Research Interests

Hamiltonian dynamical systems is a modern subject with very old roots. It is located on the intersection between Mathematics and Physics. It is of fundamental importance in modelling natural phenomena, which also explains its paramount histroical role. The following list gives a brief description of my research interests. A list of possible PhD projects is also available.

Topology of Integrable Systems

Doubly Pinched Torus An integrable Hamiltonian System defines a singular foliation. The regular fibres of this foliation are tori. The objective of this research is to understand the geometry and topology of this foliation and in particular its singular fibres. The singular fibres can for example be pinched tori, as illustrated in the figure. This torus has two pinches, corresponding to two equilibrium points of focus-focus type joined by a heteroclinic connection. In the neighborhood of a pinched torus the regular torus bundle is non-trivial, it has monodromy. This has been found in many examples [27,32,36]. Quantisation of these system in the semiclassical limit gives lattices of eigenstates that do not posses a global basis, and hence no global quantum numbers, see below. The computation of the image of the energy-momentum map with all its singularities gives the complete classification of all possible types of motion of an integrable system. It does not, however, contain information about the dynamics on the invariant tori; for this it has to be supplemented by the frequency map:

Singularities of Frequency Maps

Resonances In KAM theory two conditions guarantee the persistence of invariant tori under perturbations: A Diophantine condition on the frequencies and a nondegeneracy condition ensuring that the frequency map is regular. If the latter condition is violated we have vanishing twist. If this occurs at a fixed point we can think of this as vanishing anharmonicity of a non-linear oscillator. Vanishing twist can lead to instability and unusual bifurcations. In [20] we have shown that vanishing twist at the origin is generic in one parameter families of Hamiltonian systems with two degrees of freedom. The goal of this research is to understand the dynamical implications of singularities of frequency maps in Hamiltonian systems in general. In [35] we show that near focus-focus points the twist always vanishes. This applies in particular to the Hamiltonian-Hopf Bifurcation [40]. And even in the basic saddle-centre bifurcation of periodic orbits there is vanishing twist [43].

Dynamics of (non-) rigid bodies

Torus 1 Escape Diagram Rigid bodies are fascinating mechanical systems with many applications. Understanding the dynamics of rigid bodies is one of the fundamental objectives of classical mechanics, and there are still many open questions. Since my thesis on the Kovalevskaya top I have been interested in rigid body dynamics. Recently a levitating magnetic top was invented and in [16] we determined its stability. In [34] we investigate the Nekhoroshev stability of the Levitron. The numerical integration of the Levitron is best performed using a Poisson integrator [39]. This works for a wider class of coupled symmetric tops. Using this integrator the long term stability near the indefinite elliptic relative equilibrium can be investigated, which leads to the pictures on the right.

Using modern methods from geometric mechanics we are now in position to obtain a good understanding of geometric phases in the dynamics of non-rigid bodies. This can explain how, e.g., a diver can perform a twist without angular momentum around the corresponding axis, or how a cat always lands on its feet, turning in free fall. This is a new project to be expanded in the future.

Torus 2 Torus 3 Torus 4 Torus 5 The Kovalevskaya top is one of the three integrable cases of the dynamics of a rigid body with a fixed point. It is one of the most complicated integrable systems with a few degrees of freedom. In my thesis I studied the action variables of this systems. In this process we produced a movie about the Kovalevskaya top together with the Institut für den Wissenschaftlichen Film (IWF), Göttingen. Above on the left are some stills from it showing projections of invariant tori, the complete movie is now available online (german version) from the German National Library of Science and Technology (TIB). More information is on my web-page on the Kovalevskaya top.

Quantum Monodromy

Sol flower Many fundamental models in quantum mechanics are obtained from classically integrable systems. The quantization of integrable systems leads to the study of the joint spectrum of commuting operators. This joint spectrum is locally a lattice, but only recently has it been realized that globally this need not be true. This phenomenon is called Quantum Monodromy, and it leads to the non-existence of global quantum numbers. The underlying classical system exhibits Hamiltonian Monodromy and its action variables are multivalued. We have shown that Quantum Monodromy occurs in physically relevant systems [27,30,37] and calculated their spectra. Recently we have constructed examples that allow for arbitrary monodromy matrices, answering one of the outstanding questions in the field [44]. The picture on the left is a lattice with monodromy given by the cat map obtained from this example.

(Integrable) Water Wave Equations

The KdV equation is the most important shallow water wave model equation. It is obtained from the full Euler equations by an asymptotic expansion. Many equations have been suggested as improvements of the KdV equation. We have shown [26] that the integrable Camassa-Holm equation is a higher order shallow water wave equation, which has advantages over other models. A major tool in this work is a nonlocal normal form transformation for PDEs, which is explored systematically in [33].

Symplectic Maps and the Variational Principle

Horseshoe 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. I have used the variational approach for extensive numerical studies of periodic orbits and their bifurcations [19,22]. We now investigate the construction of symbolic dynamics not only in the ergodic case [12] but also in the most complicated regime of coexisting regular and chaotic motion. The use of high performance computing and specially developed numerical methods now gives the opportunity to obtain a complete overview of periodic orbits for one parameter families of maps. In [25] we have done this for the family of Limaçon billiards. Using numerical continuation we were able to extend symbolic dynamics from a hyperbolic limit into a regime with stable periodic orbits [41]. This lead to a conjecture about the coding of rotational elliptic orbits of the Hénon map.

Semiclassical Quantization

There are two reasons to supplement quantum mechanics with semiclassical methods: Efficiency in the computation of spectra and the hope to gain understanding through the classical point of view. Initially focusing on integrable quantum billiards we have shown [10] how to modify the Berry-Tabor trace formula in the presence of a separatrix, and later extended some of these results to higher dimension [17]. Currently we study a family of (non-integrable) billiards and use minimal action orbits for its quantization.

Collaborators

Students

  • Nathan Duignan
  • Joachim Worthington
  • William Tong (PhD, Sydney, 2015): Coupled Rigid Body Dynamics with Application to Diving
  • Danya Rose (PhD, Sydney, 2015): Geometric phase and periodic orbits of the equal-mass, planar three-body problem with vanishing angular momentum
  • George Papadopoulos (MSc, Sydney, 2013): Semi-global symplectic invariants of the Euler top
  • Sven Schmidt (PhD, Loughborough, 2008): Resonant Elliptic Equilibria in Hamiltonian Systems
  • Chris Davison (PhD, Loughborough, 2006): Geodesic Flow on the Ellipsoid with Equal Semi-Axes