PhD projects in Geometry and Mechanics
The main focus in this area is the application of geometry and topology to Hamiltonian mechanics and its manifestations in areas such as molecular dynamics and spacecraft control. Topics range from aspects of symplectic and multi-symplectic geometry, via their applications to the theory and numerical computation of Hamiltonian ordinary and partial differential equations, through to studies of particular systems. Examples of applications include molecular and gravitational N-body problems, space and underwater vehicles, and optical fibres. The dynamics and control of spacecraft is studied in interdisciplinary projects with the Surrey Space Centre. More details, including staff members working in this area, can be found on the geometry and mechanics pages.
Below are some examples of PhD projects in geometry and mechanics. Alternative projects might be formulated following discussions with individual staff members, just contact the staff member or PhD admissions tutor Dr Gianne Derks.
Soliton Switching in Fibres (Supervisor: Dr Gianne Derks)
For the optical transmission of data across a cable, one can use two (or more) coupled fibre cables. Experiments have shown that if a certain type of signal is put at one end of the cable, it will go to the other end of this cable and hardly anything happens in the other cable. However, if one puts other types of signals on the cable, the signal will switch to the other cable. This gives a convenient way of sending data consisting of zeros and ones. In this project we will aim for a better understanding of this experimentally observed process by investigating the family of soliton-like solutions, especially issues like existence, stability, bifurcations and invariant manifolds will be investigated.
Methods for Obtaining Discrete Symmetries (Supervisor: Prof Peter Hydon)
Differential equations are widely used as models of real systems. To understand the behaviour of a system, it is usually necessary to find the symmetries of the model differential equation. Recent work at Surrey has led to the first systematic technique for obtaining discrete symmetries. The purpose of this project is to simplify and extend the technique, and to create a Maple package that will systematically obtain the discrete symmetries of a given differential equation. This project combines differential equation techniques, linear algebra, group theory and Maple programming. It is expected that the resulting package will be distributed with future releases of Maple, and will be used by those working in many areas of mathematics and theoretical physics.
Theory and numerics of reversing symmetry breaking bifurcations of Hamiltonian relative periodic orbits (Supervisor: Dr Claudia Wulff)
Relative periodic orbits (RPOs) of symmetric dynamical systems are
periodic orbits for the symmetry reduced system. Both the theory and
numerics of periodic orbits of general dynamical systems is
well-developed. But additional structure such as time-reversing and
preserving symmetries and symplecticity changes the generic behaviour
of dynamical systems dramatically. Recently there has been a lot of
progress in the development of a bifurcation theory for symmetric and
Hamiltonian systems, but the theory is far from being complete. In
particular the bifurcation theory of reversible symmetric periodic and
relative periodic orbits, is still in its beginning, as is the numerical
analysis of symmmetry breaking bifurcations of reversible periodic
orbits. The topic of this project is to analyze reversing symmetry breaking
bifurcations of RPOs theoretically and to derive and implement
numerical methods for their detection and computation within the
package SYMPERCON of Wulff, Schebesch, Schilder.
The results will be applied to various symmetric Hamiltonian
systems, in particular to N-body systems.
Multi-symplectic structures and nonlinear PDEs (Supervisor: Prof Thomas J. Bridges)
An understanding of nonlinear PDEs is one of the great challenges for
the twenty-first century. In this project the geometry of nonlinear
PDEs is studied. Multi-symplectic structure, which generalises
classical symplectic geometry, is used as a backbone for the analysis.
The project has three parts:
(I) Develop the geometry of the total exterior algebra bundle for smooth two-dimensional manifolds and construct Dirac operators on this bundle as a basis for canonical hyperbolic and elliptic PDEs.
(II) Use the calculus of variations to find solutions of multi-symplectic Hamiltonian PDEs, using the Mountain Pass Theorem and related functional analytical tools.
(III) Apply numerical methods to either part (I) or part (II).
