The Geometry of the Group of Symplectic Diffeomorphism (Lectures in Mathematics. ETH Zürich)
Edition: 1 edition
Publisher: Birkhäuser Basel
Published: April 20, 2001
Language: eng
Format: Paperback
Number of pages: 133
ISBN-13: 9783764364328
Subjects: Classical mechanics; Differential & Riemannian geometry; Differential Geometry; Global Analysis; Mathematics; Science/Mathematics; Geometry - Differential; Geometry; Mathematics / General; Functional Analysis
First sentence: In this chapter we sum up some well-known preliminary facts about the group of Hamiltonian diffeomorphisms.
Description: The group of symplectic diffeomorphisms of a symplectic manifold plays a fundamental role both in geometry and classical mechanics. What is the minimal amount of energy required in order to generate a given mechanical motion? This variational problem admits an interpretation in terms of a remarkable geometry on the group discovered by Hofer in 1990. Hofer's geometry serves as a source of interesting problems and gives rise to new methods and notions which extend significantly our vision of the symplectic world. In the past decade this new geometry has been intensively studied in the framework of symplectic topology with the use of modern techniques such as Gromov's theory of pseudo-holomorphic curves, Floer homology and Guillemin-Sternberg-Lerman theory of symplectic connections. Furthermore, it opens up the intriguing prospect of using an alternative geometric intuition in dynamics. The book provides an essentially self-contained introduction into these developments and includes recent results on diameter, geodesics and growth of one-parameter subgroups in Hofer's geometry, as well as applications to dynamics and ergodic theory. It is addressed to researchers and students from the graduate level onwards.