Geometric Constraints and Equilibrium in Mechanics

This lecture explores the concept of geometric constraints and equilibrium in mechanics, using the example of a hemispherical slide. The instructor discusses the forces involved, acceleration in spherical coordinates, and the application of Newton's second law. The stability of equilibrium points is analyzed by perturbing around them, determining stability based on the resulting forces. Different equilibrium positions are identified, and their stability conditions are examined. The lecture concludes with a detailed discussion on stability analysis, highlighting the conditions for stable and unstable equilibrium points, and the concept of an oscillatory motion around equilibrium.