GCM26: Solving the Anharmonic Oscillator using Canonical Transformations

I introduce the important phase space transformations known as canonical transformations (or, amongst mathematicians, symplectomorphisms). These CTs have special properties: they leave the new action invariant, and the form of Hamilton's equations, with respect to a new Hamiltonian (known as the Kamiltonian) is unchanged. We show that CTs form a group, and talk about the Poisson Bracket formulation of Hamilton's equations, as well as the Maxwell-type relations which are the necessary and sufficient condition that any transformation be canonical. Finally, we apply the generating function of the second kind to solve the anharmonic oscillator without solving any nonlinear differential equations, but rather by just some bootstrap tricks and approximations.