Geometric mechanics and Lagrangian reduction

The purpose of this thesis is two-fold: Firstly, to contribute to the tools available to geometric mechanics; secondly, to apply the geometric perspective to two particular problems. The thesis falls into three parts. The first part deals with the dynamics of charged molecular strands (CMS). The second part contributes general tools for use in geometric mechanics. The third part develops a new geometric modelling technique and applies it to image dynamics. Part I develops equations of motion for the dynamical folding of CMS (such as DNA). The CMS are modelled as flexible continuous filamentary distributions of interacting rigid charge conformations, and their dynamics are derived via a modified Hamilton-Pontryagin variational formulation. The new feature is the inclusion of nonlocal screened Coulomb interactions, or Lennard-Jones potentials between pairs of charges. The CMS equations are shown to arise from a form of Lagrangian reduction initially developed for complex fluids. Subsequently, the equations are also shown to arise from Lagrange-Poincaré reduction of a field theory. This dual interpretation of the CMS equations motivates the undertakings of Part II. In Part II, a general treatment of Lagrange-Poincaré (LP) reduction theory is undertaken. The LP equations are cast into a field theoretic context together with their associated constrained variational principle. An integrability/reconstruction condition is established that relates solutions of the original problem with those of the reduced problem. The new contribution of the LP framework is to unify the Lagrange-Poincaré field reduction with the canonical theory, which involves a single independent variable, and to extend LP field reduction to the general fibre bundle setting. The Kelvin-Noether theorem is generalised in two new ways; from the Euler- Poincaré to the LP setting, and from the canonical to the field setting. The importance of the extended Kelvin-Noether theorem is elucidated by an application to the CMS problem, yielding new qualitative insight into molecular strand dynamics. Finally, Part III gives a full geometric development of a new technique called un-reduction, that uses the canonical LP reduction back-to-front. Application of un-reduction leads to new developments in image dynamics