Definition:Canonical Transformation

Definition
Let $\left({x, \mathbf y, \mathbf p, H}\right)$ be canonical variables.

Let $\left({x, \mathbf Y, \mathbf P, H^*}\right)$ be another set of canonical variables.

A mapping between these is a canonical transformation :


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

imply:


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