Rigidity and Flexibility of Periodic Hamiltonian Flows

An old problem in classical mechanics is the existence of periodic flows within specific classes of Hamiltonian systems such as geodesic and magnetic flows, and central forces. In the last years, interest in this problem has been revitalized since recent research has unveiled a deep relationship between periodic Hamiltonian flows and systolic questions in symplectic and contact geometry. While only trivial examples of periodic flows among magnetic and central systems exist, Zoll and, later, Guillemin have shown that there are many exotic examples among geodesic flows on the two-sphere. Following Guillemin's approach, the goal of this talk is to show how the Nash-Moser implicit function theorem can be used to construct magnetic flows on the two-torus which are periodic for a single value of the energy. This is joint work with Luca Asselle and Massimiliano Berti.