Conservative Dynamical Systems

We introduce and study major results in conservative dynamical systems such as Hamiltonian systems and twist maps. We will mainly focus on the existence of invariant curves and the dynamical behaviour close to them. Through Arnold-Liouville integrability and Aubry-Mather sets we will first study the existence of these curves in different systems. After that, we will consider what happen when we perturb an integrable system: which invariant curves remain (KAM and weak KAM theory)? How close does the dynamics stay close to an integrable curve after the perturbation (Arnold diffusion, Nekhoroshev estimates)? References - Mathematical methods of classical mechanics, V.I. Arnold - Introduction to the modern theory of dynamical systems, A. Katok and B. Hasselblatt - The principle of least action in geometry and dynamics, K.F. Siburg