749:
2148:
31:
1949:
762:
2584:
2826:
3848:
2143:{\displaystyle {}^{\mathrm {N} }\mathbf {a} ^{\mathrm {Q} }={}^{\mathrm {N} }\mathbf {a} ^{\mathrm {P} }+{}^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }\times \left({}^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }\times \mathbf {r} ^{\mathrm {PQ} }\right)+{}^{\mathrm {N} }{\boldsymbol {\alpha }}^{\mathrm {B} }\times \mathbf {r} ^{\mathrm {PQ} }}
1887:
3208:, walking people, etc., usually rotate according to changes in the direction of the velocity: they move forward with respect to their own orientation. Then, if the body follows a closed orbit in a plane, the angular velocity integrated over a time interval in which the orbit is completed once, is an integer times 360°. This integer is the
1366:
2432:
2636:
1048:) which has a fixed orientation relative to the body (i.e. rotates together with the body), relative to another basis set (or coordinate system), from which the motion of the rigid body is observed. For instance, a basis set with fixed orientation relative to an airplane can be defined as a set of three orthogonal
1708:
3353:
contains only proper rotations. In the opposite case an object is called achiral: the mirror image is a copy, not a different object. Such an object may have a symmetry plane, but not necessarily: there may also be a plane of reflection with respect to which the image of the object is a rotated
897:
of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, namely that all its particles maintain the same distance relative to each other. If the body is rigid, it is sufficient to describe the position of at least
2409:
1575:
1161:, respectively. Indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation (roto-translation) of the body starting from a hypothetic reference position (not necessarily coinciding with a position actually taken by the body during its motion).
3370:
For a (rigid) rectangular transparent sheet, inversion symmetry corresponds to having on one side an image without rotational symmetry and on the other side an image such that what shines through is the image at the top side, upside down. We can distinguish two cases:
1770:
1267:
2579:{\displaystyle {}^{\mathrm {N} }\mathbf {a} ^{\mathrm {R} }={}^{\mathrm {N} }\mathbf {a} ^{\mathrm {Q} }+{}^{\mathrm {B} }\mathbf {a} ^{\mathrm {R} }+2{}^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }\times {}^{\mathrm {B} }\mathbf {v} ^{\mathrm {R} }}
1207:
involves rotation, the instantaneous velocity of any two points on the body will generally not be the same. Two points of a rotating body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous
1457:
2821:{\displaystyle {\boldsymbol {\psi }}(t,\mathbf {r} _{0})=\mathbf {a} (t,\mathbf {r} _{0})-{\boldsymbol {\omega }}(t)\times \mathbf {v} (t,\mathbf {r} _{0})={\boldsymbol {\psi }}_{c}(t)+{\boldsymbol {\alpha }}(t)\times A(t)\mathbf {r} _{0}}
2198:
1602:
1762:
3379:
the sheet surface with the image has a symmetry axis - in this case the two sides are the same, and the mirror image of the object is also the same, again after a rotation by 180° about the axis perpendicular to the mirror
2316:
2301:
1480:
2256:
1926:
is the position vector from P to Q., with coordinates expressed in N (or a frame with the same orientation as N.) This relation can be derived from the temporal invariance of the norm distance between P and Q.
3375:
the sheet surface with the image is not symmetric - in this case the two sides are different, but the mirror image of the object is the same, after a rotation by 180° about the axis perpendicular to the mirror
906:
position relative to the three selected particles is known. However, typically a different, mathematically more convenient, but equivalent approach is used. The position of the whole body is represented by:
3140:
103:
1882:{\displaystyle {}^{\mathrm {N} }\mathbf {v} ^{\mathrm {Q} }={}^{\mathrm {N} }\!\mathbf {v} ^{\mathrm {P} }+{}^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }\times \mathbf {r} ^{\mathrm {PQ} }.}
127:
1199:
of its linear position. Thus, it is the velocity of a reference point fixed to the body. During purely translational motion (motion with no rotation), all points on a rigid body move with the same
3287:
of the body, each component of the angular momentum is a product of a moment of inertia (a principal value of the inertia tensor) times the corresponding component of the angular velocity; the
3055:
3007:
1361:{\displaystyle {}^{\mathrm {N} }\!{\boldsymbol {\omega }}^{\mathrm {B} }={}^{\mathrm {N} }\!{\boldsymbol {\omega }}^{\mathrm {D} }+{}^{\mathrm {D} }\!{\boldsymbol {\omega }}^{\mathrm {B} }.}
3184:
1924:
3088:
2955:
1144:
1261:
The angular velocity of a rigid body B in a reference frame N is equal to the sum of the angular velocity of a rigid body D in N and the angular velocity of B with respect to D:
2858:
1385:
1240:. The relationship between orientation and angular velocity is not directly analogous to the relationship between position and velocity. Angular velocity is not the
2894:
1462:
The norm of a position vector is the spatial distance. Here the coordinates of all three vectors must be expressed in coordinate frames with the same orientation.
