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  "body": "\n## Definition\n\nLet $(M, \\omega)$ be a [[symplectic manifold|CAN490EPRT]] of dimension $2n$.\nLet $L \\subseteq M$ be a submanifold with inclusion map $\\iota \\colon L \\to M$.\nThe submanifold $L$ is *Lagrangian* if:\n\n1. $\\dim L = n$, and\n2. $\\iota^* \\omega = 0$, that is $\\omega_p(v, w) = 0$ for all $p \\in L$ and $v, w \\in T_p L$.\n\n## Remarks\n\nExample: in $\\R^4$ with $\\omega_0 = dx_1 \\wedge dy_1 + dx_2 \\wedge dy_2$, the plane $L = \\set{y_1 = y_2 = 0}$.\nIt has dimension $2$, and on it $dy_1 = dy_2 = 0$, so $\\iota^* \\omega_0 = 0$.\n\nNon-example: the plane $P = \\set{x_2 = y_2 = 0}$ in the same space has dimension $2$, but $\\omega_0$ restricts to $dx_1 \\wedge dy_1 \\ne 0$ on it, so clause 2 fails.\nSuch a plane is called symplectic.\n\nClause 2 alone forces $\\dim L \\le n$, so Lagrangian submanifolds are the isotropic submanifolds of largest dimension.\nIn a cotangent bundle $T^*Q$ with its canonical form, the graph of a $1$-form $\\alpha$ on $Q$ is Lagrangian exactly when $\\alpha$ is [[closed|0ZV14XRNF0]].\nA diffeomorphism $\\phi \\colon M \\to M$ is a [[symplectomorphism|A8TS2SNX9Z]] exactly when its graph is Lagrangian in $M \\times M$ with the form $\\mathrm{pr}_1^* \\omega - \\mathrm{pr}_2^* \\omega$.\n",
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