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  "body": "\n## Summary\n\nA Lagrangian submanifold of a symplectic manifold is a submanifold of half the dimension on which the symplectic form vanishes.\nLagrangian submanifolds are the largest submanifolds on which the form vanishes identically.\nThey include the zero section and the graphs of closed $1$-forms in cotangent bundles, and the graphs of [[symplectomorphisms|A8TS2SNX9Z]].\nThey are central objects of symplectic topology, Floer theory and mirror symmetry.\n\n## Explanations\n\n### L1 human\n\nA symplectic form measures areas of small surfaces inside a manifold of even dimension $2n$.\nA Lagrangian submanifold has dimension $n$ and is so placed that every small surface inside it has zero symplectic area.\nIn the phase plane with position $q$ and momentum $p$, every curve is Lagrangian.\nIn $\\R^4$ with coordinates $(q_1, q_2, p_1, p_2)$, the plane where both momenta are zero is Lagrangian.\n",
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