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  "body": "\n## Summary\n\nA Poisson manifold is a [[smooth manifold|F45VQ47GPV]] with a bracket on its smooth functions.\nThe bracket has the algebraic properties of the [[Poisson bracket|2S0MXDT4YH]] of a [[symplectic manifold|CAN490EPRT]].\nEvery symplectic manifold is a Poisson manifold, but the bracket of a Poisson manifold may be degenerate.\nPoisson manifolds are the phase spaces of mechanics with constraints or symmetry, and they are the classical limits that deformation quantization starts from.\n\n## Explanations\n\n### L1 human\n\nIn mechanics the Poisson bracket of two observables tells how one changes along the motion generated by the other.\nA Poisson manifold keeps only the rules this bracket obeys: it is antisymmetric, satisfies the Jacobi identity, and behaves like a derivative in each slot.\nSuch a manifold splits into pieces, called symplectic leaves, each of which is an ordinary symplectic manifold.\nFor example, on $\\R^3$ the bracket with $\\{x, y\\} = z$, $\\{y, z\\} = x$, $\\{z, x\\} = y$ comes from the rotation group, and its leaves are the spheres centered at the origin and the origin itself.\n",
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