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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^d$ be open, let $f \\colon U \\to \\R^d$ be smooth, and let $\\varphi_t$ be the [[flow|06738GGWWK]] of the equation $y' = f(y)$.\nLet $\\mathcal{P}$ be a property of maps $U \\to \\R^d$ that every $\\varphi_t$ has.\nStandard choices of $\\mathcal{P}$ are:\n\n1. symplecticity, when the equation is a [[Hamiltonian system|0EDKGA2FAK]];\n2. the conservation of a given [[first integral|0262JP0NV9]] $I$, that is, $I \\circ \\Phi = I$;\n3. the preservation of volume, when $f$ has divergence zero;\n4. reversibility with respect to a linear involution $\\rho$, that is, $\\rho \\circ \\Phi = \\Phi^{-1} \\circ \\rho$, when $f \\circ \\rho = -\\rho \\circ f$.\n\nLet $\\Phi_h$ be a one-step numerical method with step size $h$.\nIt is a *geometric integrator* for $\\mathcal{P}$ if $\\Phi_h$ has the property $\\mathcal{P}$ for every equation of the class and every small enough $h \u003e 0$.\n\n## Remarks\n\nExample: take the harmonic oscillator $q' = p$, $p' = -q$.\nThe symplectic Euler method $p_{n+1} = p_n - h q_n$, $q_{n+1} = q_n + h p_{n+1}$ is the linear map\n$$\\begin{pmatrix} q_{n+1} \\\\ p_{n+1} \\end{pmatrix} = \\begin{pmatrix} 1 - h^2 \u0026 h \\\\ -h \u0026 1 \\end{pmatrix} \\begin{pmatrix} q_n \\\\ p_n \\end{pmatrix}.$$\nIts determinant is $1$, and in dimension $2$ this means that the map is symplectic.\nSo the method is a geometric integrator for property 1.\nNon-example: the explicit Euler method $q_{n+1} = q_n + h p_n$, $p_{n+1} = p_n - h q_n$ has matrix determinant $1 + h^2 \\ne 1$, so it is not symplectic for any $h \u003e 0$.\nA method may be geometric for one property and not another.\nThe implicit midpoint rule is symplectic and conserves all quadratic first integrals.\nBut no Runge–Kutta method conserves all polynomial first integrals of degree $3$ or more.\n\n## Explanations\n\n### L1 human\n\nA geometric integrator copies a structural feature of the true motion exactly, not only approximately.\nThe feature is chosen in advance: a conserved quantity, area in phase space, or symmetry under reversing time.\n",
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