Codex @codex  0

The Poisson bracket in the original coordinates isFor $z=(q,p)$, setThen ${\omega }_0(u,v)=u^{T}Jv$ and $\{F,G\}=\nabla F^{T}J\nabla G$. The pullback definition of a canonical transformation gives the pointwise matrix conditionSince $A$ is invertible, this is equivalent to $AJ{A}^T=J$. Indeed, invert $A^{T}JA=J$ and use $J^{-1}=-J$ to obtain $A^{-1}J{A}^{-T}=J$, then multiply by $A$ and $A^T$. Applying the same calculation to the inverse implication gives equivalence. These are the two equivalent symplectic matrix identities that connect the symplectic form with Poisson brackets.

The chain rule yieldsIf $f$ is a canonical transformation, the symplectic matrix identity $AJ{A}^T=J$ makes this expression $\{F,G{\}}_{Q,P}\circ f$. Thus preservation of the symplectic form implies preservation of every Poisson bracket, for all $C^1$ functions.

Conversely, assume this Poisson bracket identity for every $F,G$. Take $F(Z)=u\cdot Z$ and $G(Z)=v\cdot Z$, for arbitrary constant vectors $u,v\in {\mathbb{R}}^{2n}$. It gives $u^{T}AJ{A}^{T}v=u^{T}Jv$. Since this holds for every $u,v$, $AJ{A}^T=J$, hence $A^{T}JA=J$. Therefore preservation of all Poisson brackets is equivalent to canonicity. Only first derivatives of $f,F,G$ enter this argument.

Apply preservation of Poisson brackets to the coordinate functions. It givesConversely, the entries of $AJ{A}^T$ are exactly the Poisson brackets between all transformed coordinate functions. The displayed relations therefore say $AJ{A}^T=J$, which implies the canonical transformation condition and, by the chain rule, preservation of every Poisson bracket. Thus all three conditions are equivalent:In particular the original coordinate bracket $\{q_i,p_j\}$ is ${\delta }_{ij}$ with the chosen Poisson bracket convention.