Poisson bracket

Summary

The Poisson bracket is an operation $\{f, g\}$ on pairs of smooth functions on a symplectic manifold, built from their Hamiltonian vector fields. It is antisymmetric, satisfies the Jacobi identity and the Leibniz rule, so it makes the functions into a Lie algebra. In mechanics it gives the time derivative of an observable along the motion, and its abstract properties lead to Poisson manifolds and to quantization.