Explore the intricate dynamics of trajectories near Lagrange points L1, L2, and L3 in the three-body problem through this comprehensive lecture. Delve into the four categories of motion for particles with energy just above the Lagrange point, including Lyapunov orbits, asymptotic trajectories, transit orbits, and non-transit orbits. Examine the projection of these orbits in position space and their appearance in the x-y plane. Learn about the analytical solution in Cartesian coordinates, categorization by x-coordinate motion, and the concept of velocity wedges. Apply this knowledge through MATLAB simulations for the Earth-Moon system, discover methods for finding Lagrange point locations, and explore techniques for entering orbit around L1. Extend your understanding to the Sun-Jupiter system and gain insights into the circular restricted three-body problem's applications in space mission design and dynamical astronomy.