Definition:Canonical Transformation

Definition
Let a mapping between canonical variables $\left({ x, \langle y_i \rangle_{1 \le i \le n}, \langle p_i \rangle_{1 \le i \le n}, H } \right) \to \left({ x, \langle Y_i \rangle_{1 \le i \le n}, \langle P_i \rangle_{1 \le i \le n}, H^* } \right)$ be such that equations


 * $ \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}$

Then this a mapping is called a canonical transformation.