840:
40:
853:
1328:
966:
2037:
The computation rules for infinitesimal rotation matrices are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals. It turns
985:
it undergoes a changing velocity and acceleration at any time. When dealing with the rotation of a body, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the body's motion, so for
1317:
2033:
194:
1148:
1200:
1842:
1494:
1930:
1585:
1923:
1705:
1666:
218:
1751:
1338:
In three dimensions, angular displacement is an entity with a direction and a magnitude. The direction specifies the axis of rotation, which always exists by virtue of the
1229:
1070:
2931:
1880:
1797:
1617:
1440:
1413:
1256:
1774:
1323:: Euler's rotation theorem. A great circle transforms to another great circle under rotations, leaving always a diameter of the sphere in its original position.
45:
The angle of rotation from the black ray to the green segment is 60°, from the black ray to the blue segment is 210°, and from the green to the blue segment is
2323:
3021:
2186:
1365:
for addition. Nevertheless, when dealing with infinitesimal rotations, second order infinitesimals can be discarded and in this case commutativity appears.
2250:
1388:
Given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix. Being
986:
example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal and negligible.
2175:
2118:
884:
473:
2316:
592:
1496:. When this product is performed having a very small difference between both frames we will obtain a matrix close to the identity.
565:
1002:, rotating counterclockwise. It becomes important to then represent the position of particle P in terms of its polar coordinates (
150:
2309:
2293:
2266:
2155:
1109:
547:
994:
In the example illustrated to the right (or above in some mobile versions), a particle or body P is at a fixed distance
1262:
17:
2214:
877:
3057:
3032:
2993:
839:
1164:
213:
2028:{\displaystyle dL_{x}={\begin{bmatrix}1&0&0\\0&1&-d\theta \\0&d\theta &1\end{bmatrix}}.}
1802:
1708:
468:
208:
1445:
2335:
2285:
1517:
1509:
1270:
1545:
1084:
or degrees. Using radians provides a very simple relationship between distance traveled around the circle (
870:
857:
618:
541:
463:
306:
1889:
3333:
2077:
1674:
1339:
537:
338:
1633:
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1624:
1311:
757:
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572:
432:
365:
2964:
2769:
2623:
2142:
1299:
511:
1720:
2736:
1620:
1205:
1157:, the angular displacement is given by the distance traveled around the circumference - which is 2π
787:
631:
2747:
1043:
737:
501:
2907:
2807:
777:
782:
599:
2206:
2130:
1849:
1779:
981:
When a body rotates about its axis, the motion cannot simply be analyzed as a particle, as in
2882:
2752:
2490:
2087:
1628:
1290:
792:
767:
453:
271:
3204:
3129:
2757:
1590:
1521:
1418:
1391:
1238:
1014:
is changing, while the value of the radius remains the same. (In rectangular coordinates (
812:
772:
676:
668:
658:
448:
441:
197:
2172:"The NIST Guide for the Use of the International System of Units, Special Publication 811"
807:
8:
2935:
2664:
2612:
2332:
922:
587:
528:
506:
251:
246:
241:
141:
1759:
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333:
301:
261:
3328:
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2708:
2467:
2289:
2275:
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2210:
2199:
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2082:
2057:
1540:
1358:
1232:
727:
684:
641:
636:
577:
353:
343:
236:
27:
Displacement measured angle-wise when a body is showing circular or rotational motion
3041:
3011:
2977:
2696:
2652:
2607:
2171:
2067:
2062:
2052:
942:
822:
802:
747:
742:
688:
663:
518:
376:
321:
296:
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2092:
1536:
1382:
1374:
1362:
1347:
982:
934:
817:
762:
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707:
626:
100:
50:
3049:
2985:
39:
3323:
3230:
3194:
3119:
2900:
2787:
2677:
2578:
2301:
2258:
1281:
1031:
950:
930:
844:
752:
653:
370:
54:
1442:
two matrices, the angular displacement matrix between them can be obtained as
945:. Angular displacement may be signed, indicating the sense of rotation (e.g.,
3317:
3104:
2972:
2686:
1669:
1525:
914:
732:
559:
1276:
Angular displacement may be signed, indicating the sense of rotation (e.g.,
3179:
3165:
3065:
3001:
2726:
2509:
2420:
1378:
1285:
1086:
954:
797:
722:
411:
291:
117:
108:
2114:
1037:, which becomes related to the angular position through the relationship:
2547:
1266:
104:
2150:(9th ed.), International Bureau of Weights and Measures, Dec 2022,
1351:
1089:
1030:
vary with time.) As the particle moves along the circle, it travels an
965:
582:
256:
1357:
Despite having direction and magnitude, angular displacement is not a
2640:
2567:
2555:
1277:
946:
604:
2040:
the order in which infinitesimal rotations are applied is irrelevant
2944:
2843:
2597:
2456:
2410:
1882:
representing an infinitesimal three-dimensional rotation about the
1529:
1316:
938:
523:
406:
381:
1327:
3270:
2072:
496:
349:
266:
124:
96:
91:
2115:"ISO 80000-3:2019 Quantities and units — Part 3: Space and time"
3184:
3153:
3109:
3087:
2541:
2440:
2405:
1343:
1099:
1081:
926:
555:
401:
311:
61:
1153:
For example, if a body rotates 360° around a circle of radius
3218:
3143:
3077:
2681:
2587:
2582:
2519:
2494:
1499:
In the limit, we will have an infinitesimal rotation matrix.
