Define what it means for the transformation $\mathbb{R}^{2 n} \rightarrow \mathbb{R}^{2 n}$ given by

$$\left(q_{i}, p_{i}\right) \mapsto\left(Q_{i}\left(q_{j}, p_{j}\right), P_{i}\left(q_{j}, p_{j}\right)\right), \quad i, j=1, \ldots, n$$

to be canonical. Show that a transformation is canonical if and only if

$$\left\{Q_{i}, Q_{j}\right\}=0, \quad\left\{P_{i}, P_{j}\right\}=0, \quad\left\{Q_{i}, P_{j}\right\}=\delta_{i j}$$

Show that the transformation $\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by

$$Q=q \cos \epsilon-p \sin \epsilon, \quad P=q \sin \epsilon+p \cos \epsilon$$

is canonical for any real constant $\epsilon$. Find the corresponding generating function.