DOCUMENTA MATHEMATICA, Extra Volume ICM II (1998), 401-410

Leonid Polterovich

Title: Geometry on the Group of Hamiltonian Diffeomorphisms

The group of Hamiltonian diffeomorphisms $\Ham(M,\Omega)$ of a symplectic manifold $(M,\Omega)$ 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 isometries of the symplectic structure $\Omega$. From the point of view of mechanics, $\Ham(M,\Omega)$ is the group of all admissible motions. It was discovered by H. Hofer ([H1], 1990) that this group carries a natural Finsler metric with a non-degenerate distance function. Intuitively speaking, the distance between a given Hamiltonian diffeomorphism $f$ and the identity transformation is equal to the minimal amount of energy required in order to generate $f$. This new geometry has been intensively studied for the past 8 years in the framework of modern symplectic topology. It serves as a source of refreshing problems and gives rise to new methods and notions. Also, it opens up the intriguing prospect of using an alternative geometric intuition in Hamiltonian dynamics. In the present note we discuss these developments.

1991 Mathematics Subject Classification: 58Dxx (Primary) 58F05 53C15 (Secondary)

Keywords and Phrases:

Full text: dvi.gz 21 k, dvi 48 k, ps.gz 71 k.