918:
391:
386:
328:
1288:. In the ISQ/SI, angular displacement is used to define the
49:. A complete rotation about the center point is equal to 1
3300:
3260:
2833:
2499:
2430:
2395:
396:
359:
189:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
58:
1955:
2910:
1933:
1892:
1852:
1805:
1782:
1762:
1723:
1677:
1636:
1593:
1548:
1448:
1421:
1394:
1241:
1208:
1167:
1112:
1046:
153:
1334:: A rotation represented by an Euler axis and angle.
1143:{\displaystyle \theta ={\frac {s}{r}}\mathrm {rad} }
2245:
2230:
2925:
2198:
2170:Thompson, Ambler; Taylor, Barry N. (2020-03-04) .
2027:
1917:
1874:
1836:
1791:
1768:
1745:
1699:
1660:
1611:
1579:
1502:
1488:
1434:
1407:
1350:to determine direction). This entity is called an
1250:
1223:
1194:
1142:
1064:
188:
2109:
2107:
3315:
2331:
2196:
1265:(ISQ), formalized in the international standard
1373:Several ways to describe rotations exist, like
2176:National Institute of Standards and Technology
2119:International Organization for Standardization
2104:
1714:An infinitesimal rotation matrix has the form
2317:
2163:
1010:). In this particular example, the value of
878:
2197:Kleppner, Daniel; Kolenkow, Robert (1973).
2135:
1195:{\displaystyle \theta ={\frac {2\pi r}{r}}}
2324:
2310:
2274:
1711:), which is not itself a rotation matrix.
1342:; the magnitude specifies the rotation in
885:
871:
76:rotational displacement, angle of rotation
38:
1837:{\displaystyle A\in {\mathfrak {so}}(n).}
1736:
1080:Angular displacement may be expressed in
1489:{\displaystyle \Delta A=A_{f}A_{0}^{-1}}
1326:
1315:
964:
1075:
941:(revolves or spins) around a centre or
14:
3316:
1643:
1580:{\displaystyle R^{\mathsf {T}}=R^{-1}}
1555:
1305:
2305:
1520:or differential rotation matrix is a
1269:(Space and time), and adopted in the
567:Newton's law of universal gravitation
2169:
1918:{\displaystyle {\mathfrak {so}}(3),}
1368:
1261:The above definition is part of the
1898:
1895:
1817:
1814:
1700:{\displaystyle {\mathfrak {so}}(n)}
1683:
1680:
548:Mechanics of planar particle motion
156:
24:
1661:{\displaystyle A^{\mathsf {T}}=-A}
1449:
1263:International System of Quantities
1136:
1133:
1130:
901:(symbol θ, ϑ, or φ) – also called
25:
3345:
2231:Goldstein, Poole & Safko 2002
2144:The International System of Units
1508:This section is an excerpt from
852:
851:
838:
171:
2348:Linear/translational quantities
1503:Infinitesimal rotation matrices
937:, etc.) through which the body
2223:
2190:
1909:
1903:
1828:
1822:
1746:{\displaystyle I+d\theta \,A,}
1709:special orthogonal Lie algebra
1694:
1688:
1606:
1600:
1280:); it may also be greater (in
949:); it may also be greater (in
13:
1:
2353:Angular/rotational quantities
2098:
1518:infinitesimal rotation matrix
1510:Infinitesimal rotation matrix
1361:because it does not obey the
1271:International System of Units
1224:{\displaystyle \theta =2\pi }
474:Koopman–von Neumann mechanics
2201:An Introduction to Mechanics
1202:which easily simplifies to:
542:Non-inertial reference frame
7:
2046:
1587:representing an element of
1346:about that axis (using the
1065:{\displaystyle s=r\theta .}
469:Appell's equation of motion
339:Inertial frame of reference
10:
3350:
2926:{\displaystyle {\dot {m}}}
2350:
2239:
1886:-axis, a basis element of
1799:is vanishingly small, and
1507:
1312:Three-dimensional rotation
1309:
1298:=θ/(2π rad), a ratio-type
989:
960:
2624:specific angular momentum
2352:
2347:
2342:
2253:; Safko, John L. (2002),
1300:quantity of dimension one
1161:- divided by the radius:
969:Rotation of a rigid body
115:
90:
80:
70:
37:
32:
2276:Wedderburn, Joseph H. M.
