I have some questions regarding Canonical transformation and hamiltonian systems.
I will upload an image with the text from wikipedia:
How can I obtain same results... I have no idea how they obtained the last two equations from image.
Can you help me, please?
Thanks
It must be $\dot Q_m = \frac{\partial H}{\partial P_m}$ by definition [Remember that a restricted canonical transformation maps $H(q,p) \mapsto H(Q(q,p), P(q,p))$, i.e. preserves the functional form of the Hamiltonian.]. Subtract the expression of $\frac{\partial H}{\partial P_m}$ as given to the second line for $\dot Q_m$. The result must be zero and this is only possible if the the coefficients of $\partial H / \partial q$ and $\partial H / \partial p$ vanish identically, i.e. iff the relations you are looking for hold. Note that when differentiating with respect $q_j, p_j, Q_m, P_m$, the others $q_k, p_k, Q_l, P_l$ stay fixed.