10.2: General and Canonical Transformations

In the Hamiltonian approach, we’re in phase space with a coordinate system having positions and momenta on an equal footing. It is therefore possible to think of more general transformations than the point transformation (which was restricted to the position coordinates).

We can have transformations that mix up position and momentum variables:

\begin{equation}
Q_{i}=Q_{i}\left(p_{i}, q_{i}, t\right), \quad P_{i}=P_{i}\left(p_{i}, q_{i}, t\right)
\end{equation}

where \(\begin{equation}
\left(p_{i}, q_{i}\right)
\end{equation}\) means the whole set of the original variables.

In those original variables, the equations of motion had the nice canonical Hamilton form,

\begin{equation}
\dot{q}_{i}= \dfrac{\partial H}{\partial p_{i}}, \quad \dot{p}_{i}=- \dfrac{\partial H }{ \partial q_{i}}
\end{equation}

Things won’t usually be that simple in the new variables, but it does turn out that many of the “natural” transformations that arise in dynamics, such as that corresponding to going forward in time, do preserve the form of Hamilton’s canonical equations, that is to say

\begin{equation}
\dot{Q}_{i}=\partial H^{\prime} / \partial P_{i}, \quad \dot{P}_{i}=-\partial H^{\prime} / \partial Q_{i}, \text { for the new } H^{\prime}(P, Q) .
\end{equation}

A transformation that retains the canonical form of Hamilton’s equations is said to be canonical.

(Jargon note: these transformations are occasionally referred to as contact transformations.)