Attempts to explain and predict the dynamics of physical systems which can simultaneously rotate in space and undergo shape deformations have had a long and illustrious history. Indeed they have driven much of the development of theoretical mechanics. Early highlights include the work of Kepler, Newton, Lagrange, Jacobi, Dirichlet, Dedekind, Riemann and Poincaré (among many others) on the N-body problems of celestial mechanics and on rotating self-gravitating fluid masses. The continuous rotational and translational symmetry groups of these systems play central roles in both their behaviour and the methods used to analyse them. Early work by Jacobi on the `elimination of nodes' in N-body problems is now recognized as a forerunner of the method of phase space reduction. Noether's Theorem, which associates conserved quantities to continuous symmetries, is of fundamental importance in many physical applications. More recently, in the 20th century, the application of theoretical mechanics to the study of molecular and atomic spectra has also highlighted the importance of discrete symmetries, corresponding to permutations of identical particles, inversion in space and time-reversal, and the role that group representation theory plays in describing these.
Over the last two decades Noether's Theorem and the principle of phase space reduction have been generalised and explored in great depth in classical mechanics within the frameworks of differential and symplectic geometry. The result is a very general and powerful set of ideas and techniques that can be applied to a wide range of problems from particle and continuum mechanics. Successes have included detailed studies of the highly nontrivial stability properties of relative equilibria (REs), theories for `geometric phases' associated to reduction and the development of new theoretical concepts and practical techniques for the control of mechanical systems.
Despite these major successes, relatively little effort has been put into understanding more complex dynamics of mechanical systems with symmetry. For example there has been very little research on the existence and stability of relative periodic orbits (RPOs) and virtually no research on symmetric chaos in Hamiltonian systems. One of the central beliefs underlying this project is that major breakthroughs can be achieved by combining highly successful ideas and techniques from `geometric mechanics' with those from the theory of general equivariant dynamical systems. For such systems symmetries do not imply the existence of conserved quantities which constrain the dynamics and because of this they are intrinsically simpler to work with and a formidable array of techniques has been developed. These techniques have been tremendously successful in applications to symmetry breaking bifurcations, pattern formation and the chaotic behaviour of symmetric systems. In particular recent work on bifurcations from relative equilibria and relative periodic orbits, on skew-product flows and on dynamics near heteroclinic cycles suggests that there are many equally interesting dynamical phenomena waiting to be discovered in the Hamiltonian context, which in turn will have important implications for mechanical systems.
Investigation of these phenomena in symmetric Hamiltonian systems is of more than academic interest. It is of fundamental importance to our understanding of, for example, atomic and molecular spectra. Modern experimental techniques are now able to detect very detailed structure in these spectra and reveal regularities and patterns that call for theoretical explanations. These spectra can be very accurately reproduced by model quantum Hamiltonians, but such quantum models do not explain the patterns that are typically observed. Deeper understanding can be obtained by studying associated classical Hamiltonians and then applying the quantum-classical correspondence principle. There have been a number of notable recent successes in this direction, but many qualitative universal phenomena in quantum systems related to singularities of classical systems (such as the existence and behaviour of band structures) are still not well understood.
Other areas of fundamental importance include the dynamics of fluids, elastic bodies, optical media and other continuum systems. Modern geometric methods dovetail perfectly with classical analytical techniques to give new insights into the dynamics of such systems. These in turn have important consequences for areas such as climatology and weather prediction. However recent stability and bifurcation studies (of planar free boundary liquid drops and elliptical and ellipsoidal motions of gases, for example) have revealed a wealth of new phenomena that are still poorly understood.
The central aim of this project is to develop new theoretical and numerical tools that can be used to predict, describe and explain the classical dynamics of mechanical systems with symmetries. These will be applied to the classical dynamics of atoms and molecules, the motion of fluids and elastic bodies and the stability and bifurcation of optical pulses. For atoms and molecules the quantum-classical correspondence principle will be used to translate the classical behaviour into predictions and explanations of their quantum mechanical spectra. Conversely studies of these examples will be used to drive the development of new theoretical and numerical techniques that can be applied to a much wider range of systems and which could lead to fundamental breakthroughs in a number of different areas of mathematics, physics and chemistry.