MAT 595 Hamiltonian Dynamics and Symplectic Topology

Spring 2009, March 16 - May 6.

Tuesday and Thursday 9:00 -- 10:20 in 341 Altgeld Hall.


Overview of the course

Hamiltonian dynamical systems are the general mathematical framework which describe classical mechanical systems such as a charged particle moving under the influence of an electromagnetic field, or the motion of celestial bodies under their mutual gravitational attraction. The first part of this course will be a survey of Hamiltonian dynamics with an emphasis on the presentation of many examples; from billiards to geodesic flows.

Since energy is conserved in classical mechanical systems, Hamiltonian flows are highly recurrent. A tremendous amount of research has been devoted to the study of the orbits of these flows which are genuinely periodic. This includes a large part of the work of Poincare, and one can trace the roots of a great deal of modern mathematics to the study of these periodic motions. In the second part of the course we will discuss the variational approach to detecting periodic orbits and will survey several of the landmark results concerning their existence.

In recent times, the long-studied search for periodic orbits has been linked to the new and dynamic field of Symplectic Topology. Roughly speaking, this is the study of global phenomena for symplectic manifolds, which are of a similar nature to standard results in differential topology but which can not be established by standard topological methods. One of the fundamental results in this field is Gromov's Nonsqueezing theorem which asserts that there is no diffeomorphism of R^2n which preserves the standard symplectic two-form and maps a ball of radius 1 into a cylinder of the form D^2(r) x R^(2n-2) for r < 1. In the third part of this course, we will present the proof of this remarkable result by Hofer and Zehnder which uses only the variational principle for periodic orbits. Borrowing a poignant phrase of Vladimir Arnold, these orbits act as symplectic ribs which represent the desired obstruction to the symplectic embedding of the unit ball into a thin cylinder.


Prerequisites

There are no formal prerequisites for this mini-course, but basic differential geometry will be helpful. If you have taken Math 518 (formerly Math 520) you should be well equipped for this class.


Office Hours

Wednesday 1:00 to 3:00, or by appointment.


Reference (suggested)

``Symplectic Invariants and Hamiltonian Dynamics" by H. Hofer and E. Zehnder, Birkhauser, 1994.


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This page last modified by Ely Kerman
Friday, 20-Aug-2008 13:12:53 EST
Email corrections and comments to ekerman@math.uiuc.edu