Canonical coordinates

Canonical coordinates are generalized positions and their conjugate momenta on phase space. They express Hamiltonian mechanics in the standard form of Hamilton’s equations.

Generalized coordinates: Coordinates that describe a system’s configuration using variables adapted to its constraints. They supply the position variables paired with momenta in canonical coordinates.
Hamilton’s equations: First-order differential equations that determine how canonical positions and momenta evolve under a Hamiltonian. Canonical coordinates are defined by their appearance in this standard pair of equations.
Action-angle variables: Canonical coordinates in which integrable Hamiltonian motion is described by constant actions and evolving angles. They are a specialized canonical coordinate system that makes integrable motion transparent.
Generalized velocity: The time derivative of a generalized coordinate describing a system’s motion. Velocities describe motion but do not replace conjugate momenta in Hamilton’s canonical pairs.
Canonical momentum: The momentum conjugate to a generalized coordinate, defined from the system’s Lagrangian. Each canonical position is paired with this variable to form a phase-space coordinate pair.
Hamiltonian vector field: The phase-space vector field generated by a Hamiltonian through the symplectic structure. It packages the motion that Hamilton’s equations express in canonical coordinates.
Darboux theorem: A theorem stating that every symplectic manifold has local coordinates in which its symplectic form is standard. It guarantees local canonical coordinates on a broad class of phase spaces.
Noncanonical coordinates: Phase-space variables whose Poisson brackets do not have the standard canonical form. They can describe Hamiltonian systems, but their equations require nonstandard coordinate brackets.
Phase space: A space whose points specify a system’s complete state through positions and momenta. Canonical coordinates provide a standard coordinate system for this state space.
Hamilton–Jacobi equation: A first-order nonlinear partial differential equation for Hamilton’s principal function. Its solutions can generate canonical coordinates in which dynamics becomes simpler.

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