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The Geometry of the Group of Symplectic Diffeomorphism (Paperback, 2001 ed.)
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The Geometry of the Group of Symplectic Diffeomorphism (Paperback, 2001 ed.)
Series: Lectures in Mathematics. ETH Zurich
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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."
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