2414:
where Q is the point fixed in B that is instantaneously coincident with R at the instant of interest. This relation is often combined with the relation for the
1703:{\displaystyle {}^{\mathrm {N} }\mathbf {a} ^{\mathrm {P} }={\frac {^{\mathrm {N} }\mathrm {d} }{\mathrm {d} t}}({}^{\mathrm {N} }\mathbf {v} ^{\mathrm {P} }).}
793:
3298:
Possible motions in the absence of external forces are translation with constant velocity, steady rotation about a fixed principal axis, and also torque-free
2156:
3835:
1379:
For any set of three points P, Q, and R, the position vector from P to R is the sum of the position vector from P to Q and the position vector from Q to R:
382:
1721:
3473:
501:
2404:{\displaystyle {}^{\mathrm {N} }\mathbf {v} ^{\mathrm {R} }={}^{\mathrm {N} }\mathbf {v} ^{\mathrm {Q} }+{}^{\mathrm {B} }\mathbf {v} ^{\mathrm {R} }}
474:
1584:
operator indicates that the derivative is taken in reference frame N. The result is independent of the selection of O so long as O is fixed in N.
1570:{\displaystyle {}^{\mathrm {N} }\mathbf {v} ^{\mathrm {P} }={\frac {{}^{\mathrm {N} }\mathrm {d} }{\mathrm {d} t}}(\mathbf {r} ^{\mathrm {OP} })}
1149:
In general, when a rigid body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as
2261:
921:
of the body, namely the position of one of the particles of the body, specifically chosen as a reference point (typically coinciding with the
748:
2219:
3503:
2589:
where Q is the point fixed in B that instantaneously coincident with R at the instant of interest. This equation is often combined with
456:
3272:
is independent of the rotational motion. At any time it is equal to the total mass of the rigid body times the translational velocity.
2860:
represents the position of the point/particle with respect to the reference point of the body in terms of the local coordinate system
3790:
3538:
786:
1236:
at all times. During purely rotational motion, all points on the body change position except for those lying on the instantaneous
3355:
3873:
3099:
122:
59:
3816:
117:
377:
3283:
times the angular velocity. When the angular velocity is expressed with respect to a coordinate system coinciding with the
3234:) to describe the linear motion of the body (the linear position, velocity and acceleration vectors depend on the choice).
3405:
1943:
in N with respect to time, the acceleration in reference frame N of a point Q fixed on a rigid body B can be expressed as
2916:. Think of this matrix as three orthogonal unit vectors, one in each column, which define the orientation of the axes of
372:
3878:
779:
766:
527:
450:
3305:
The net external force on the rigid body is always equal to the total mass times the translational acceleration (i.e.,
215:
3813:
3014:
2966:
446:
247:
1229:
860:, a perfectly rigid body does not exist; and objects can only be assumed to be rigid if they are not moving near the
3306:
666:
555:
481:
341:
274:
3151:
1895:
826:
420:
3062:
2929:
2310:
If the point R is moving in the rigid body B while B moves in reference frame N, then the velocity of R in N is
2905:
2897:
2426:
The acceleration in reference frame N of the point R moving in body B while B is moving in frame N is given by
696:
540:
3621:
has a different meaning. In both contexts, the word "linear" is related to the word "line". In mathematics, a
3408:
of a rigid body with one point fixed (i.e., a body with zero translational motion) is given by the underlying
902:
particles. This makes it possible to reconstruct the position of all the other particles, provided that their
3852:
3809:
extensively in place of material accelerations as they simplify the equations and allow for compact notation.
1096:
3338:
3230:
Any point that is rigidly connected to the body can be used as reference point (origin of coordinate system
877:
686:
646:
410:
3309:
holds for the translational motion, even when the net external torque is nonzero, and/or the body rotates).
1764:
in the reference frame N, the velocity of Q in N can be expressed as a function of the velocity of P in N:
3585:
3432:
1041:
1021:
691:
508:
3868:
3468:
3458:
3257:
1151:
701:
676:
362:
180:
1452:{\displaystyle \mathbf {r} ^{\mathrm {PR} }=\mathbf {r} ^{\mathrm {PQ} }+\mathbf {r} ^{\mathrm {QR} }.}
3279:
with respect to the center of mass is the same as without translation: at any time it is equal to the
2834:
1371:
In this case, rigid bodies and reference frames are indistinguishable and completely interchangeable.
