Definition:Canonical Transformation

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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 if and only if:

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

imply:

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


Sources