Geometric mechanics and Lagrangian reduction
Author(s)
Ellis, David
Type
Thesis
Abstract
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.
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.
Date Issued
2011-05
Date Awarded
2011-06
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Holm, Darryl
Sponsor
EPSRC
Creator
Ellis, David
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)