Hamiltonian Dynamics
6-10 November 2006
DynamIC: Dynamical Systems at Imperial College
Organiser: Jeroen Lamb

This session of the UK Dynamical Systems Graduate School on Hamiltonian Dynamics is designed to provide an introduction to the subject, as well as highlighting some more advanced topics. It is meant to be accessible to PhD students of all levels and thus has no non-standard pre-requisites. In line with the general philosophy of the Graduate School it's aim is primarily to present an introduction to the main ideas and results so that post-graduate students in all areas of dynamics have an opportunity to become familiar with such notions and become aware of the central definition and open problems. It is not therefore directed specifically at students already working in topics related to those discussed in the lectures. Quite the opposite ! On the other side, lectures may present topics in a way which are likely to be new even to students working in related areas.

Precise schedule of the lectures will be posted here closer to the time. For information on options for accomodation, please follow this link

Geometry and dynamics of perturbed harmonic oscillators
Richard Cushman (University of Calgary)

The course centers around the problem of finding and analyzing integrable Hamiltonian systems coming from Hamiltonian perturbations of the harmonic oscillator, using the techniques of normal form and reduction. Such systems include perturbations of the three dimensional Kepler problem, which of course there are a legion. These integrable systems typically have a complicated toral geometry, i.e. monodromy. There is even a form of monodromy for the unbounded states.

Exponentially small phenomena in Hamiltonian systems
Vassili Gelfreich (University of Warwick)

This is an introduction to the theory of exponentially small phenomena in Hamiltonian systems. In analytic theory, an analytic perturbation often causes qualitative changes in Dynamics, which are exponentially small compared to the size of the perturbation. In particular we will study dynamics near a resonant torus in a near-integrable Hamiltonian system and bifurcations at strongly resonant periodic orbits. We will also see that exponentially small phenomena are responsible for exponentially small speed of Arnold diffusion.

Kolmogorov-Arnold-Moser (KAM) theory
Bob Rink (Imperial College London)

Existence proofs and continuation results for periodic solutions in systems of ordinary differential equations are often in one way or another an application of the implicit function theorem. Quasi-periodic solutions generalize the notion of periodic solutions and in physics these solutions occur for instance in integrable Hamiltonian systems and their perturbations. A famous example from astronomy is the restricted three-body problem that I hope to discuss during these lectures: the existence of quasi-periodic solutions in this dynamical system enables one to prove the stability of the famous Lagrangean equilibria. The main difficulty of quasi-periodic solutions has to do with the fact that the functional-analytic problem one wants to solve to prove their existence, usually exibits the problem of "small devisors". This really just means that the derivative of a certain infinite-dimensional operator has arbitrarily small eigenvalues, so that the assumptions of the ordinary implicit function theorem fail to hold. In the case that the frequencies of the quasi-periodic solution are very irrational (we say that they are "Diophantine"), this problem can be sometimes be overcome though by an ingenious construction similar to Newton's quadratic iteration scheme. This is the topic of Kolmogorov-Arnol'd-Moser (KAM) theory. The main purpose of these lectures is to understand this construction and the conditions under which it works.