Canonical Transformations: Hamilton-Jacobi Equation

This lecture covers canonical transformations, focusing on the Hamilton-Jacobi equation. It explains the concept of canonical transformations, the symplectic group, and the volume element. The lecture delves into the theory of differential equations, emphasizing the importance of partial derivatives and the determination of constants. It also discusses the harmonic oscillator and the ambiguity in the solutions. The instructor demonstrates how to calculate the constants and solve the Hamilton-Jacobi equation. The lecture concludes with examples and applications of canonical transformations in physics.