2205:. McGraw-Hill. pp.
1875:{\displaystyle A=L_{x},}
1792:{\displaystyle d\theta }
1776:is the identity matrix,
1621:special orthogonal group
1340:Euler's rotation theorem
632:Rotating reference frame
464:Hamilton–Jacobi equation
2748:rotational acceleration
907:rotational displacement
573:Newton's laws of motion
433:Newton's laws of motion
2927:
2078:Infinitesimal rotation
2029:
1919:
1876:
1838:
1793:
1770:
1747:
1701:
1662:
1613:
1581:
1490:
1436:
1409:
1335:
1324:
1252:
1225:
1196:
1144:
1066:
978:
600:Simple harmonic motion
513:Euler's laws of motion
307:D'Alembert's principle
190:
2928:
2088:Second moment of area
2030:
1920:
1877:
1839:
1794:
1771:
1748:
1702:
1663:
1629:skew-symmetric matrix
1614:
1612:{\displaystyle SO(n)}
1582:
1491:
1437:
1435:{\displaystyle A_{f}}
1410:
1408:{\displaystyle A_{0}}
1330:
1319:
1291:number of revolutions
1253:
1251:{\displaystyle 2\pi }
1226:
1197:
1145:
1067:
968:
454:Hamiltonian mechanics
272:Statistical mechanics
191:
2908:
2758:angular acceleration
2530:angular displacement
2281:Lectures on Matrices
1931:
1890:
1850:
1803:
1780:
1760:
1721:
1675:
1634:
1591:
1546:
1446:
1419:
1392:
1239:
1206:
1165:
1110:
1076:Definition and units
1044:
899:angular displacement
677:Angular acceleration
669:Rotational frequency
449:Lagrangian mechanics
442:Analytical mechanics
198:Second law of motion
151:
33:Angular displacement
2848:weighted position:
2665:rotational velocity
2613:kinematic viscosity
2333:Classical mechanics
2255:Classical Mechanics
1627:of a rotation is a
1485:
1306:In three dimensions
1093:) and the distance
973:about a fixed axis
911:rotary displacement
529:Harmonic oscillator
507:Equations of motion
142:Classical mechanics
136:Part of a series on
3334:Sign (mathematics)
2923:
2257:(third ed.),
2247:Goldstein, Herbert
2025:
2016:
1915:
1872:
1834:
1789:
1766:
1743:
1697:
1658:
1609:
1577:
1486:
1468:
1432:
1405:
1336:
1325:
1248:
1221:
1192:
1140:
1062:
979:
845:Physics portal
459:Routhian mechanics
334:Frame of reference
186:
119:SI base units
18:Angles of rotation
3311:
3310:
3306:
3305:
2920:
2873:moment of inertia
2295:978-0-8218-3204-2
2268:978-0-201-65702-9
2251:Poole, Charles P.
2174:(2008 ed.).
2157:978-92-822-2272-0
2083:Linear elasticity
2058:Angular frequency
1769:{\displaystyle I}
1541:orthogonal matrix
1375:rotation matrices
1369:Rotation matrices
1190:
1127:
1097:from the centre (
998:from the origin,
903:angle of rotation
895:
894:
642:Centrifugal force
637:Centripetal force
593:Euler's equations
578:Relative velocity
354:Moment of inertia
184:
158:
131:
130:
47:210° − 60° = 150°
16:(Redirected from
3341:
3298:
3294:
3282:
3278:
3258:
3254:
3242:
3238:
3216:
3212:
3202:
3192:
3177:
3173:
3163:
3141:
3137:
3127:
3117:
3102:
3098:
3085:
3061:
3055:
3047:
3036:
3030:
3019:
3012:angular momentum
2997:
2991:
2983:
2968:
2962:
2952:
2939:
2933:
2932:
2930:
2929:
2924:
2922:
2921:
2913:
2886:
2880:
2864:
2841:
2822:
2817:
2795:
2773:
2767:
2750:
2734:
2712:
2706:
2697:angular velocity
2694:
2675:
2674:
2661:
2660:
2653:rotational speed
2649:
2648:
2631:
2620:
2605:
2595:
2576:
2575:
2558:
2553:
2539:
2527:
2507:
2488:
2487:
2481:
2475:
2464:
2438:
2418:
2403:
2387:
2380:
2345:
2344:
2326:
2319:
2312:
2303:
2302:
2298:
2271:
2234:
2227:
2221:
2220:
2204:
2194:
2188:
2185:
2183:
2182:
2167:
2161:
2160:
2149:
2139:
2133:
2129:
2127:
2126:
2111:
2068:Angular velocity
2063:Angular position
2053:Angular distance
2034:
2032:
2031:
2026:
2021:
2020:
1946:
1945:
1924:
1922:
1921:
1916:
1902:
1901:
1885:
1881:
1879:
1878:
1873:
1868:
1867:
1846:For example, if
1843:
1841:
1840:
1835:
1821:
1820:
1798:
1796:
1795:
1790:
1775:
1773:
1772:
1767:
1752:
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1749:
1744:
1706:
1704:
1703:
1698:
1687:
1686:
1667:
1665:
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1659:
1648:
1647:
