Vortrag: "Autonomous Tonelli Hamiltonian Systems on twisted cotangent bundles"

Abstract: Tonelli Hamiltonian systems play an important role at the crossroads between dynamical systems, symplectic geometry, differential geometry and physics, as they can be used to study several problems coming e.g. from hydrodynamics, electromagnetism, nuclear physics, etc. Roughly speaking, they are a natural generalization of geodesic flows and, as such, share some common properties with them (e.g conservation of energy). However, they differ quite remarkably from the latter class of flows for many reasons: The dynamics depends strongly on the energy; in particular its properties change quite drastically when crossing some special energy values. Moreover, there are examples of energy levels without periodic orbits. From a symplectic geometry point of view, their study is made difficult by the fact that energy levels are - in most of the cases - not of contact type. In this talk we quickly recall how such systems are defined and briefly describe their properties, with particular attention to the existence and multiplicity of periodic orbits. If time permits, we will also present some of the open questions in the field.