Explore periodic systems and motion in this comprehensive lecture on Hamiltonian and nonlinear dynamics. Delve into the analysis of time-dependent systems, focusing on periodic time-dependence. Discover parametric resonance through the motion of a pendulum with a vibrating pivot, progressing from simple "square-wave" forcing to more realistic sinusoidal forcing, leading to the Mathieu equation. Investigate the fascinating Kapitza pendulum and learn about vibration-induced stability of the inverted pendulum. Gain insights into resonance tongues of instability, forcing response diagrams, and the geometry of stroboscopic Poincaré maps for forced systems. This in-depth exploration covers essential concepts in dynamical systems, nonlinear dynamics, and mechanics, providing a solid foundation for understanding complex periodic phenomena.