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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] and let $C^\\infty(M)$ be the algebra of smooth real functions on $M$.\nA *Poisson bracket* on $M$ is a map $\\{\\cdot, \\cdot\\} \\colon C^\\infty(M) \\times C^\\infty(M) \\to C^\\infty(M)$ such that for all $f, g, h \\in C^\\infty(M)$:\n\n1. it is bilinear over $\\R$;\n2. it is antisymmetric: $\\{f, g\\} = -\\{g, f\\}$;\n3. it satisfies the Jacobi identity: $\\{f, \\{g, h\\}\\} + \\{g, \\{h, f\\}\\} + \\{h, \\{f, g\\}\\} = 0$;\n4. it satisfies the Leibniz rule: $\\{f, gh\\} = \\{f, g\\}\\, h + g\\, \\{f, h\\}$.\n\nA *Poisson manifold* is a smooth manifold with a Poisson bracket.\n\n## Remarks\n\nClauses 1 to 3 say that $(C^\\infty(M), \\{\\cdot, \\cdot\\})$ is a Lie algebra over $\\R$.\nBy clause 4, there is a unique bivector field $\\pi$, a section of $\\Lambda^2 TM$, with $\\{f, g\\} = \\pi(df, dg)$; the Jacobi identity is then the condition $[\\pi, \\pi] = 0$ for the Schouten bracket.\n\nExample: every [[symplectic manifold|CAN490EPRT]] with its [[Poisson bracket|2S0MXDT4YH]] is a Poisson manifold.\nAnother example is the dual $\\mathfrak{g}^*$ of a finite-dimensional Lie algebra $\\mathfrak{g}$, with $\\{f, g\\}(\\xi) = \\langle \\xi, [df_\\xi, dg_\\xi] \\rangle$, where $df_\\xi \\in (\\mathfrak{g}^*)^* = \\mathfrak{g}$.\nOn $\\mathfrak{g}^*$ for $\\mathfrak{g} = \\mathfrak{so}(3)$ with suitable coordinates, $\\{x, y\\} = z$, $\\{y, z\\} = x$ and $\\{z, x\\} = y$.\n\nNon-example: on $\\R^3$, define a bracket by clauses 1, 2, 4 and $\\{x, y\\} = 1$, $\\{z, x\\} = x$, $\\{y, z\\} = 0$.\nThen $\\{x, \\{y, z\\}\\} + \\{y, \\{z, x\\}\\} + \\{z, \\{x, y\\}\\} = 0 + \\{y, x\\} + \\{z, 1\\} = -1$, so the Jacobi identity fails.\n",
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