Definition:Canonical Transformation

Definition
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 :


 * $\dfrac {\d y_i} {\d x} = \dfrac {\partial H} {\partial p_i},\quad\dfrac {\d p_i} {\d x}=-\dfrac {\partial H} {\partial y_i}$

imply:


 * $\dfrac {\d Y_i} {\d x}=\dfrac {\partial H^*} {\partial P_i},\quad\dfrac {\d P_i} {\d x}=-\dfrac {\partial H^*} {\partial Y_i}$