Canonical coordinates

Summary

Canonical coordinates are coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$ on a phase space in which the symplectic form takes its standard form $\sum_i dp_i \wedge dq_i$. In them the Poisson brackets of the coordinates are $\{q_i, p_j\} = \delta_{ij}$ and all others vanish, and the equations of motion take the form of Hamilton's equations. Darboux's theorem says that such coordinates exist near every point of a symplectic manifold. See the definition.