3806:
3493:
3413:
3292:
3187:
3143:
3091:
2626:
2201:
963:
721:
716:
681:
589:
585:
577:
567:
357:
350:
106:
3498:
3463:
3385:
3225:
917:
846:
496:
437:
415:
160:
155:
150:
50:
35:
3802:
3654:
3330:
1185:
1005:) and its tip at an arbitrary point of interest on the rigid body, typically coinciding with its
857:
626:
367:
242:
210:
170:
3245:
of the whole system, which generally has the simplest motion for a body moving freely in space;
3786:
3589:
3534:
3317:
3284:
3280:
2901:
2606:
1192:
1045:
1033:
1014:
1002:
990:
979:
894:
865:
636:
593:
550:
545:
486:
262:
252:
145:
3526:
3198:
In 2D, the angular velocity is a scalar, and matrix A(t) simply represents a rotation in the
3453:
3448:
3276:
2958:
2869:
1237:
1233:
1225:
1216:
1209:
1204:
1180:
934:
842:
834:
731:
711:
656:
651:
597:
572:
427:
285:
230:
205:
34:
The position of a rigid body is determined by the position of its center of mass and by its
3649:(namely, it has the same non-restricted meaning as that given, in mathematics, to the word
3416:. The configuration space of a nonfixed (with non-zero translational motion) rigid body is
1580:
where O is any arbitrary point fixed in reference frame N, and the N to the left of the d/d
876:(consisting of the point masses: electrons and nuclei) are often seen as rigid bodies (see
30:
3622:
3610:
3581:
3428:
3424:
3417:
3334:
1593:
1471:
1241:
1228:
about which it is rotating (the existence of this instantaneous axis is guaranteed by the
1196:
1037:
994:
726:
671:
621:
616:
535:
3478:
1718:
For two points P and Q that are fixed on a rigid body B, where B has an angular velocity
2193:{\displaystyle \scriptstyle {{}^{\mathrm {N} }\!{\boldsymbol {\alpha }}^{\mathrm {B} }}}
3350:
3313:
3242:
3209:
1006:
983:
922:
903:
869:
861:
753:
661:
562:
279:
225:
1244:
of orientation, because there is no such concept as an orientation vector that can be
3862:
3653:). In short, both straight and curved lines are supposed to exist. In kinematics and
3483:
1936:
1221:
822:
641:
468:
1757:{\displaystyle \scriptstyle {^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }}}
3798:
3694:
3657:, the following words refer to the same non-restricted meaning of the term "line":
3392:
the sheet surface with the image has no symmetry axis - the two sides are different
3342:
2912:, with respect to the arbitrary reference orientation of another coordinate system
2618:
1025:
971:
706:
631:
320:
200:
3395:
the sheet surface with the image has a symmetry axis - the two sides are the same
17:
3690:
3614:
3488:
1049:
3763:
Kane, Thomas; Levinson, David (1996). "2-8 One Point Moving on a Rigid Body".
3745:
Kane, Thomas; Levinson, David (1996). "2-7 Two Points Fixed on a Rigid Body".
3634:
3602:
3572:
In general, the position of a point or particle is also known, in physics, as
3299:
1245:
1029:
959:
165:
27:
Physical object which does not deform when forces or moments are exerted on it
3248:
a point such that the translational motion is zero or simplified, e.g. on an
1224:
at which the orientation of the rigid body is changing and the instantaneous
966:
quantities describing the motion of a rigid body, such as linear and angular
3202:-plane by an angle which is the integral of the angular velocity over time.
899:
873:
513:
3584:, the "radius" joining the rotating point with the center of rotation), or
3847:
3686:
3436:
3409:
3346:
3269:
1200:
1170:
1157:
1010:
975:
967:
926:
830:
432:
315:
290:
3707:
Kane, Thomas; Levinson, David (1996). "2-4 Auxiliary
Reference Frames".
2296:{\displaystyle {}^{\mathrm {N} }{\boldsymbol {\alpha }}^{\mathrm {B} }.}
3725:
Kane, Thomas; Levinson, David (1996). "2-6 Velocity and
Acceleration".
3213:
3205:
2251:{\displaystyle {}^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }}
807:
405:
258:
175:
3829:
3618:
3609:
means "along a straight or curved line" (the path of the particle in
3561:
3288:
2258:
in a fixed reference frame N, and thus the same angular acceleration
2208:
Angular velocity and acceleration of two points fixed on a rigid body
464:
310:
220:
3237:
However, depending on the application, a convenient choice may be:
1592:
The acceleration of point P in reference frame N is defined as the
3834:
Prof. Dr. Dennis M. Kochmann, Dynamics
Lecture Notes, ETH Zurich.
