\begin{definition}[Definition:Canonical Transformation]
Let $\paren{x,\mathbf y,\mathbf p,H}$ be canonical variables.
Let $\paren{x,\mathbf Y,\mathbf P,H^*}$ be another set of canonical variables.
A mapping between these is a '''canonical transformation''' {{iff}}:
:$\dfrac {\d y_i} {\d x} = \dfrac {\partial H} {\partial p_i},\quad\dfrac {\d p_i} {\d x}=-\dfrac {\partial H} {\partial y_i}$
imply:
:$\dfrac {\d Y_i} {\d x}=\dfrac {\partial H^*} {\partial P_i},\quad\dfrac {\d P_i} {\d x}=-\dfrac {\partial H^*} {\partial Y_i}$
\end{definition}