Analytical Mechanics

OVERVIEW OF CHAPTER 6

Canonical transformations are transformations from one set of canonically conjugate variables q, p to another conjugate set Q, P. A transformation is said to be canonical if, after the transformation, Hamilton's equations are still the correct dynamical equations for the time development of the new variables. The new Hamiltonian may look quite different from the old one. It may prove easier to solve the EOM in terms of the new variables Q, P. The concept of a generating function is introduced, which gives an “automatic” method for producing canonical transformations. There are four types of generating functions for canonical transformations. It will be explained how these different generating functions are connected by Legendre transformations.

Poisson brackets will be introduced, which are invariant under canonical transformations. If Hamilton's dynamics is formulated in terms of Poisson brackets, we have a coordinate-free way to express the equations of motion. The close resemblance of Poisson brackets used in classical mechanics to commutators of operators in quantum mechanics is not an accident, since Poisson brackets played a fundamental role in the invention of quantum mechanics.

We proceed from the general notion of a generating function to the special generating function S. which produces a canonical transformation leading to the Hamilton–Jacobi equation. The Hamilton–Jacobi equation leads to a geometric picture of dynamics relating the dynamics to wave motion. […]