Past exam of the theoretical physics I course of the University of Cambridge / 2012 / Paper 1 / 2 / a

A one-degree-of-freedom canonical transformation requires the Poisson bracket $\{x,p{\}}_{X,P}=1$. To first order,The product of the off-diagonal derivatives contributes only at second order. Consequentlywith ${\alpha }_3$ and ${\beta }_1$ unconstrained at this order. The self-brackets vanish identically. This is perturbative preservation of the symplectic form, not exact canonicity of the truncated polynomial map.