Local calculation of Hofer’s metric and applications to beam physics. (English) Zbl 1192.70022

Erdelyi, Bela

J. Geom. Symmetry Phys. 9, 9-31 (2007).

Throughout this paper, the author uses methods of symplectic geometry and topology. The unifying concept behind these methods is the flows of Hamiltonian systems. The systems of interest, e.g. particle accelerators and other beam physics systems, can be often modeled as Hamiltonian systems, and the time evolution of these systems (i.e. the flows, or the orbits) can be considered as curves in the space of Hamiltonian symplectic maps. The geometric properties of these curves with respect to Hofer’s metric are deeply related to the Hamiltonian dynamics, and in the first part of the paper the author develops several aspects of this relationship.
The Hofer’s metric is a very interesting way of measuring distances between compactly supported Hamiltonian symplectic maps. Unfortunately, it is not known how to compute it in general, for example for symplectic maps far away from each other. But it is known that the Hofer’s metric is locally flat, and it can be computed by the so-called oscillation norm of the difference between Poincaré generating functions of symplectic maps close to identity. Here the author applies the local flatness of Hofer’s metric to the solution of an optimal symplectic approximation problem in Hamiltonian nonlinear dynamics. Some applications to beam physics are outlined.

Reviewer: Marian Ioan Munteanu (Iaşi)

MSC:

70G45	Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70H15	Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics
53Z05	Applications of differential geometry to physics
78A35	Motion of charged particles

Keywords:

Hamiltonian dynamics; symplectic map; flow; flatness