Explore integrable and non-integrable Hamiltonian systems, KAM tori, Poincare sections, and Poisson brackets in this advanced dynamics lecture. Delve into the complexities of Hamiltonian and nonlinear dynamics, examining the KAM theorem, Poincare maps, and Poisson systems. Gain insights into the behavior of dynamical systems, from integrable cases to chaotic scenarios, and understand the fundamental concepts that underpin modern approaches to analyzing complex mechanical systems.