A twist condition for magnetic flows on the two-sphere.
The magnetic flow on the two-dimensional sphere \(S^2\) is determined by a Riemannian metric \(g\) and a two-form \(\sigma\), each on \(S^2\). The triple \((S^2,g,\sigma)\) is called a magnetic system. The goal of this thesis is to find a condition on the magnetic system so that the magnetic flow has infinitely many periodic orbits. We will use existing results that allow us to view the problem in a contact geometric context where the dynamics of the Reeb flow corresponds to the dynamics of the magnetic flow on different kinetic energy level sets. We will then construct a global surface of section that is an annulus and find a condition for which the first return map of the surface of section satisfies the requirements of the Poincaré-Birkhoff Theorem which then implies that the first return map has infinitely many fixed points and the Reeb flow therefore has an infinite number of periodic Reeb orbits.Back to list
Author : Raphael Schlarb