= Canonical completion of independent commuting functions
{title2=$\{Q_i,Q_j\}=0$}
Independent functions $Q_1,\ldots,Q_n$ on a $2n$-dimensional <symplectic manifold> can locally form the position half of canonical coordinates exactly when $\{Q_i,Q_j\}=0$ throughout a neighbourhood. Their commuting <Hamiltonian vector fields> span the fibres of $Q$. Flow from a transverse section gives coordinates $(Q,t)$ with $\omega=\sum dt_i\wedge dQ_i+\beta(Q)$. Write $\beta=d\alpha$ locally and take $P_i=-t_i-\alpha_i(Q)$. Then $\omega=\sum dQ_i\wedge dP_i$.
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