[NOTES/CM-10004] Infinitesimal Canonical Transformations --- Examples

Space translations
Under translations for a system of particles we have
\begin{equation}
\vec{r}_\alpha \longrightarrow \vec{r}_\alpha^\prime
= \vec{r}_\alpha + \vec{\epsilon}, \qquad \vec{p}_\alpha
\longrightarrow \vec{p}_\alpha^\prime =\vec{p}_\alpha
\end{equation}
It is easily seen that the above equations define a canonical transformation. The generator of infinitesimal transformation corresponding to the above transformation is total momentum \(\vec{P}= \sum_\alpha \vec{p}_\alpha\).

Rotations
Under an infinitesimal rotation, angle \(\Delta \theta\), about axis given by unit vector \(\hat{n}\) the changes in position and momenta are given by
\begin{eqnarray} \Delta \vec{r} &=& - \Delta \theta (\hat{n}\times \vec{r})\\ \Delta \vec{p} &=& - \Delta \theta (\hat{n}\times \vec{p}) \end{eqnarray}
The corresponding constant of motion is \(\vec{L} = \vec{r}\times \vec{p}\).
Starting with
\begin{eqnarray} \Delta \vec{r} &=& - \Delta \theta (\hat{n}\times \vec{r})\\ \Delta \vec{p} &=& - \Delta \theta (\hat{n}\times \vec{p}) \end{eqnarray}
Computing the Poisson brackets with \(\vec{r}\) and with \(\vec{p}\) we get
\begin{eqnarray} \Delta \vec{r} = -\Delta \theta (\hat{n}\times \vec{r}),\qquad \Delta \vec{p} = - \Delta \theta (\hat{n}\times \vec{p}) \end{eqnarray}
These equations correspond to \eqref{EQ04} and \eqref{EQ05}. All the above statements trivially generalize to system of many particles.

Time Evolution as Canonical Transformation
The change in a function \(F(q,p)\) in time \(\Delta t\) is given by
\begin{eqnarray} \Delta F &=&F(q+\Delta q, p + \Delta p) - F(q,p)\\ &=& \{F, H\}_\text{PB} \Delta t. \end{eqnarray} Comparing the above equation with \eqref{EQ10}, we see that the time evolution is a canonical transformation with the Hamiltonian as the generator of infinitesimal time translation.