The Geometry of the Group of Symplectic Diffeomorphism

Leonid Polterovich

ISBN 10: 3764364327 / ISBN 13: 9783764364328

Published by Springer, Basel, Birkhäuser Basel, Birkhäuser Mrz 2001, 2001

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This item is printed on demand - it takes 3-4 days longer - Neuware -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. For a geometer, at least under some assumptions on the manifold M, this is just the connected component of the identity in the group of all symplectic diffeomorphisms. From the viewpoint of mechanics, Ham(M,O) is the group of all admissible motions. What is the minimal amount of energy required in order to generate a given Hamiltonian diffeomorphism I An attempt to formalize and answer this natural question has led H. Hofer [HI] (1990) to a remarkable discovery. It turns out that the solution of this variational problem can be interpreted as a geometric quantity, namely as the distance between I and the identity transformation. Moreover this distance is associated to a canonical biinvariant metric on Ham(M, 0). Since Hofer's work this new ge ometry has been intensively studied in the framework of modern symplectic topology. In the present book I will describe some of these developments. Hofer's geometry enables us to study various notions and problems which come from the familiar finite dimensional geometry in the context of the group of Hamiltonian diffeomorphisms. 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. 136 pp. Englisch. Seller Inventory # 9783764364328

Bibliographic Details

Title: The Geometry of the Group of Symplectic ...

Publisher: Springer, Basel, Birkhäuser Basel, Birkhäuser Mrz 2001

Publication Date: 2001

Binding: Taschenbuch

Condition: Neu

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Synopsis:

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. For a geometer, at least under some assumptions on the manifold M, this is just the connected component of the identity in the group of all symplectic diffeomorphisms. From the viewpoint of mechanics, Ham(M,O) is the group of all admissible motions. What is the minimal amount of energy required in order to generate a given Hamiltonian diffeomorphism I? An attempt to formalize and answer this natural question has led H. Hofer [HI] (1990) to a remarkable discovery. It turns out that the solution of this variational problem can be interpreted as a geometric quantity, namely as the distance between I and the identity transformation. Moreover this distance is associated to a canonical biinvariant metric on Ham(M, 0). Since Hofer's work this new ge­ ometry has been intensively studied in the framework of modern symplectic topology. In the present book I will describe some of these developments. Hofer's geometry enables us to study various notions and problems which come from the familiar finite dimensional geometry in the context of the group of Hamiltonian diffeomorphisms. 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.

Product 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. This 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.