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  "body": "\n## Statement\n\nLet $a \\colon \\R^d \\to \\R^d$ be continuous, and consider the equation $x'' = a(x)$.\nFix a step size $h \u003e 0$ and write $a_n = a(x_n)$.\nBeeman's algorithm computes, from $x_n$, $v_n$, $a_n$ and $a_{n-1}$,\n$$x_{n+1} = x_n + h v_n + \\frac{h^2}{6} (4 a_n - a_{n-1}),$$\n$$v_{n+1} = v_n + \\frac{h}{6} (2 a_{n+1} + 5 a_n - a_{n-1}).$$\nThen for all $n \\ge 2$ for which the sequences are defined,\n$$x_{n+1} = 2 x_n - x_{n-1} + h^2 a_n.$$\nSo Beeman's algorithm produces the same positions as the [[Störmer–Verlet method|ESWE8Y3JD4]], once both start from the same $x_1$ and $x_2$.\n\n## Remarks\n\nSubtract the position formula at step $n - 1$ from the one at step $n$:\n$$x_{n+1} - 2x_n + x_{n-1} = h (v_n - v_{n-1}) + \\frac{h^2}{6} (4 a_n - 5 a_{n-1} + a_{n-2}).$$\nThe velocity formula gives $h (v_n - v_{n-1}) = \\frac{h^2}{6}(2 a_n + 5 a_{n-1} - a_{n-2})$, and the sum is $h^2 a_n$.\nThe difference from Verlet lies only in the velocities, which Beeman's formula computes more accurately than the central difference $(x_{n+1} - x_{n-1})/(2h)$; this improves the computed kinetic energy.\nThe velocity formula above is the implicit form; a common explicit variant predicts $v_{n+1}$ first when the force depends on the velocity.\n\n## Explanations\n\n### L1 human\n\nBeeman's method is a molecular dynamics integrator that uses the forces at the current and previous steps.\nIts positions turn out to be exactly those of the simpler Verlet method.\nWhat it adds is a more accurate formula for the velocities, so that the energy is measured more accurately.\n\n### L2 human\n\nThe identity shows that Beeman's method inherits the stability and the long-time behaviour of Verlet's positions.\nIt is a multistep method, so it needs a starting procedure to produce $a_{-1}$ or $x_1$.\nIn practice velocity Verlet is more common, because it gives similar accuracy with less storage.\n",
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