1646:
1618:
1616:
1615:
1610:
1586:
1584:
1583:
1578:
1576:
1575:
1560:
1559:
1558:
1524:representing an
1495:
1493:
1492:
1487:
1484:
1476:
1467:
1466:
1441:
1439:
1438:
1433:
1431:
1430:
1414:
1412:
1411:
1406:
1404:
1403:
1257:
1255:
1254:
1249:
1230:
1228:
1227:
1222:
1201:
1199:
1198:
1193:
1191:
1186:
1175:
1149:
1147:
1146:
1141:
1139:
1128:
1120:
1071:
1069:
1068:
1063:
943:axis of rotation
887:
880:
873:
860:
855:
854:
847:
843:
842:
748:Johann Bernoulli
743:Daniel Bernoulli
664:Tangential speed
568:
544:
519:Fictitious force
514:
366:Mechanical power
356:
297:Angular momentum
195:
193:
192:
187:
185:
183:
175:
174:
165:
160:
159:
133:
132:
120:
48:
42:
30:
29:
21:
3349:
3348:
3344:
3343:
3342:
3340:
3339:
3338:
3314:
3313:
3312:
3307:
3296:
3295:
3290:
3280:
3279:
3274:
3256:
3255:
3250:
3240:
3239:
3234:
3214:
3213:
3208:
3198:
3188:
3175:
3174:
3169:
3157:
3139:
3138:
3133:
3123:
3113:
3100:
3099:
3097:
3091:
3081:
3059:
3056:
3053:
3045:
3034:
3031:
3025:
3022:angular impulse
3015:
2995:
2992:
2989:
2981:
2966:
2963:
2958:
2948:
2937:
2934:
2912:
2911:
2909:
2906:
2905:
2904:
2884:
2881:
2876:
2849:
2842:
2837:
2820:
2818:
2811:
2796:
2791:
2771:
2768:
2761:
2751:
2746:
2735:
2730:
2710:
2707:
2700:
2690:
2676:
2668:
2663:
2656:
2651:
2644:
2639:
2632:
2627:
2622:
2616:
2606:
2601:
2591:
2577:
2571:
2566:
2554:
2551:
2546:
2540:
2533:
2523:
2508:
2503:
2493:
2483:
2477:
2471:
2466:
2460:
2439:
2434:
2419:
2414:
2404:
2399:
2383:
2376:
2338:
2330:
2296:
2269:
2242:
2237:
2228:
2224:
2217:
2195:
2191:
2180:
2178:
2168:
2164:
2158:
2147:
2141:
2140:
2136:
2124:
2122:
2113:
2112:
2105:
2101:
2093:Unwrapped phase
2049:
2044:
2043:
2015:
2014:
2009:
2001:
1995:
1994:
1983:
1978:
1972:
1971:
1966:
1961:
1951:
1950:
1941:
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1932:
1929:
1928:
1894:
1893:
1891:
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1887:
1883:
1863:
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1813:
1812:
1804:
1801:
1800:
1781:
1778:
1777:
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1722:
1719:
1718:
1679:
1678:
1676:
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1672:
1642:
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1637:
1635:
1632:
1631:
1592:
1589:
1588:
1568:
1564:
1554:
1553:
1549:
1547:
1544:
1543:
1537:rotation matrix
1513:
1505:
1477:
1472:
1462:
1458:
1447:
1444:
1443:
1426:
1422:
1420:
1417:
1416:
1399:
1395:
1393:
1390:
1389:
1383:charts on SO(3)
1371:
1363:commutative law
1348:right-hand rule
1314:
1308:
1240:
1237:
1236:
1231:. Therefore, 1
1207:
1204:
1203:
1176:
1174:
1166:
1163:
1162:
1129:
1119:
1111:
1108:
1107:
1078:
1045:
1042:
1041:
992:
983:circular motion
963:
891:
850:
837:
836:
829:
828:
827:
702:
694:
693:
673:
627:Circular motion
621:
611:
610:
609:
566:
536:
533:
512:
491:
483:
482:
479:
478:
436:
426:
418:
417:
416:
375:
371:Mechanical work
364:
348:
286:
278:
277:
276:
231:
223:
200:
176:
170:
166:
164:
155:
154:
152:
149:
148:
118:
83:
73:
66:
65:
46:
28:
23:
22:
15:
12:
11:
5:
3347:
3337:
3336:
3331:
3326:
3309:
3308:
3304:
3303:
3284:
3268:
3266:
3263:
3244:
3228:
3226:
3222:
3221:
3182:
3151:
3149:
3146:
3107:
3095:
3075:
3073:
3069:
3068:
3039:
3009:
3007:
3004:
2975:
2942:
2919:
2916:
2901:Mass flow rate
2898:
2894:
2893:
2891:
2889:
2870:
2867:
2865:
2846:
2831:
2827:
2826:
2824:
2805:
2803:
2800:
2798:
2785:
2783:
2779:
2778:
2776:
2755:
2744:
2741:
2739:
2724:
2722:
2718:
2717:
2715:
2684:
2637:
2634:
2610:
2585:
2564:
2560:
2559:
2544:
2517:
2515:
2512:
2497:
2454:
2452:
2448:
2447:
2445:
2443:
2428:
2425:
2423:
2408:
2393:
2389:
2388:
2381:
2374:
2371:
2368:
2365:
2362:
2359:
2355:
2354:
2351:
2349:
2343:
2340:
2339:
2329:
2328:
2321:
2314:
2306:
2300:
2299:
2294:
2272:
2267:
2259:Addison Wesley
2241:
2238:
2236:
2235:
2222:
2215:
2189:
2162:
2156:
2134:
2117:(2 ed.).
