Notes on Hamiltonian dynamical systems A major role in the development of the theory of classical dynamical systems is played by the Hamiltonian formulation of the equations of dynamics. This chapter is intended to provide a basic knowledge of the formalism assuming that the Lagrangian formalism is known. A reader already familiar with the canonical formalism may want to skip the present chapter Starting with the basics of Hamiltonian dynamics and canonical transformations this text follows the historical development of the theory culminating in recent results: the Kolmogorov Arnold Moser theorem Nekhoroshevs theorem and superexponential stability. Its analytic approach allows students to learn about perturbation methods leading to advanced results. Key topics covered include Liouvilles theorem the proof of Poincarés non integrability theorem and the nonlinear dynamics in the neighbourhood of equilibria. The theorem of Kolmogorov on persistence of invariant tori and the theory of exponential stability of Nekhoroshev are proved via constructive algorithms based on the Lie series method. A final chapter is devoted to the discovery of chaos by Poincaré and its relations with integrability also including recent results on superexponential stability. Written in an accessible self contained way with few prerequisites this book can serve as an introductory text for senior undergraduate and graduate students