The Geometry of the Group of Symplectic Diffeomorphism - pr_35417

The Geometry of the Group of Symplectic Diffeomorphism

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The group of symplectic diffeomorphisms of a symplectic manifold plays a fundamental role both in geometry and classical mechanics. What is the niminal 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 extensively 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.

Product code: 9783764364328

ISBN 9783764364328
Dimensions (HxWxD in mm) H240xW170
Series Lectures in Mathematics. ETH Zurich
No. Of Pages 136
Publisher Birkhauser Verlag AG
Edition 2001 ed.
The group of Hamiltonian diffeomorphisms Ham(M, 0) of a symplectic mani fold (M, 0) plays a fundamental role both in geometry and classical mechanics. They turn out to be very different from the usual circle of problems considered in symplectic topology and thus extend significantly our vision of the symplectic world.