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  "body": "\n## Definition\n\nWork in $\\R^3$ with the [[dot product|0S52TNNTH9]] $\\cdot$ and the [[cross product|0N5N15HWZ7]] $\\times$.\n\n1. A *screw* is a pair $S = (\\omega, v)$ of vectors of $\\R^3$, an element of $\\R^6$.\n2. A *twist* is a screw read as the velocity field $u(x) = v + \\omega \\times x$ of a moving rigid body. Here $\\omega$ is the angular velocity and $v$ is the velocity of the body point at the origin.\n3. A *wrench* is a screw $(f, m)$ read as a system of forces with resultant force $f$ and moment $m$ about the origin.\n4. If $\\omega \\ne 0$, the *pitch* of $(\\omega, v)$ is $h = \\omega \\cdot v / \\abs{\\omega}^2$, and its *axis* is the line $\\set{\\omega \\times v / \\abs{\\omega}^2 + t \\omega : t \\in \\R}$.\n\nThe *reciprocal product* of a twist $(\\omega, v)$ and a wrench $(f, m)$ is $\\omega \\cdot m + v \\cdot f$, the power of the wrench on the motion.\n*Screw theory* is the study of screws, twists and wrenches with these operations.\n\n## Remarks\n\nOn the axis, the velocity is parallel to $\\omega$: for $p = \\omega \\times v / \\abs{\\omega}^2$ one finds $\\omega \\times p = \\omega (\\omega \\cdot v)/\\abs{\\omega}^2 - v$, so $u(p) = h \\omega$.\nSo the body turns about the axis and slides along it with speed $h \\abs{\\omega}$, which is the infinitesimal form of Chasles' theorem.\nExample: for $\\omega = (0, 0, 1)$ and $v = (0, -1, 2)$, the pitch is $h = 2$ and $p = \\omega \\times v = (1, 0, 0)$.\nCheck: $u(p) = (0, -1, 2) + (0, 0, 1) \\times (1, 0, 0) = (0, -1, 2) + (0, 1, 0) = (0, 0, 2) = 2\\omega$; the axis is the vertical line through $(1, 0, 0)$.\nIf $\\omega = 0$, the twist is a pure translation with velocity $v$, a screw of infinite pitch.\nNon-example: the field $u(x) = A x$ with $A = \\operatorname{diag}(1, 0, 0)$ is not a twist, since $A$ is not skew-symmetric; it stretches the body along the first axis instead of moving it rigidly.\nThe twists form the Lie algebra of the [[Euclidean group|0H3YY8CPKH]], with bracket $[(\\omega_1, v_1), (\\omega_2, v_2)] = (\\omega_1 \\times \\omega_2, \\omega_1 \\times v_2 - \\omega_2 \\times v_1)$.\n",
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