8.3 Infinitesimal canonical transformations

In an infinitesimal canonical transformation, we have:

\begin{align} Q_i&=q_i+\delta q_i\tag{8.18}\\ P_i&=p_i+\delta p_i \tag{8.19}\end{align}

where the \(\delta q_i\) and \(\delta p_i\) are small changes in the variables that are consistent with a canonical transformation (rather than an arbitrary virtual displacement). This transformation can be related to the identity transformation by:

\begin{align} F=\sum_iq_iP_i+\epsilon G(q_i,P_i) \tag{8.20}\end{align}

where \(G\) is some arbitrary function and \(\epsilon\) is small. The transformation for a canonical transformation of the second type are:

\begin{align} p_i&=\die{F}{q_i}=P_i+\epsilon\die{G}{q_i}\\ Q_i&=\die{F}{P_i}=q_i+\epsilon\die{G}{P_i} \tag{8.21}\end{align}

We can thus identify:

\begin{align} \delta q_i&=\epsilon\die{G}{P_i}\\ \delta p_i&=-\epsilon\die{G}{q_i} \tag{8.22}\end{align}

The first term can be written as:

\begin{align} \delta q_i&=\epsilon\die{G}{P_i}=\epsilon\die{G}{(p_i+\delta p_i)}=\epsilon\die{G}{p_i} \tag{8.23}\end{align}

to first order in \(\epsilon\). That is, for a small value of \(\epsilon\), the derivative with respect to \(P_i\) is almost the same as with respect to \(p_i\). From the properties of Poisson Brackets, this can also be written as:

\begin{align} \delta q_i&=\epsilon\die{G}{p_i}=\epsilon\{q_i,G\}\\ \delta p_i&=-\epsilon\die{G}{q_i}=\epsilon\{p_i,G\} \tag{8.24}\end{align}

if \(G\) is taken to be the Hamiltonian, then we have:

\begin{align} \delta q_i&=\epsilon\{q_i,H\}=\epsilon\frac{dq_i}{dt}\\ \delta p_i&=\epsilon\{q_i,H\}=\epsilon\frac{dp_i}{dt} \tag{8.25}\end{align}

and the displacement, \(\epsilon\frac{dq(p)}{dt}\) are in the direction of time. That is, the Hamiltonian is the generator of the transformation that takes the variable \(q_i\) and \(p_i\) to variables \(Q_i\) and \(P_i\) an infinitesimal time later.