 Solution

Let $M_{ij}=\partial y_i/\partial x_j$ be the Jacobian of an invertible change of phase coordinates, and set $K(y,t)=H(x(y),t)$. The chain rule gives ${\nabla }_{x}H=M^T{\nabla }_{y}K$, whenceTherefore the necessary and sufficient condition to retain Hamilton's equations for arbitrary Hamiltonians isequivalently $M^{T}JM=J$. In one degree of freedom this is the Poisson bracket condition $\{Q,P\}=Q_q{P}_p-Q_p{P}_q=1$.

Put $D=p^2+{\alpha }^2{q}^2$. On a smooth branch of the angular coordinate,Consequently $Q_q{P}_p-Q_p{P}_q=(p^2+{\alpha }^2{q}^2)/D=1$. The transformation is canonical for every real $\alpha \ne 0$ on each angular-coordinate chart. The origin is excluded and a single-valued arctangent cannot give a global chart on the punctured plane; using an angle modulo $2\pi$ accounts for the usual branch qualification.