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  "body": "\n## Statement\n\nLet $M$ be a closed (compact, without boundary) smooth manifold.\nLet $(\\omega_t)_{t \\in [0, 1]}$ be a family of [[symplectic forms|CAN490EPRT]] on $M$ that depends smoothly on $t$.\nSuppose that the [[de Rham class|0ZV14XRNF0]] $[\\omega_t] \\in H^2_{dR}(M)$ does not depend on $t$.\nThen there is a smooth family of [[diffeomorphisms|0KZZ70ZHXP]] $\\phi_t \\colon M \\to M$, $t \\in [0, 1]$, with $\\phi_0 = \\id$ and $\\phi_t^* \\omega_t = \\omega_0$ for all $t$.\n\n## Remarks\n\nSince the class is constant, $\\tfrac{d}{dt}\\omega_t$ is exact, and one can choose $1$-forms $\\sigma_t$ depending smoothly on $t$ with $\\tfrac{d}{dt}\\omega_t = d\\sigma_t$.\nNondegeneracy gives a unique time-dependent vector field $X_t$ with $\\iota_{X_t} \\omega_t = -\\sigma_t$.\nLet $\\phi_t$ be its [[flow|06738GGWWK]], which exists for all $t \\in [0, 1]$ because $M$ is compact.\nBy Cartan's formula for the [[Lie derivative|PJWK0PKN6Q]],\n$$\\frac{d}{dt} \\phi_t^* \\omega_t = \\phi_t^* \\Big( \\mathcal{L}_{X_t} \\omega_t + \\frac{d}{dt}\\omega_t \\Big) = \\phi_t^* \\big( d\\iota_{X_t}\\omega_t + d\\sigma_t \\big) = 0.$$\nThe trick is to look for the diffeomorphisms as the flow of a vector field, which turns a nonlinear problem into a linear equation for $X_t$.\n\nExample of use: two area forms on a closed connected oriented surface, both positive for the orientation and with the same total area, are related by a diffeomorphism.\nApply the theorem to $\\omega_t = (1 - t)\\omega_0 + t\\omega_1$, which stays nondegenerate because both forms are positive for the orientation, so at each point one is a positive multiple of the other.\n\nCompactness is needed for the flow to exist up to time $1$; on open manifolds one adds conditions, as in local versions near a point or a submanifold.\n\n## Explanations\n\n### L1 human\n\nIf one moves a symplectic form continuously without changing its cohomology class, nothing really changes.\nThe theorem gives a family of diffeomorphisms that carries the moving form back to the starting one.\nSo on a closed manifold, symplectic forms in a fixed class that can be joined by a path are all the same up to diffeomorphism.\n\n### L2 human\n\nThe same argument proves Darboux's theorem, the Weinstein neighbourhood theorem for [[Lagrangian submanifolds|H8ACPTMX08]], and Moser's theorem on volume forms.\nIn each case one joins two structures by a path that agrees on a set, solves $\\iota_{X_t}\\omega_t = -\\sigma_t$, and integrates.\nThe method shows that symplectic structures have no local invariants, in contrast with Riemannian metrics, which have curvature.\n",
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