If you are interested in a PhD then please let me know. My research interests are centered around Hamiltonian Systems. The following are suggestions for possible projects, but I'm open to any of your ideas.
Recently a levitating magnetic top (the Levitron) was invented and in [16] we determined for which spin rates its relative equilibrium is stable. There are still many open questions to be answered about the dynamics of this mechanical system, relating to its bifurcations, the influence of dissipation and radiation, and geometry. This projects aims to answer some of these questions using modern tools from the geometric theory of Hamiltonian Systems, numerical methods, and visualization.
While the Levitron is about the dynamics of a rigid body the human body can change its shape. The dynamics of non-ridig bodies is very interesting and recently a beautiful mathematical theory has been developed in order to explain one basic phenomenon: The change of overall orientation without angular momentum. Divers use a certain technique to perform a twist without angular momentum about the corresponding axis. This theory has not been explored in this context and this interdiciplinary projects aims to study its implications in collaboration with the Sports Science group and the New South Wales Institut of Sports.
Area preserving twist maps are probably the best understood models for Chaos in Hamiltonian systems. Recently the variational methods of Aubry-Mather theory have allowed for substantial progress. However, in four dimensions the dynamics of symplectic maps is not well understood. Using a combination of numerical and analytical tools the aim of this project is to obtain a qualitative understanding of (periodic) orbits in the neighborhood of an elliptic fixed point.
July 2008