3630:
3626:
3253:
3190:
of the rigid body (i.e. the spatial acceleration of the origin of
2864:(the rigidity of the body means that this does not depend on time)
998:
838:
819:
300:
295:
237:
29:
3812:
JPL DARTS page has a section on spatial operator algebra (link:
3633:
can be straight, and curved lines are not supposed to exist. In
3249:
1079:
is parallel to the chord line of the wing and directed forward,
850:
837:
on a rigid body remains constant in time regardless of external
305:
268:
1470:
The velocity of point P in reference frame N is defined as the
1086:
is normal to the plane of symmetry and directed rightward, and
2216:, all points on a rigid body B have the same angular velocity
3214:
amount of rotation associated with the vertices of a polygon
1040:). All these methods actually define the orientation of a
3805:
for robotic applications. The author also chooses to use
3135:{\displaystyle {\boldsymbol {\psi }}(t,\mathbf {r} _{0})}
98:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
3212:
with respect to the origin of the velocity. Compare the
868:, a rigid body is usually thought of as a collection of
3057:
represents the total acceleration of the point/particle
950:
Thus, the position of a rigid body has two components:
845:
exerted on it. A rigid body is usually considered as a
3388:
image is achiral. We can distinguish again two cases:
3345:
is different in that sense, i.e., if it has either no
2160:
1725:
3154:
3102:
3065:
3017:
2969:
2932:
2872:
2837:
2639:
2435:
2319:
2264:
2222:
2159:
1952:
1898:
1773:
1724:
1605:
1483:
1388:
1270:
1099:
62:
3815:) as well as an extensive list of references (link:
3264:
When the center of mass is used as reference point:
3663:"rectilinear" (= along a straight line, from Latin
3009:
represents the total velocity of the point/particle
1020:There are several ways to numerically describe the
3693:, the term "line" has the same meaning; namely, a
3178:
3134:
3082:
3049:
3001:
2949:
2908:(angular position) of the local coordinate system
2888:
2852:
2820:
2578:
2403:
2295:
2250:
2192:
2142:
1918:
1881:
1756:
1702:
1569:
1451:
1360:
1232:). All points on a rigid body experience the same
1138:
1013:. This reference point may define the origin of a
97:
2173:
1813:
1340:
1311:
1282:
997:with its tail at an arbitrary reference point in
3674:"curvilinear" (=along a curved line, from Latin
3050:{\displaystyle \mathbf {a} (t,\mathbf {r} _{0})}
3002:{\displaystyle \mathbf {v} (t,\mathbf {r} _{0})}
2591:Acceleration of two points fixed on a rigid body
2422:Acceleration of one point moving on a rigid body
2213:
1931:Acceleration of two points fixed on a rigid body
3337:from one to the other. A rigid body is called
3767:. Sunnyvale, California: OnLine Dynamics, Inc.
3749:. Sunnyvale, California: OnLine Dynamics, Inc.
3729:. Sunnyvale, California: OnLine Dynamics, Inc.
3711:. Sunnyvale, California: OnLine Dynamics, Inc.
3660:"linear" (= along a straight or curved line),
2633:(as opposed to material acceleration above):
787:
8:
3179:{\displaystyle {\boldsymbol {\psi }}_{c}(t)}
2416:Velocity of two points fixed on a rigid body
2306:Velocity of one point moving on a rigid body
1941:Velocity of two points fixed on a rigid body
1919:{\displaystyle \mathbf {r} ^{\mathrm {PQ} }}
1714:Velocity of two points fixed on a rigid body
958:, respectively. The same is true for other
3531:Modelling and control of robot manipulators
3525:Lorenzo Sciavicco, Bruno Siciliano (2000).
878:classification of molecules as rigid rotors
3083:{\displaystyle {\boldsymbol {\alpha }}(t)}
2950:{\displaystyle {\boldsymbol {\omega }}(t)}
1024:of a rigid body, including a set of three
794:
780:
41:
3629:. For those who adopt this definition, a
3161:
3156:
3153:
3123:
3118:
3103:
3101:
3066:
3064:
3038:
3033:
3018:
3016:
2990:
2985:
2970:
2968:
2933:
2931:
2885:
2871:
2844:
2839:
2836:
2812:
2807:
2777:
2759:
2754:
2741:
2736:
2721:
2704:
2692:
2687:
2672:
2660:
2655:
2640:
2638:
2569:
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2563:
2555:
2554:
2552:
2541:
2540:
2535:
2527:
2526:
2524:
2510:
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2504:
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2481:
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2467:
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2453:
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2440:
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2437:
2434:
2394:
2393:
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2380:
2379:
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2366:
2365:
2360:
2352:
2351:
2349:
2338:
2337:
2332:
2324:
2323:
2321:
2318:
2283:
2282:
2277:
2269:
2268:
2266:
2263:
2241:
2240:
2235:
2227:
2226:
2224:
2221:
2181:
2180:
2175:
2166:
2165:
2163:
2161:
2158:
2130:
2129:
2124:
2113:
2112:
2107:
2099:
2098:
2096:
2077:
2076:
2071:
2060:
2059:
2054:
2046:
2045:
2043:
2027:
2026:
2021:
2013:
2012:
2010:
1999:
1998:
1993:
1985:
1984:
1982:
1971:
1970:
1965:
1957:
1956:
1954:
1951:
1906:
1905:
1900:
1897:
1866:
1865:
1860:
1849:
1848:
1843:
1835:
1834:
1832:
1821:
1820:
1815:
1806:
1805:
1803:
1792:
1791:
1786:
1778:
1777:
1775:
1772:
1745:
1744:
1739:
1731:
1730:
1726:
1723:
1687:
1686:
1681:
1673:
1672:
1670:
1655:
1648:
1641:
1640:
1634:
1624:
1623:
1618:
1610:
1609:
1607:
1604:
1554:
1553:
1548:
1533:
1526:
1519:
1518:
1516:
1512:
1502:
1501:
1496:
1488:
1487:
1485:
1482:
1474:in N of the position vector from O to P:
1436:
1435:
1430:
1416:
1415:
1410:
1396:
1395:
1390:
1387:
1348:
1347:
1342:
1333:
1332:
1330:
1319:
1318:
1313:
1304:
1303:
1301:
1290:
1289:
1284:
1275:
1274:
1272:
1269:
1191:The linear velocity of a rigid body is a
1130:
1117:
1104:
1098:
79:
73:
64:
63:
61:
3316:is simply the sum of translational and
1220:is a vector quantity that describes the
3517:
3474:Euler's equations (rigid body dynamics)
3157:
3104:
3067:
2934:
2778:
2755:
2705:
2641:
2536:
2278:
2236:
2176:
2108:
2055:
2022:
1844:
1740:
1588:Mathematical definition of acceleration
1343:
1314:
1285:
1139:{\displaystyle b_{3}=b_{1}\times b_{2}}
113:
49:
3758:
3756:
3740:
3738:
3736:
3720:
3718:
3533:(2nd ed.). Springer. p. 32.
