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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $J = \\begin{pmatrix} 0 \u0026 I_n \\\\ -I_n \u0026 0 \\end{pmatrix}$, where $I_n$ is the $n \\times n$ identity matrix.\nFor a smooth function $H \\colon \\R^{2n} \\to \\R$, consider the [[Hamiltonian system|0EDKGA2FAK]] $y' = J \\nabla H(y)$ with $y = (q, p)$, that is, $q' = \\nabla_p H$ and $p' = -\\nabla_q H$.\nA one-step method assigns to each such $H$ and each small step size $h \u003e 0$ a map $\\Phi_h$, and computes $y_{k+1} = \\Phi_h(y_k)$.\nThe method is a *symplectic integrator* if $\\Phi_h$ is a symplectic map whenever $H$ is smooth and $\\Phi_h$ is defined and smooth.\nThis means\n$$D\\Phi_h(y)^T \\, J \\, D\\Phi_h(y) = J \\quad \\text{for all } y.$$\n\n## Remarks\n\nFor $n = 1$, a map of the plane is symplectic exactly when its Jacobian determinant is $1$, that is, when it preserves area.\nExample: for $H = \\frac12 (p^2 + q^2)$, the symplectic Euler method $p_{k+1} = p_k - h q_k$, $q_{k+1} = q_k + h p_{k+1}$ gives $q_{k+1} = (1 - h^2) q_k + h p_k$, with Jacobian determinant $(1 - h^2) \\cdot 1 - h \\cdot (-h) = 1$.\nIn general the symplectic Euler method is $p_{k+1} = p_k - h \\nabla_q H(q_k, p_{k+1})$, $q_{k+1} = q_k + h \\nabla_p H(q_k, p_{k+1})$, and it is symplectic for every $H$.\nNon-example: the explicit [[Euler method|0JARTXX4XP]] for the same $H$ is $q_{k+1} = q_k + h p_k$, $p_{k+1} = p_k - h q_k$, with Jacobian determinant $1 + h^2 \\ne 1$, so it is not symplectic.\nA Runge–Kutta method with coefficients $a_{ij}$, $b_i$ is symplectic when $b_i a_{ij} + b_j a_{ji} = b_i b_j$ for all $i, j$; the implicit midpoint rule is an example.\n",
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