Canonical coordinates

In mathematics and classical mechanics, canonical coordinates are sets of coordinates which can be used to describe a physical system at any given point in time (locating the system within phase space). Canonical coordinates are used in the Hamiltonian formulation of classical mechanics. A closely related concept also appears in quantum mechanics; see the Stone–von Neumann theorem and canonical commutation relations for details.

As Hamiltonian mechanics is generalized by symplectic geometry and canonical transformations are generalized by contact transformations, so the 19th century definition of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition of coordinates on the cotangent bundle of a manifold.

In classical mechanics, canonical coordinates are coordinates ${\displaystyle q_{i}\,}$ and ${\displaystyle p_{i}\,}$ in phase space that are used in the Hamiltonian formalism. The canonical coordinates satisfy the fundamental Poisson bracket relations:

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