3580:of a line, or line segment (e.g., in
3431:in three dimensions (combinations of
2904:with determinant 1, representing the
1257:Addition theorem for angular velocity
476:Newton's law of universal gravitation
7:
3824:Introduction to Statics and Dynamics
3822:Andy Ruina and Rudra Pratap (2015).
3797:This reference effectively combines
3556:Introduction to Statics and Dynamics
3554:Andy Ruina and Rudra Pratap (2015).
893:The position of a rigid body is the
1466:Mathematical definition of velocity
457:Mechanics of planar particle motion
65:
38:(at least six parameters in total).
2625:of a rigid body is defined as the
2570:
2556:
2542:
2528:
2511:
2497:
2483:
2469:
2455:
2441:
2395:
2381:
2367:
2353:
2339:
2325:
2284:
2270:
2242:
2228:
2182:
2167:
2134:
2131:
2114:
2100:
2081:
2078:
2061:
2047:
2028:
2014:
2000:
1986:
1972:
1958:
1910:
1907:
1870:
1867:
1850:
1836:
1822:
1807:
1793:
1779:
1746:
1732:
1688:
1674:
1656:
1649:
1642:
1625:
1611:
1558:
1555:
1534:
1527:
1520:
1503:
1489:
1440:
1437:
1420:
1417:
1400:
1397:
1349:
1334:
1320:
1305:
1291:
1276:
25:
3641:is used as a synonym of the term
3846:
3354:version. The latter applies for
3329:Two rigid bodies are said to be
3291:is the inertia tensor times the
3119:
3034:
3019:
2986:
2971:
2853:{\displaystyle \mathbf {r} _{0}}
2840:
2808:
2737:
2722:
2688:
2673:
2656:
2564:
2505:
2477:
2449:
2389:
2361:
2333:
2125:
2072:
1994:
1966:
1901:
1861:
1816:
1787:
1682:
1619:
1549:
1497:
1431:
1411:
1391:
1248:to obtain the angular velocity.
761:
760:
747:
80:
3625:is often defined as a straight
2204:of B in the reference frame N.
1184:are measured with respect to a
3527:"§2.4.2 Roll-pitch-yaw angles"
3173:
3167:
3129:
3108:
3077:
3071:
3044:
3023:
2996:
2975:
2944:
2938:
2882:
2876:
2803:
2797:
2788:
2782:
2771:
2765:
2747:
2726:
2715:
2709:
2698:
2677:
2666:
2645:
1694:
1666:
1564:
1544:
1093:is given by the cross product
1:
1375:Addition theorem for position
383:Koopman–von Neumann mechanics
3333:(not copies) if there is no
2612:, attached to the body, the
451:Non-inertial reference frame
3367:= 1 is inversion symmetry.
1165:Linear and angular velocity
929:of the body), together with
889:Linear and angular position
829:is zero or negligible. The
378:Appell's equation of motion
248:Inertial frame of reference
3895:
3826:. Oxford University Press.
3558:. Oxford University Press.
3223:
3783:Robot Dynamics Algorithms
3781:Roy Featherstone (1987).