2102:
2100:
2097:
2096:
2095:
2090:
2085:
2080:
2075:
2070:
2065:
2060:
2055:
2048:
2045:
2036:
2035:
2024:
2019:
2013:
2010:
2008:
2005:
2002:
2000:
1997:
1996:
1993:
1990:
1987:
1984:
1982:
1979:
1977:
1974:
1973:
1970:
1967:
1965:
1962:
1960:
1957:
1956:
1954:
1949:
1944:
1940:
1936:
1914:
1911:
1908:
1905:
1900:
1897:
1871:
1866:
1862:
1858:
1855:
1833:
1830:
1827:
1824:
1819:
1816:
1811:
1808:
1788:
1785:
1765:
1754:
1753:
1742:
1739:
1735:
1732:
1729:
1726:
1696:
1693:
1690:
1685:
1682:
1657:
1654:
1651:
1645:
1640:
1608:
1605:
1602:
1599:
1596:
1574:
1571:
1567:
1563:
1557:
1552:
1514:
1506:
1504:
1501:
1483:
1480:
1475:
1471:
1465:
1461:
1457:
1454:
1451:
1429:
1425:
1402:
1398:
1370:
1367:
1310:Main article:
1307:
1304:
1284:) than a full
1282:absolute value
1247:
1244:
1220:
1217:
1214:
1211:
1189:
1185:
1182:
1179:
1173:
1170:
1151:
1150:
1138:
1135:
1132:
1126:
1123:
1118:
1115:
1077:
1074:
1073:
1072:
1061:
1058:
1055:
1052:
1049:
991:
988:
962:
959:
953:) than a full
951:absolute value
893:
892:
890:
889:
882:
875:
867:
864:
863:
862:
861:
848:
831:
830:
826:
825:
820:
815:
810:
805:
800:
795:
790:
785:
780:
775:
770:
765:
760:
755:
750:
745:
740:
735:
730:
725:
720:
715:
710:
704:
703:
700:
699:
696:
695:
692:
691:
672:
671:
666:
661:
656:
654:Coriolis force
651:
650:
649:
639:
634:
629:
623:
622:
617:
616:
613:
612:
608:
607:
602:
597:
596:
595:
590:
580:
575:
570:
563:
552:
551:
550:
545:
532:
531:
526:
521:
516:
509:
504:
499:
493:
492:
489:
488:
485:
484:
481:
480:
477:
476:
471:
466:
461:
456:
451:
445:
439:
437:
430:
427:
424:
423:
420:
419:
415:
414:
409:
404:
399:
394:
389:
384:
379:
373:
368:
362:
357:
346:
341:
336:
331:
326:
325:
324:
319:
309:
304:
299:
294:
288:
287:
284:
283:
280:
279:
275:
274:
269:
264:
259:
254:
249:
244:
239:
233:
232:
229:
228:
225:
224:
222:
221:
216:
211:
205:
202:
201:
196:
182:
179:
173:
169:
163:
145:
144:
138:
137:
129:
128:
122:
113:
112:
94:
88:
87:
84:
82:Common symbols
81:
78:
77:
74:
71:
68:
67:
44:
43:
35:
34:
26:
9:
6:
4:
3:
2:
3346:
3335:
3332:
3330:
3327:
3325:
3322:
3321:
3319:
3302:
3293:
3288:
3285:
3277:
3272:
3269:
3267:
3264:
3262:
3253:
3248:
3245:
3237:
3232:
3229:
3227:
3224:
3223:
3220:
3211:
3206:
3201:
3196:
3191:
3186:
3183:
3181:
3172:
3167:
3162:
3161:
3155:
3152:
3150:
3147:
3145:
3136:
3131:
3126:
3121:
3116:
3111:
3108:
3106:
3094:
3089:
3084:
3079:
3076:
3074:
3071:
3070:
3067:
3063:
3051:
3043:
3040:
3038:
3029:
3023:
3018:
3013:
3010:
3008:
3005:
3003:
2999:
2987:
2979:
2976:
2974:
2970:
2961:
2956:
2951:
2946:
2943:
2941:
2917:
2914:
2902:
2899:
2896:
2895:
2892:
2890:
2888:
2879:
2874:
2871:
2868:
2866:
2863:
2860:
2856:
2852:
2847:
2845:
2840:
2835:
2832:
2829:
2828:
2825:
2816:
2815:
2809:
2806:
2804:
2801:
2799:
2794:
2789:
2786:
2784:
2781:
2780:
2777:
2775:
2766:
2765:
2759:
2756:
2754:
2749:
2745:
2742:
2740:
2738:
2733:
2728:
2725:
2723:
2720:
2719:
2716:
2714:
2705:
2704:
2698:
2693:
2688:
2687:angular speed
2685:
2683:
2679:
2673:
2672:
2666:
2659:
2654:
2647:
2642:
2638:
2635:
2630:
2625:
2619:
2614:
2611:
2609:
2604:
2599:
2594:
2589:
2586:
