Explore Hamilton-Jacobi theory and its application in finding optimal canonical transformations to solve Hamilton's equations. Delve into the Hamilton-Jacobi equation, a partial differential equation for Hamilton's principal function S, and understand its geometric interpretation. Learn about common simplifications for time-independent Hamiltonians and the relationship between S and the action integral. Apply these concepts to practical examples, including the simple harmonic oscillator and Kepler's two-body problem. Gain insights into solving complex dynamical systems and enhancing your understanding of advanced Hamiltonian mechanics.