2605:is the origin of a local
1230:Euler's rotation theorem
1001:(the origin of a chosen
993:can be represented by a
541:Rotating reference frame
373:Hamilton–Jacobi equation
3469:Infinitesimal rotations
1195:quantity, equal to the
1036:(also referred to as a
1034:direction cosine matrix
847:continuous distribution
482:Newton's laws of motion
342:Newton's laws of motion
3874:Rigid bodies mechanics
3180:
3136:
3084:
3051:
3003:
2951:
2890:
2889:{\displaystyle A(t)\,}
2854:
2822:
2580:
2405:
2297:
2252:
2194:
2144:
1920:
1883:
1758:
1704:
1596:in N of its velocity:
1571:
1453:
1362:
1140:
833:between any two given
509:Simple harmonic motion
422:Euler's laws of motion
216:D'Alembert's principle
99:
39:
3807:spatial accelerations
3459:Differential rotation
3258:ball and socket joint
3256:, at the center of a
3181:
3146:of the point/particle
3137:
3085:
3052:
3004:
2952:
2891:
2855:
2823:
2581:
2406:
2298:
2253:
2195:
2145:
1939:the equation for the
1921:
1884:
1759:
1705:
1572:
1454:
1363:
1252:Kinematical equations
1141:
363:Hamiltonian mechanics
181:Statistical mechanics
100:
33:
3855:at Wikimedia Commons
3576:, as opposed to the
3494:Rigid transformation
3414:rotation group SO(3)
3363:, of which the case
3293:angular acceleration
3188:spatial acceleration
3152:
3144:spatial acceleration
3100:
3092:angular acceleration
3063:
3015:
2967:
2930:
2870:
2835:
2637:
2627:spatial acceleration
2433:
2317:
2262:
2220:
2202:angular acceleration
2157:
1950:
1896:
1771:
1722:
1603:
1481:
1386:
1268:
1097:
586:Angular acceleration
578:Rotational frequency
358:Lagrangian mechanics
351:Analytical mechanics
107:Second law of motion
60:
3879:Rotational symmetry
3505:Classical Mechanics
3499:Geometric Mechanics
3464:Rigid body dynamics
3406:configuration space
3400:Configuration space
3386:through and through
3307:Newton's second law
3226:Rigid body dynamics
1242:time rate of change
1197:time rate of change
1017:fixed to the body.
438:Harmonic oscillator
416:Equations of motion
51:Classical mechanics
45:Part of a series on
3423:, the subgroup of
3176:
3132:
3080:
3047:
2999:
2947:
2886:
2850:
2818:
2576:
2401:
2293:
2248:
2190:
2189:
2140:
1916:
1879:
1754:
1753:
1700:
1567:
1449:
1358:
1186:frame of reference
1136:
858:special relativity
814:, also known as a
754:Physics portal
368:Routhian mechanics
243:Frame of reference
95:
40:
3851:Media related to
3590:coordinate system
3425:direct isometries
3318:rotational energy
3094:of the rigid body
2961:of the rigid body
2902:orthogonal matrix
2607:coordinate system
1664:
1542:
1203:. However, when
1046:coordinate system
1015:coordinate system
1003:coordinate system
866:quantum mechanics
804:
803:
551:Centrifugal force
546:Centripetal force
502:Euler's equations
487:Relative velocity
263:Moment of inertia
93:
67:
18:Rigid-body motion
16:(Redirected from
3886:
3850:
3827:
3801:with rigid body
3796:
3769:
3768:
3760:
3751:
3750:
3742:
3731:
3730:
3722:
3713:
3712:
3704:
3698:
3667:= straight, and
3599:
3593:
3578:angular position
3570:
3564:
3559:
3551:
3545:
3544:
3522:
3454:Axes conventions
3449:Angular velocity
3277:angular momentum
3185:
3183:
3182:
3177:
3166:
3165:
3160:
3141:
3139:
3138:
3133:
3128:
3127:
3122:
3107:
3089:
3087:
3086:
3081:
3070:
3056:
3054:
3053:
3048:
3043:
3042:
3037:
3022:
3008:
3006:
3005:
3000:
2995:
2994:
2989:
2974:
2959:angular velocity
2956:
2954:
2953:
2948:
2937:
2920:with respect to
2895:
2893:
2892:
2887:
2859:
2857:
2856:
2851:
2849:
2848:
2843:
2827:
