[NOTES/CM-10011] Infinitesimal Canonical Transformation

We can imagine a canonical transformation where the changes in coordinates and momenta are infinitesimal. Thus we can write
\begin{eqnarray}\label{EQ01} Q_k&=& q_k + \Delta q_k \text{ where } \Delta q_k= \epsilon \chi(q,p)\\ P_k&=&p_k + \Delta p_k \text{ where } \Delta p_k = \epsilon\phi(q,p). \end{eqnarray}
where \(\epsilon\) is an infinitesimal constant.It can be proved that such a transformation is canonical if and only if there exists a function \(G(q,p)\) such that
\begin{eqnarray} \chi(q,p) &=& \{q, G(q,p)\}_\text{PB} = \pp[G]{p_k}\\ \phi(q,p) &=& \{p,G(q,p)\}_\text{PB} = -\pp[G]{q_k}. \end{eqnarray}
Thus
\begin{eqnarray}\label{EQ04} \Delta q_k &=& \epsilon \{q, G(q,p)\}_\text{PB},\\ \Delta p_k &=& \epsilon\{p, G(q,p)\}_\text{PB} \label{EQ05}. \end{eqnarray}
and the function \(G(q,p)\) is called generator of infinitesimal canonical transformation \eqref{EQ01}. Under the canonical transformation change in any function \(F(q,p)\) is given by
\begin{eqnarray} \Delta F(q,p) &=& F(q+\Delta q, p+\Delta p)\\ &=& \sum_k\Big(\pp[F(q,p)]{q_k}\Delta q_k + \pp[F(q,p)]{p_k} \Delta p_k\Big)\\ &=& \epsilon\sum_k\Big(\pp[F(q,p)]{q_k}\pp[G]{q_k} - \pp[F(q,p)]{p_k} \pp[G]{p_k}\Big)\\ &=& \epsilon\{F(q,p), G(q,p)\}_\text{PB}\label{EQ10} \end{eqnarray}