If you are interested in a PhD in Hamiltonian Dynamical Systems 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.
The gravitational 3-body problem is arguably the most important problem in dynamical systems, and it is far from being understood. There are many possible projects: 1) The computation of normally hyperbolic invariant manifolds and their use in space flight. 2) The algebraic computation of Birkhoff normal form at relative equilibria. 3) The description of the dynamics as a billiard using discrete Lagrangian dynamics. 4) The study of singularities of the Lie-Poisson structure obtained from reduction using invariants. 5) The description of the reduced invariant manifolds and their bifurcations in four spatial dimensions. 6) The numerical continuation of periodic orbits from various limiting cases, specifically the restricted 3-body problem or the problem of 2 fixed centres.
A levitating magnetic top (the Levitron) was invented in 1983 and in  we gave the first complete analysis determining 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 geometry of its momentum map, and also the influence of dissipation and radiation. This projects aims to answer some of these questions using modern tools from geometric mechanics.
The dynamics of non-ridig bodies (e.g. the human body) 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. The key concept is the so-called geometric phase which appears when reconstructing symmetry reduced dynamics. William Tong's 2016 PhD developed these ideas for springboard and platform diving. This interdisciplinary project aims to extend and apply these ideas to aerial skiing in collaboration with the New South Wales Institute 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 still not well understood. Using a combination of numerical and analytical tools the aim of this project is to obtain a qualitative and quantitative understanding of (periodic) orbits in the neighborhood of an elliptic fixed point.