MASESS II
Lecture Courses
Ratiu & Perlmutter Shkoller Reich
Bridges & Derks Roberts & Wulff Broer & Fassò

Download lecture notes for some courses from here

Phase space reduction and reconstruction
Tudor Ratiu (Lausanne) and Matthew Perlmutter (Lisbon)

Description in preparation

Geometry and analysis in hydrodynamics
Steve Shkoller (UC Davis)

This course will survey recent analytic and geometric developments in fluid dynamics. We shall present a Lagrangian approach to well-posedness theory for the Lagrangian averaged Navier-Stokes (LANS-alpha) and Lagrangian averaged Euler (LAE-alpha) equations, which is based on the development of new subgroups of the volume-preserving diffeomorphism group. We shall also present a complete derivation of these models of fluid turbulence. Additional topics will include a new formulation for two phase Navier-Stokes flow, with moving interface whose motion is by mean curvature. Finally, we shall explain recent developments in the theory of liquid crystals which are Navier-Stokes fluids with polymers; we shall prove that the classical Ericksen-Leslie model of liquid crystals is well-posed.

Numerical Methods for Hamiltonian PDEs
Sebastian Reich (Imperial College)

This series of lectures will start out with a review of standard methods for truncating Hamiltonian PDEs to a finite-dimensional Hamiltonian ODE. We will then move on to discuss recent developments in the area of multi-symplectic integration of dispersive wave equations. The final part of the series will be devoted to numerical methods for the rotating shallow-water equations.

Multi-Symplectic Geometry and Mechanics
Tom Bridges and Gianne Derks (Surrey)

These lectures will introduce the concept of multi-symplecticity and show how it is a useful framework for analyzing nonlinear PDEs.

The concept of multi-symplecticity will be first introduced as a directional concept where it occurs most naturally, leading to a collection of pre-symplectic forms on a given manifold. In this setting much of the theory of finite-dimensional symplectic manifolds can be used as a guide for setting up the framework. The framework will then be related to Lagrangian field theory and the Cartan form, and to higher-order symplectic structures in field theory.

Symmetries in multi-symplectic systems occur often and lead naturally to a generalisation of relative equilibria with interesting consequences and interpretations, as will be shown in the lectures.

In the examples, it will be shown how the multi-symplecticity leads to new results about the bifurcation and stability of patterns. Examples will include:
  • the stability of solitary waves,
  • the stability of periodic patterns and quasiperiodic patterns,
  • the dimension breaking of nonlinear elliptic PDEs.

Stability and Bifurcations of Relative Equilibria
Mark Roberts (Warwick/Surrey) and Claudia Wulff (Berlin/Warwick)

This course will provide an introduction to old and new results on the stability and bifurcations of relative equilibria (invariant group orbits) of symmetric Hamiltonian systems. Topics will include Energy-Casimir methods for equilibria of Poisson systems, Energy-Momentum methods for relative equilibria, the local structure of Hamiltonian vector fields near relative equilibria, and generic bifurcation of families of relative equilibria parametrised by conserved quantities. Particular emphasis will be placed on recent results for noncompact symmetry groups and applications to systems with Euclidean symmetry.

KAM theory and Nekhoroshev stability
Henk Broer (Groningen) and Franceso Fassò (Padova)

Part I: KAM theory
  1. KAM-theory: multiperiodicity in dissipative and conservative systems
  2. Survey of a KAM proof
  3. Survey of applications of KAM theory
Periodicity is a well-known dynamical phenomenon. The problem is whether multiperiodicity can also occur in a visible and persistent way. The statement that it does when the frequencies are rationally independent (quasiperiodicity) is the heart of the Kolmogorov-Arnold-Moser Theory. We describe this theory in various contexts, particularly for dissipative and conservative systems. A proof in one of the simplest cases will be outlined. The course concludes with a survey of applications of KAM theory.

Recommended text: HW Broer, MB Sevryuk, GB Huitema. Quasi-periodicity in Families of Dynamical Systems: Order amidst Chaos. LNM 1645, Springer-Verlag, 1996. [It is recommended that participants buy a copy of this book.]
Part II: Nekhoroshev theory for superintegrable Hamiltonian systems.
Integrable Hamiltonian systems which have ``additional'' integrals of motion -- more integrals of motion than degrees of freedom -- are called superintegrable (or degenerate). Many of the classical integrable systems of mechanics are superintegrable (Kepler, the free rigid body, any system of resonant harmonic oscillators, ...). Superintegrable systems have a rich geometry, which is however hidden if they are regarded as standard completely integrable systems, namely, looking for an invariant Lagrangian fibration. The fact is that there are two invariant fibrations which are naturally defined in the phase space, not just one: the fibration by the (isotropic) invariant tori and its (coisotropic) symplectic ortogonal. Hence, the geometry of a superintegrable system is that of a dual pair. This geometric structure plays a central role for understanding the dynamics of a small perturbation of the system. According to Nekhoroshev theory, (i) the coisotropic fibration remains nearly invariant for times which grow exponentially fast with the inverse of the perturbation parameter, but (ii) motions can depart from the invariant tori on relatively short time scales. This gives rise to the possibility of a certain type of ``slow chaotic'' movements on the symplectic leaves of the base of the fibration by the invariant tori. It is conjectured that this phenomenon may play an important role in the long time dynamics of rigid bodies. The course will introduce the geometric setting and the results of Nekhoroshev theory; examples will come mainly from rigid body dynamics.

See also the list of individual lectures.

James Montaldi
Last modified: Wed Aug 29 16:57:51 BST 2001