# University of Toronto's Symplectic Geometry Seminar

March 15, 2004, 2:10 - 3 PM

SS5017A

## Sergei Tabachnikov

###
Pennsylvania State University

##
Magnetic flows and magnetic billiards, Finsler metrics
and a magnetic analog of Hilbert's fourth problem

** Abstract: **
The motion of a charged particle in a magnetic field has become a
popular object of study in mathematical physics, dynamical systems,
symplectic and differential geometry. In this talk, magnetic
billiards will be interpreted as billiards in Finsler geometry; this
interpretation has geometric consequences, such as the existence of
magnetic "whispering galleries".
I will also characterize the Finsler metrics in the plane whose
geodesics are circles of a fixed radius. This is a magnetic analog of
Hilbert's Fourth problem asking to describe the Finsler metrics whose
geodesics are straight lines. Both problems have solutions in terms
of special symplectic structures on the spaces of extremals. The
magnetic analog of Hilbert's fourth problem is closely related to
another well studied subject, the Pompeiu problem.