Generating Function of a Canonical Transformation - Examples and the Big Picture

Explore the concept of generating functions for canonical transformations in this comprehensive lecture from a course on Hamiltonian and nonlinear dynamics. Delve into the construction of canonical transformations, their relationship to gauge invariance in action integrals, and their central role in studying Hamiltonian systems. Examine practical examples, including the harmonic oscillator and near-identity transformations related to Hamiltonian flow maps. Gain insights into Hamilton's canonical equations derived from the principle of least action, various types of generating functions, and the big picture of canonical transformations in advanced dynamics.

Summary so far.
Hamilton's canonical equations from the principal of least action.
Generating function approach to canonical transformations.
Harmonic oscillator example.
Aside: photon energy and momentum looks like harmonic oscillator in quantum mechanics.
Different kinds of generating functions.
Near-identity transformations and flow map of Hamilton's equations.
Summary / big picture of canonical transformations .