Poisson bracket

A bilinear, antisymmetric operation on functions that satisfies the Jacobi identity and a product rule. It encodes a Poisson manifold’s geometry and gives classical observables their equations of motion.

Poisson manifold: A manifold equipped with a bracket on smooth functions that is antisymmetric, satisfies the Jacobi identity, and obeys the product rule. The bracket is the structure that gives this manifold its name and geometry.
Smooth function: A function with derivatives of every order on its domain. Poisson brackets act on smooth functions, typically observables on a manifold.
Hamiltonian: A function that generates time evolution in Hamiltonian mechanics, usually representing total energy. Bracketing an observable with the Hamiltonian gives its rate of change.
Commutator: The operation [A,B] = AB − BA, measuring the failure of two operators to commute. Quantum commutators parallel Poisson brackets, with a factor of iℏ in the classical limit.
Hamiltonian mechanics: A formulation of classical mechanics in which a system’s state evolves according to a Hamiltonian function. The bracket expresses Hamiltonian evolution as the bracket of an observable with the Hamiltonian.
Differential: The linear approximation to a function’s change, represented by a covector at each point. The bracket pairs derivatives of its two input functions.
Observable: A function on a system’s state space representing a measurable physical quantity. Poisson brackets describe how classical observables change and interact.
Dirac bracket: A modified bracket used in constrained Hamiltonian systems to enforce second-class constraints. It replaces the ordinary Poisson bracket when constraints prevent unrestricted phase-space motion.
Jacobi identity: An identity requiring a bracket’s cyclically nested operations to sum to zero. This identity is what makes Poisson brackets form a Lie algebra.
Cotangent bundle: The manifold formed by collecting all cotangent spaces over the points of another manifold. Classical phase space is often a cotangent bundle with its canonical bracket.

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