2584:
2580:
2574:
2569:
2565:
2562:
2561:
2557:
2549:
2545:
2543:
2538:
2537:
2531:
2526:
2521:
2518:
2516:
2513:
2511:
2506:
2501:
2498:
2496:
2492:
2486:
2480:
2474:
2469:
2463:
2458:
2455:
2453:
2450:
2449:
2446:
2444:
2442:
2437:
2432:
2429:
2426:
2424:
2422:
2417:
2412:
2409:
2407:
2402:
2397:
2394:
2391:
2390:
2386:
2382:
2379:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2356:
2346:
2341:
2337:
2334:
2327:
2322:
2320:
2315:
2313:
2308:
2307:
2304:
2297:
2291:
2287:
2283:
2282:
2277:
2273:
2270:
2264:
2260:
2256:
2252:
2248:
2244:
2243:
2232:
2226:
2218:
2216:9780070350489
2212:
2208:
2203:
2202:
2193:
2187:
2177:
2173:
2166:
2159:
2153:
2146:
2145:
2138:
2131:
2120:
2116:
2110:
2108:
2103:
2094:
2091:
2089:
2086:
2084:
2081:
2079:
2076:
2074:
2071:
2069:
2066:
2064:
2061:
2059:
2056:
2054:
2051:
2050:
2041:
2022:
2017:
2011:
2006:
2003:
1998:
1991:
1988:
1985:
1980:
1975:
1968:
1963:
1958:
1952:
1947:
1942:
1938:
1934:
1927:
1926:
1925:
1912:
1906:
1869:
1864:
1860:
1856:
1853:
1844:
1831:
1825:
1809:
1806:
1786:
1783:
1763:
1740:
1737:
1733:
1730:
1727:
1724:
1717:
1716:
1715:
1712:
1710:
1691:
1671:
1670:tangent space
1655:
1652:
1649:
1638:
1630:
1626:
1622:
1603:
1597:
1594:
1572:
1569:
1565:
1561:
1550:
1542:
1538:
1533:
1531:
1527:
1523:
1519:
1511:
1500:
1497:
1481:
1478:
1473:
1469:
1463:
1459:
1455:
1452:
1427:
1423:
1400:
1396:
1386:
1384:
1380:
1376:
1366:
1364:
1360:
1355:
1353:
1349:
1345:
1341:
1333:
1329:
1322:
1318:
1313:
1303:
1301:
1297:
1293:
1292:
1287:
1283:
1279:
1274:
1272:
1268:
1264:
1259:
1245:
1242:
1234:
1218:
1215:
1212:
1209:
1187:
1183:
1180:
1177:
1171:
1168:
1160:
1156:
1124:
1121:
1116:
1113:
1106:
1105:
1104:
1102:
1101:
1096:
1092:
1091:
1088:
1083:
1059:
1056:
1053:
1050:
1047:
1040:
1039:
1038:
1036:
1033:
1029:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
987:
984:
976:
972:
967:
958:
956:
952:
948:
944:
940:
936:
932:
928:
924:
920:
916:
915:physical body
912:
908:
904:
900:
888:
883:
881:
876:
874:
869:
868:
866:
865:
859:
849:
846:
841:
835:
834:
833:
832:
824:
821:
819:
816:
814:
811:
809:
806:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
779:
776:
774:
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
705:
698:
697:
690:
686:
682:
678:
675:
674:
670:
667:
665:
662:
660:
657:
655:
652:
648:
645:
644:
643:
640:
638:
635:
633:
630:
628:
625:
624:
620:
615:
614:
606:
603:
601:
598:
594:
591:
589:
586:
585:
584:
581:
579:
576:
574:
571:
569:
564:
561:
557:
554:
553:
549:
546:
543:
539:
535:
534:
530:
527:
525:
522:
520:
517:
515:
510:
508:
505:
503:
500:
498:
495:
494:
487:
486:
475:
472:
470:
467:
465:
462:
460:
457:
455:
452:
450:
447:
446:
444:
443:
438:
435:
434:
429:
428:
422:
421:
413:
410:
408:
405:
403:
400:
398:
395:
393:
390:
388:
385:
383:
380:
378:
374:
372:
369:
367:
363:
361:
358:
355:
351:
347:
345:
342:
340:
337:
335:
332:
330:
327:
323:
320:
318:
315:
314:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
289:
282:
281:
273:
270:
268:
265:
263:
260:
258:
255:
253:
250:
248:
245:
243:
240:
238:
235:
234:
227:
226:
220:
217:
215:
212:
210:
207:
