Hamilton's equations for a system with $n$ degrees of freedom can be written in vector form as

$$\dot{\mathbf{x}}=J \frac{\partial H}{\partial \mathbf{x}}$$

where $\mathbf{x}=\left(q_{1}, \ldots, q_{n}, p_{1}, \ldots, p_{n}\right)^{T}$ is a $2 n$-vector and the $2 n \times 2 n$ matrix $J$ takes the form

$$J=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right)$$

where 1 is the $n \times n$ identity matrix. Derive the condition for a transformation of the form $x_{i} \rightarrow y_{i}(\mathbf{x})$ to be canonical. For a system with a single degree of freedom, show that the following transformation is canonical for all nonzero values of $\alpha$ :

$$Q=\tan ^{-1}\left(\frac{\alpha q}{p}\right), \quad P=\frac{1}{2}\left(\alpha q^{2}+\frac{p^{2}}{\alpha}\right)$$