Simple Pendulum: Constraints and Equations of Motion

This lecture covers the analysis of a simple pendulum in Cartesian and polar coordinates, discussing constraints, forces, and degrees of freedom. The resolution of the mathematical pendulum equations is explored, highlighting the complexity and inelegance of the process. The lecture also delves into the stability of geometric constraints and equilibrium points. The instructor demonstrates the application of Newton's laws in polar coordinates, leading to the derivation of the harmonic oscillator equation. Various exercises, including a ball rolling on a sphere and a mass on a half-sphere, are solved to illustrate the concepts discussed.