Conservation of Mechanical Energy: Stability and Equilibria

This lecture covers the conservation of mechanical energy in systems with conservative forces, demonstrating how the mechanical energy remains constant. It explores the stability of equilibrium points by analyzing the potential energy function and the behavior of systems near stable equilibrium points. The lecture also discusses the approximation of potential energy near stable equilibrium points and the motion of systems in the vicinity of these points. Examples such as the mathematical pendulum and the looping motion are used to illustrate the concepts of mechanical energy conservation and stability. The lecture concludes with a discussion on the visualization of trajectories in phase space and the conditions for non-detachment in dynamic systems.