Canonical Transformations for Hamiltonian Systems
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 $g, h \in G$.