Canonical coordinates

Remarks

Equivalently, the Poisson brackets of the coordinate functions are $$\{q_i, q_j\} = 0, \qquad \{p_i, p_j\} = 0, \qquad \{q_i, p_j\} = \delta_{ij}.$$ Some authors use the opposite sign convention $\omega = \sum_i dq_i \wedge dp_i$; the notion of canonical coordinates is the same up to this convention.

Example: on $M = \R \times (0, \infty)$ with $\omega = dp \wedge dq$, put $Q = \log p$ and $P = -q p$. Then $dP \wedge dQ = (-p \, dq - q \, dp) \wedge \frac{dp}{p} = -dq \wedge dp = dp \wedge dq$, so $(Q, P)$ are canonical coordinates.

Non-example: on $\R^2$ with $\omega = dp \wedge dq$, the chart $(Q, P) = (2q, p)$ gives $dP \wedge dQ = 2 \, dp \wedge dq \ne \omega$, so it is not canonical, although it is a valid chart.

Darboux's theorem states that every point of a symplectic manifold has a neighbourhood with canonical coordinates.