206:
204:
203:
199:
180:
177:
167:
161:
147:
146:
143:
140:
139:
135:
134:
126:
123:
121:
114:
110:
106:
102:
98:
95:
93:
89:
85:
79:
75:
69:
63:
60:
56:
52:
41:
36:
31:
19:
3291:
3275:
3251:
3235:
3209:
3199:
3189:
3170:
3159:
3158:
3134:
3124:
3114:
3092:
3082:
3027:
3016:
2959:
2949:
2877:
2861:
2858:
2854:
2850:
2838:
2813:
2812:
2808:angular jerk
2792:
2763:
2762:
2731:
2727:acceleration
2702:
2701:
2691:
2670:
2669:
2657:
2645:
2628:
2617:
2602:
2592:
2572:
2535:
2534:
2529:
2524:
2504:
2491:displacement
2484:
2478:
2472:
2461:
2435:
2415:
2400:
2384:
2377:
2280:
2254:
2225:
2200:
2192:
2179:. Retrieved
2165:
2143:
2137:
2123:. Retrieved
2039:
1845:
1755:
1713:
1625:differential
1534:
1515:
1498:
1387:
1385:for others.
1379:Euler angles
1372:
1356:
1337:
1331:
1320:
1295:
1289:
1275:
1260:
1158:
1154:
1152:
1098:
1094:
1087:circular arc
1085:
1079:
1034:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
999:
995:
993:
980:
974:
970:
910:
906:
902:
898:
896:
687: /
683: /
681:displacement
680:
679: /
540: /
502:Displacement
440:
431:
425:Formulations
412:Virtual work
352: /
292:Acceleration
285:Fundamentals
109:angular unit
107:, etc. (any
92:SI unit
2548:solid angle
1267:ISO 80000-3
823:von Neumann
490:Core topics
72:Other names
3318:Categories
3283:m s, N m s
3259:m s,
3205:Lagrangian
3130:Lagrangian
2370:Dimensions
2358:Dimensions
2181:2023-07-17
2132:(11 pages)
2125:2019-10-23
2099:References
1526:infinitely
1352:axis-angle
1233:revolution
1032:arc length
758:d'Alembert
738:Maupertuis
701:Scientists
583:Rigid body
257:Kinematics
3299:ms,
2918:˙
2641:frequency
2568:frequency
2038:out that
2007:θ
1992:θ
1986:−
1810:∈
1787:θ
1734:θ
1653:−
1570:−
1479:−
1450:Δ
1278:clockwise
1258:radians.
1246:π
1219:π
1210:θ
1181:π
1169:θ
1114:θ
1057:θ
947:clockwise
803:Liouville
685:frequency
605:Vibration
322:potential
247:Continuum
242:Celestial
219:Textbooks
3329:Rotation
3243:m s, N s
3066:J s
3062:m s
3037:m s
2998:m s
2969:m s
2945:momentum
2598:velocity
2468:position
2457:distance
2411:absement
2336:SI units
2278:(1934),
2047:See also
1535:While a
1530:rotation
1332:Figure 2
1321:Figure 1
858:Category
783:Hamilton
768:Lagrange
763:Clairaut
728:Horrocks
689:velocity
659:Pendulum
647:reactive
619:Rotation
588:dynamics
538:Inertial
524:Friction
407:Velocity
382:Momentum
262:Kinetics
252:Dynamics
230:Branches
214:Timeline
3271:rotatum
3050:actergy
2986:actergy
2955:impulse
2875::
2626::
2556:rad, sr
2240:Sources
2233:, §4.8)
2073:Azimuth
1668:in the
1623:), the
1344:radians
1082:radians
1022:) both
990:Example
961:Context
939:rotates
931:degrees
927:radians
917:is the
913:– of a
818:Koopman
778:Poisson
773:Laplace
718:Huygens
713:Galileo
558: (
497:Damping
350:Inertia
344:Impulse
317:kinetic
267:Statics
237:Applied
209:History
125:radians
101:degrees
97:radians
86:θ, ϑ, φ
62:radians
3185:energy
3166:moment
3154:torque
3110:energy
3088:weight
3060:
3042:action
3035:
2996:
2978:action
2967:
2938:
2885:
2857:⟩ = ∑
2821:
2772:
2711:
2292:
2265:
2213:
2154:
2121:. 2019
1756:where
1539:is an
1528:small
1522:matrix
1381:. See
1359:vector
1273:(SI).