2825:
2824:
2819:
2817:
2816:
2811:
2781:
2764:
2763:
2758:
2746:
2745:
2740:
2725:
2708:
2697:
2696:
2691:
2676:
2665:
2664:
2659:
2644:
2597:Other quantities
2585:
2583:
2582:
2577:
2575:
2574:
2573:
2567:
2561:
2560:
2559:
2553:
2547:
2546:
2545:
2539:
2533:
2532:
2531:
2525:
2516:
2515:
2514:
2508:
2502:
2501:
2500:
2494:
2488:
2487:
2486:
2480:
2474:
2473:
2472:
2466:
2460:
2459:
2458:
2452:
2446:
2445:
2444:
2438:
2410:
2408:
2407:
2402:
2400:
2399:
2398:
2392:
2386:
2385:
2384:
2378:
2372:
2371:
2370:
2364:
2358:
2357:
2356:
2350:
2344:
2343:
2342:
2336:
2330:
2329:
2328:
2322:
2302:
2300:
2299:
2294:
2289:
2288:
2287:
2281:
2275:
2274:
2273:
2267:
2257:
2255:
2254:
2249:
2247:
2246:
2245:
2239:
2233:
2232:
2231:
2225:
2199:
2197:
2196:
2191:
2188:
2187:
2186:
2185:
2179:
2172:
2171:
2170:
2164:
2149:
2147:
2146:
2141:
2139:
2138:
2137:
2128:
2119:
2118:
2117:
2111:
2105:
2104:
2103:
2097:
2091:
2087:
2086:
2085:
2084:
2075:
2066:
2065:
2064:
2058:
2052:
2051:
2050:
2044:
2033:
2032:
2031:
2025:
2019:
2018:
2017:
2011:
2005:
2004:
2003:
1997:
1991:
1990:
1989:
1983:
1977:
1976:
1975:
1969:
1963:
1962:
1961:
1955:
1925:
1923:
1922:
1917:
1915:
1914:
1913:
1904:
1888:
1886:
1885:
1880:
1875:
1874:
1873:
1864:
1855:
1854:
1853:
1847:
1841:
1840:
1839:
1833:
1827:
1826:
1825:
1819:
1812:
1811:
1810:
1804:
1798:
1797:
1796:
1790:
1784:
1783:
1782:
1776:
1763:
1761:
1760:
1755:
1752:
1751:
1750:
1749:
1743:
1737:
1736:
1735:
1709:
1707:
1706:
1701:
1693:
1692:
1691:
1685:
1679:
1678:
1677:
1671:
1665:
1663:
1659:
1653:
1652:
1647:
1646:
1645:
1635:
1630:
1629:
1628:
1622:
1616:
1615:
1614:
1608:
1576:
1574:
1573:
1568:
1563:
1562:
1561:
1552:
1543:
1541:
1537:
1531:
1530:
1525:
1524:
1523:
1517:
1513:
1508:
1507:
1506:
1500:
1494:
1493:
1492:
1486:
1458:
1456:
1455:
1450:
1445:
1444:
1443:
1434:
1425:
1424:
1423:
1414:
1405:
1404:
1403:
1394:
1367:
1365:
1364:
1359:
1354:
1353:
1352:
1346:
1339:
1338:
1337:
1331:
1325:
1324:
1323:
1317:
1310:
1309:
1308:
1302:
1296:
1295:
1294:
1288:
1281:
1280:
1279:
1273:
1238:axis of rotation
1234:angular velocity
1217:Angular velocity
1210:axis of rotation
1181:angular velocity
1145:
1143:
1142:
1137:
1135:
1134:
1122:
1121:
1109:
1108:
935:angular position
872:. For instance,
856:In the study of
796:
789:
782:
769:
764:
763:
756:
752:
751:
657:Johann Bernoulli
652:Daniel Bernoulli
573:Tangential speed
477:
453:
428:Fictitious force
423:
275:Mechanical power
265:
206:Angular momentum
104:
102:
101:
96:
94:
92:
84:
83:
74:
69:
68:
42:
21:
3894:
3893:
3889:
3888:
3887:
3885:
3884:
3883:
3859:
3858:
3843:
3821:
3793:
3780:
3777:
3772:
3765:Dynamics Online
3762:
3761:
3754:
3747:Dynamics Online
3744:
3743:
3734:
3727:Dynamics Online
3724:
3723:
3716:
3709:Dynamics Online
3706:
3705:
3701:
3600:
3596:
3582:circular motion
3574:linear position
3571:
3567:
3553:
3552:
3548:
3541:
3524:
3523:
3519:
3515:
3445:
3429:Euclidean group
3402:
3384:A sheet with a
3360:
3335:proper rotation
3327:
3228:
3222:
3186:represents the
3155:
3150:
3149:
3142:represents the
3117:
3098:
3097:
3090:represents the
3061:
3060:
3032:
3013:
3012:
2984:
2965:
2964:
2957:represents the
2928:
2927:
2868:
2867:
2838:
2833:
2832:
2806:
2753:
2735:
2686:
2654:
2635:
2634:
2599:
2562:
2551:
2534:
2523:
2503:
2492:
2475:
2464:
2447:
2436:
2431:
2430:
2424:
2387:
2376:
2359:
2348:
2331:
2320:
2315:
2314:
2308:
2276:
2265:
2260:
2259:
2234:
2223:
2218:
2217:
2210:
2174:
2162:
2155:
2154:
2123:
2106:
2095:
2070:
2053:
