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  "body": "\n## Summary\n\nThe Poisson bracket is an operation $\\{f, g\\}$ on pairs of smooth functions on a [[symplectic manifold|CAN490EPRT]], built from their [[Hamiltonian vector fields|C5EH3B4BEP]].\nIt is antisymmetric, satisfies the Jacobi identity and the Leibniz rule, so it makes the functions into a Lie algebra.\nIn mechanics it gives the time derivative of an observable along the motion, and its abstract properties lead to [[Poisson manifolds|A2KQ49Y4NF]] and to quantization.\n\n## Explanations\n\n### L1 human\n\nIn mechanics, a state is a point $(q, p)$ of positions and momenta, and an observable is a function of the state.\nThe Poisson bracket of two observables measures how one changes when the system flows along the energy defined by the other.\nOn the plane, $\\{f, g\\} = \\frac{\\partial f}{\\partial q} \\frac{\\partial g}{\\partial p} - \\frac{\\partial f}{\\partial p} \\frac{\\partial g}{\\partial q}$.\nFor example $\\{q, p\\} = 1$, which is the classical shadow of the quantum rule that position and momentum do not commute.\nA quantity $g$ is conserved by the motion with energy $H$ exactly when $\\{g, H\\} = 0$.\n",
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