= Solution
A one-degree-of-freedom <canonical transformation> requires the <Poisson bracket> $\{x,p\}_{X,P}=1$. To first order,
$$
\{x,p\}_{X,P}
=(1+2\alpha_1X+2\alpha_2P)(1+2\beta_2X+2\beta_3P)+O(\alpha\beta)
=1+2(\alpha_1+\beta_2)X+2(\alpha_2+\beta_3)P+O(2).
$$
The product of the off-diagonal <derivatives> contributes only at second order. Consequently
$$
\boxed{\beta_2=-\alpha_1,\qquad\beta_3=-\alpha_2,}
$$
with $\alpha_3$ 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.
Back to article page
