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 :


 * $\displaystyle \frac{ \mathrm d y_i }{ \mathrm d x } = \frac{ \partial H }{ \partial p_i}, \quad \frac{ \mathrm d p_i }{ \mathrm d x}= -\frac{ \partial H }{ \partial y_i}$

imply


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