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  "body": "\n## Summary\n\nA symplectomorphism is a diffeomorphism between symplectic manifolds that carries one symplectic form to the other.\nThese maps are the isomorphisms of symplectic geometry.\nIn mechanics they are the canonical transformations, and the time-$t$ maps of Hamiltonian flows are symplectomorphisms.\nUnlike isometries of a Riemannian manifold, the symplectomorphisms of a symplectic manifold form an infinite-dimensional group.\n\n## Explanations\n\n### L1 human\n\nIn the plane, a symplectomorphism is a smooth change of shape that keeps every area the same, with orientation.\nA rotation keeps areas, and so does a shear that slides each vertical line up or down.\nStretching by $2$ in one direction and keeping the other direction fixed doubles areas, so it is not a symplectomorphism.\nIn higher dimensions a symplectomorphism keeps a whole family of areas, one for each pair of coordinates, not only volume.\n",
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