Canonical Transformations for Hamiltonian Systems

These notes are my own summary of Chapter 1.3 of Notes on Dynamical Systems by Moser and Zehnder

Hamiltonian Systems

In this section we consider Hamiltonian systems on the phase space $\mathbb{R}^{2n}$. We denote the variables of the phase space as $(\mathbf{q}, \mathbf{p})$, where $\mathbf{q} \in \mathbb{R}^n$ is the “position” variable and $\mathbf{p} \in \R^{2n}$ is the “momentum” variables. Every smooth function $H : \mathbb{R}^{2n} \to \mathbb{R}$ defines a vector field on the phase space via the

Canonical Transformations

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 $g, h \in G$.