1100:radius
1090:length
856:
808:Appell
793:Cauchy
788:Jacobi
733:Halley
723:Newton
708:Kepler
560:linear
556:Motion
402:Torque
377:Moment
312:Energy
302:Couple
57:, or 2
3324:Angle
3287:power
3247:power
3217:m s,
3178:m s,
3142:m s,
3103:m s,
3078:force
2588:speed
2520:angle
2209:–89.
2148:(PDF)
1707:(the
1619:(the
935:turns
923:units
919:angle
909:, or
813:Gibbs
798:Routh
753:Euler
392:Speed
387:Space
329:Force
127:(rad)
105:turns
53:, 360
3231:yank
3195:work
3120:work
2834:mass
2788:jerk
2500:area
2431:time
2396:time
2290:ISBN
2263:ISBN
2211:ISBN
2152:ISBN
1415:and
1286:turn
1026:and
955:turn
921:(in
897:The
397:Time
360:Mass
3265:MLT
3180:N m
3148:MLT
3101:kg
3006:MLT
3002:J s
2973:N s
2819:rad
2797:m s
2770:rad
2737:m s
2709:rad
2633:m s
2608:m s
2542:rad
2421:m s
2286:AMS
2207:288
1516:An
1377:or
1235:is
1103:):
925:of
116:In
3320::
3297:kg
3289::
3281:kg
3273::
3257:kg
3249::
3241:kg
3233::
3225:MT
3215:kg
3207::
3203:,
3197::
3193:,
3187::
3176:kg
3168::
3164:,
3156::
3140:kg
3132::
3128:,
3122::
3118:,
3112::
3090::
3086:,
3080::
3072:MT
3064:,
3058:kg
3052::
3048:,
3046:𝒮
3044::
3033:kg
3024::
3020:,
3014::
3000:,
2994:kg
2988::
2984:,
2982:𝒮
2980::
2971:,
2965:kg
2957::
2953:,
2947::
2936:kg
2903::
2897:MT
2883:kg
2869:ML
2844:kg
2836::
2810::
2790::
2760::
2729::
2699::
2695:,
2689::
2682:Hz
2680:,
2667::
2662:,
2655::
2650:,
2643::
2615::
2600::
2596:,
2590::
2583:Hz
2581:,
2570::
2550::
2532::
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2522::
2502::
2489:,
2482:,
2476:,
2470::
2465:,
2459::
2433::
2413::
2398::
2288:,
2284:,
2261:,
2249:;
2106:^
1532:.
1354:.
1302:.
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1018:,
1006:,
957:.
933:,
929:,
905:,
103:,
99:,
51:tr
3301:W
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3276:P
3261:W
3252:P
3236:Y
3219:J
3210:L
3200:W
3190:E
3171:M
3160:τ
3144:J
3135:L
3125:W
3115:E
3105:N
3096:g
3093:F
3083:F
3054:ℵ
3028:L
3026:Δ
3017:L
2990:ℵ
2960:J
2950:p
2940:s
2915:m
2887:m
2878:I
2862:x
2859:m
2855:x
2853:⟨
2851:M
2839:m
2830:M
2823:s
2814:ζ
2802:T
2793:j
2782:T
2774:s
2764:α
2753:s
2743:T
2732:a
2721:T
2713:s
2703:ω
2692:ω
2678:s
2671:n
2658:n
2646:f
2636:T
2629:h
2621:,
2618:ν
2603:v
2593:v
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2525:θ
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2505:A
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2451:1
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2427:T
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2406:s
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2219:.
2184:.
2128:.
2042:.
2023:.
2018:]
2012:1
2004:d
1999:0
1989:d
1981:1
1976:0
1969:0
1964:0
1959:1
1953:[
1948:=
1943:x
1939:L
1935:d
1913:,
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1907:3
1904:(
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1896:s
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1870:,
1865:x
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1854:A
1832:.
1829:)
1826:n
1823:(
1818:o
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1807:A
1784:d
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1692:n
1689:(
1684:o
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1650:=
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1562:=
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1482:1
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1470:A
1464:f
1460:A
1456:=
1453:A
1428:f
1424:A
1401:0
1397:A
1296:N
1243:2
1216:2
1213:=
1188:r
1184:r
1178:2
1172:=
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1117:=
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1051:=
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1035:s
1028:y
1024:x
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1000:O
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181:t
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168:d
162:=
157:F
111:)
64:.
59:π
55:°
20:)
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