R. Cushman and L. Bates. Birkhauser, 1997 (ISBN -3-7643-5485-2)
| These lectures represent a crash course in geometric mechanics. I will start from scratch with the definition of Hamilton's equations on manifolds, go through elementary symplectic and Poisson geometry, and variational principles and Lagrangian systems. Then the theory of the momentum map will be reviewed, including issues related to non-equivariance. Reduction theory follows, first Lie-Poisson and Euler Poincare reduction, then the general method for any symplectic or Poisson manifold in the regular context. The cotangent bundle will be analyzed separately and the the two main theorems on cotangent bundle reduction explained. A brief review of semidirect product reduction appears an an application. Eventually, we will get to the theory of singular reduction and present at least the statements of the key results. If time permits, various dynamic aspects such as stability, persistence, and bifurcations will also be presented. |
| These lectures will cover the classical results on the bifurcation and stability of periodic solutions of a Hamiltonian system of equations. Normal forms and perturbation arguments will be used for the bifurcation results and Liapunov functions and KAM theory will be used for the stability results. The main example will be the restricted three-body problem. |
| This course will consist of 3 lectures describing the numerical methods used for integrating particular classes of hamiltonian systems. |
| The goal of this course is to give the student a hands on modern treatment of the following classical integrable systems: the 2-dimensional harmonic oscillator, the Euler top, and the spherical pendulum. These systems will be analyzed using the technique of reduction of symmetry. |
| The lectures will be based on the discussion given in the book below. It is strongly suggested that students look at this book before taking the course. There will be exercises handed out whose solution can be carried out using techniques given in the lectures. |
Global aspects of classical integrable systems.
R. Cushman and L. Bates. Birkhauser, 1997 (ISBN
-3-7643-5485-2)
| I will discuss bifurcations of equilibria (and relative equilibria) in Hamiltonian systems, and in particular the effects of symmetry on such bifurcations. The theory will be elucidated by some of my favourite examples. |
We will discuss methods for reducing the number of degrees of
freedom in fluid dynamics models based on approximating
Hamilton's principle. Making approximations in Hamilton's
principle that respect the symmetries of the original Lagrangian
yields simplified dynamical equations whose solutions preserve the
conservation laws and circulation theorems of the original
dynamical system.
This is a powerful and fruitful way of making
simplifications (e.g., coarse graining, WKB
approximations, or collective coordinates) that retains
mathematical structure. We will discuss an ordered
sequence of applications of this procedure to problems
that yield solitons, vortices, defect motion and other
interesting reduced dynamics for simple and complex
fluids.
These applications are relevant in geophysics, material science
and turbulence studies. The examples include
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See also the list of individual lectures.