Pennsylvania State University
IMA Participating Institutions
Summer Program for Graduate Students on Dynamical Systems & Applications

Talk abstract:

Hamiltonian Dynamics, Variational Principles and Symplectic Invariants

June 7-11, 1999

Helmut H.W. Hofer
Courant Institute
New York University
251 Mercer Street
New York, NY 10011
hofer@CIMS.NYU.EDU
http://www.math.nyu.edu/~hofer

There is a very intricate relationship between (dynamical systems) questions in Hamiltonian dynamics and (geometric) questions in symplectic geometry. For example, the problem of finding periodic orbits on a prescribed energy surface is closely related to the problem of designing an energy-efficient transport for open sets in phase space. Exploring this mysterious connection leads to important concepts One is that of symplectic capacities. These are non-volume related invariants of open sets which are preserved under Hamiltonian flows. The first such invariant was introduced by Gromov in 1985. The second concept is concerned with the notion of energy of a symplectic map. This idea leads to a bi-invariant metric on the (infinite-dimensional) group of compactly supported Hamiltonian diffeomorphisms. Autonomous Hamiltonian flows are geodesics for this metric (but not the unique ones) and periodic orbits are related to conjugate points. Already the mere existence of a bi-invariant metric has surprising consequences. Exploring the relationship between capacity and energy leads to interesting applications to Hamiltonian dynamics. It also stresses the practical use of the variational principles underlying Hamiltonian dynamics.

This course will be based on the Lecture Notes (c) below. As a prerequisite knowledge of the material in b) below, pages 35-66, or the equivalent material in (a) would be very helpful.

Suggested Literature:

a) H. Hofer and E. Zehnder

Symplectic invariants and Hamiltonian dynamics
Birkhuser Advanced Texts.
Birkhuser Verlag, Basel, 1994. xiv+341 pp. ISBN 3-7643-5066-0

b) D. McDuff and D. Salamon

Introduction to Symplectic Topology
Oxford Mathematical Monographs
Oxford University Press, 1995, ISBN: 0 19 851177 9

c) H. Hofer

Hamiltonian Dynamics, Variational Principle and Symplectic Invariants (Lecture Notes) postscript (898kb)    pdf (492kb)

Tentative List of Topics

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