a fully Hamiltonian perturbation theory for these and related problems and there have been very few applications of this theory to the stability and dynamics of rod and shells. A study of REs and RPOs within the context of this theory is one of the aims of this project.

4.4      The history of affine (or pseudo) rigid body models of elastic and fluid bodies goes back to the work of Dedekind, Dirichlet and Riemann on self-gravitating fluid masses. Recent work has included RE stability calculations for potentials coming from elasticity theory and Nekhoroshev stability theory for REs. The parallel quantum theory has also been studied, motivated by the possibility that affine rigid bodies may serve as a useful model for atomic nuclei. The network programme includes investigations of: (a) how the global structure of the set of REs depends on the potential energy function; (b) bifurcations of RPOs; (c) the existence of heteroclinic cycles and symmetric chaos; (d) quantum manifestations of classical phenomena; (e)non-holonomic affine rigid bodies.

4.5      Linear instability results for solitons and front solutions in evolutionary Hamiltonian wave equations have been obtained recently by introducing the multi-symplectic Evans function. This project will look at the influence of non-Hamiltonian perturbations on such solutions. It will also look at the application of these theories to pulses in optical fibres, especially the so called `soliton switching' effect in coupled fibres.

Research method

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.


2000-06-22