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  "body": "\n## Definition\n\nLet $(M, \\omega)$ and $(N, \\eta)$ be [[symplectic manifolds|CAN490EPRT]].\nA *symplectomorphism* from $M$ to $N$ is a [[diffeomorphism|0KZZ70ZHXP]] $\\phi \\colon M \\to N$ such that $\\phi^* \\eta = \\omega$, that is\n$$\\eta_{\\phi(p)}\\big(d\\phi_p(v), d\\phi_p(w)\\big) = \\omega_p(v, w)$$\nfor all $p \\in M$ and $v, w \\in T_p M$.\n\n## Remarks\n\nExample: on $\\R^2$ with $\\omega = dx \\wedge dy$, let $\\phi(x, y) = (x, y + f(x))$ for a smooth function $f$.\nIt is a diffeomorphism with inverse $(x, y) \\mapsto (x, y - f(x))$, and $\\phi^* \\omega = dx \\wedge (dy + f'(x)\\, dx) = dx \\wedge dy$.\n\nNon-example: $\\psi(x, y) = (2x, y)$ on the same plane is a diffeomorphism, but $\\psi^* \\omega = 2\\, dx \\wedge dy \\ne \\omega$.\n\nA linear map of $(\\R^{2n}, \\omega_0)$ is a symplectomorphism exactly when it lies in the [[symplectic group|X12FS2JAP0]] $Sp(2n, \\R)$.\nThe flow of a [[Hamiltonian vector field|C5EH3B4BEP]] consists of symplectomorphisms.\nA symplectomorphism preserves the volume form $\\omega^n / n!$, so it preserves volume; in dimension $2$ the converse holds for orientation-preserving diffeomorphisms.\n",
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