Groups of Canonical Transformations

These notes are my own summary of Chapter 1.5 of Notes on Dynamical Systems by Moser and Zehnder

Canonical Transformations Indexed by a Group

By now we’ve established that canonical transformations are those change-of-coordinates on $\mathbb{R}^{2n}$ that preserve the Hamiltonian nature of a dynamical system. Often, when we are studying a specific Hamiltonian it will have certain symmetries that implicitly define a group of canonical transformations that let us view the system in many different coordinates. This often leads to a clever way of studying the system, as long as we use the symmetries judiciously.

For now we will simply define what a group of canonical transformations is, and later explain how to find such a group for a given Hamiltonian. As Moser and Zehnder do, we will only consider two very simple types of groups initially:

  • translation groups:
  • matrix groups:

Most of these groups will be Lie groups, and the ones we consider will have the advantage that their Lie algebras are both easy to compute and quite explicit.

Let $G$ be a group, and assume there exists $\psi : G \times \mathbb{R}^{2n} \to \mathbb{R}^{2n}$. Write $\psi^g = \psi(g, \cdot) : \mathbb{R}^{2n} \to \mathbb{R}^{2n}$ for each $g \in G$. The family $\psi^g, g \in G$ is called a group of canonical transformations if

  • each $\psi^g$ is a canonical transformation, and
  • $\psi^{gh} = \psi^g \circ \psi^h$ for any $g, h \in G$.