Hamilton-Jacobi Equations: Phase Portrait

This lecture covers the Hamilton-Jacobi equations, focusing on the phase portrait concept and the calculation of the period of movement using action-angle variables. The instructor explains the representation of trajectories in phase space, isolated systems, and the behavior of systems with different energies. The lecture delves into the root existence in the system, type curves for movement, and the characteristics of trajectories. Emphasis is placed on understanding the phase portrait and its significance in determining the period of movement, especially for small oscillations. Various examples and illustrations are provided to clarify the concepts discussed.