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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]].\nA *symplectic form* on $M$ is a [[differential $2$-form|0E3G0GRPZ6]] $\\omega$ on $M$ such that:\n\n1. $\\omega$ is [[closed|0ZV14XRNF0]]: $d\\omega = 0$, where $d$ is the [[exterior derivative|004FRCXH6M]];\n2. $\\omega$ is nondegenerate: for each $p \\in M$ and each nonzero $v \\in T_p M$ there is $w \\in T_p M$ with $\\omega_p(v, w) \\ne 0$.\n\nA *symplectic manifold* is a pair $(M, \\omega)$ of a smooth manifold and a symplectic form on it.\n\n## Remarks\n\nExample: $\\R^{2n}$ with coordinates $(x_1, \\ldots, x_n, y_1, \\ldots, y_n)$ and $\\omega_0 = \\sum_{k=1}^n dx_k \\wedge dy_k$.\nIt has constant coefficients, so $d\\omega_0 = 0$.\nIf $v \\ne 0$ has a nonzero component $v_{x_k}$, then $\\omega_0(v, \\partial_{y_k}) = v_{x_k} \\ne 0$, and similarly if $v_{y_k} \\ne 0$ one pairs with $\\partial_{x_k}$; so $\\omega_0$ is nondegenerate.\n\nNon-example: the sphere $S^4$ has no symplectic form.\nOn a closed manifold of dimension $2n$, $\\omega^n$ is a volume form, so the class $[\\omega]^n$ is nonzero and hence $[\\omega] \\ne 0$ in $H^2_{dR}(M)$; but $H^2_{dR}(S^4) = 0$.\n\nClause 2 says that each $(T_p M, \\omega_p)$ is a [[symplectic vector space|PQ7JJY0VPH]], so $\\dim M$ is even.\nOther standard examples are oriented surfaces with an area form and cotangent bundles $T^* Q$ with the form $-d\\theta$ built from the tautological one-form $\\theta$.\nDarboux's theorem says that near each point there are coordinates in which $\\omega = \\omega_0$.\n",
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