2042:
2041:
2037:
2020:
2009:
1992:
1981:
1964:
1953:
1948:
1947:
1937:differentiating
1933:
1899:
1894:
1893:
1859:
1842:
1831:
1814:
1802:
1785:
1774:
1769:
1768:
1738:
1727:
1720:
1719:
1716:
1680:
1669:
1654:
1637:
1636:
1617:
1606:
1601:
1600:
1594:time derivative
1590:
1547:
1532:
1515:
1514:
1495:
1484:
1479:
1478:
1472:time derivative
1468:
1429:
1409:
1389:
1384:
1383:
1377:
1341:
1329:
1312:
1300:
1283:
1271:
1266:
1265:
1259:
1254:
1176:linear velocity
1167:
1126:
1113:
1100:
1095:
1094:
1092:
1085:
1078:
1071:
1064:
1057:
1038:rotation matrix
938:(also known as
913:linear position
891:
886:
800:
759:
746:
745:
738:
737:
736:
611:
603:
602:
582:
536:Circular motion
530:
520:
519:
518:
475:
445:
442:
421:
400:
392:
391:
388:
387:
345:
335:
327:
326:
325:
284:
280:Mechanical work
273:
257:
195:
187:
186:
185:
140:
132:
109:
85:
75:
58:
57:
28:
23:
22:
15:
12:
11:
5:
3892:
3890:
3882:
3881:
3876:
3871:
3861:
3860:
3857:
3856:
3842:
3841:External links
3839:
3838:
3837:
3832:
3819:
3810:
3791:
3776:
3773:
3771:
3770:
3752:
3732:
3714:
3699:
3684:
3683:
3678:= curved, and
3672:
3661:
3594:
3565:
3546:
3539:
3516:
3514:
3511:
3510:
3509:
3501:
3496:
3491:
3486:
3481:
3476:
3471:
3466:
3461:
3456:
3451:
3444:
3441:
3401:
3398:
3397:
3396:
3393:
3382:
3381:
3377:
3358:
3351:symmetry group
3326:
3323:
3322:
3321:
3314:kinetic energy
3310:
3303:
3296:
3285:principal axes
3281:inertia tensor
3273:
3262:
3261:
3246:
3243:center of mass
3224:Main article:
3221:
3218:
3210:winding number
3196:
3195:
3175:
3172:
3169:
3164:
3159:
3147:
3131:
3126:
3121:
3116:
3113:
3110:
3106:
3095:
3079:
3076:
3073:
3069:
3058:
3046:
3041:
3036:
3031:
3028:
3025:
3021:
3010:
2998:
2993:
2988:
2983:
2980:
2977:
2973:
2962:
2946:
2943:
2940:
2936:
2925:
2884:
2881:
2878:
2875:
2865:
2847:
2842:
2815:
2810:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2780:
2776:
2773:
2770:
2767:
2762:
2757:
2752:
2749:
2744:
2739:
2734:
2731:
2728:
2724:
2720:
2717:
2714:
2711:
2707:
2703:
2700:
2695:
2690:
2685:
2682:
2679:
2675:
2671:
2668:
2663:
2658:
2653:
2650:
2647:
2643:
2598:
2595:
2587:
2586:
2572:
2566:
2558:
2550:
2544:
2538:
2530:
2522:
2519:
2513:
2507:
2499:
2491:
2485:
2479:
2471:
2463:
2457:
2451:
2443:
2423:
2420:
2412:
2411:
2397:
2391:
2383:
2375:
2369:
2363:
2355:
2347:
2341:
2335:
2327:
2307:
2304:
2292:
2286:
2280:
2272:
2244:
2238:
2230:
2209:
2206:
2184:
2178:
2169:
2151:
2150:
2136:
2133:
2127:
2122:
2116:
2110:
2102:
2094:
2090:
2083:
2080:
2074:
2069:
2063:
2057:
2049:
2040:
2036:
2030:
2024:
2016:
2008:
2002:
1996:
1988:
1980:
1974:
1968:
1960:
1932:
1929:
1912:
1909:
1903:
1890:
1889:
1878:
1872:
1869:
1863:
1858:
1852:
1846:
1838:
1830:
1824:
1818:
1809:
1801:
1795:
1789:
1781:
1748:
1742:
1734:
1729:
1715:
1712:
1711:
1710:
1699:
1696:
1690:
1684:
1676:
1668:
1662:
1658:
1651:
1644:
1639:
1633:
1627:
1621:
1613:
1589:
1586:
1578:
1577:
1566:
1560:
1557:
1551:
1546:
1540:
1536:
1529:
1522:
1511:
1505:
1499:
1491:
1467:
1464:
1460:
1459:
1448:
1442:
1439:
1433:
1428:
1422:
1419:
1413:
1408:
1402:
1399:
1393:
1376:
1373:
1369:
1368:
1357:
1351:
1345:
1336:
1328:
1322:
1316:
1307:
1299:
1293:
1287:
1278:
1258:
1255:
1253:
1250:
1246:differentiated
1166:
1163:
1133:
1129:
1125:
1120:
1116:
1112:
1107:
1103:
1090:
1083:
1076:
1069:
1062:
1055:
1007:center of mass
984:kinetic energy
948:
947:
946:) of the body.
930:
923:center of mass
904:time-invariant
890:
887:
885:
882:
862:speed of light
802:
801:
799:
798:
791:
784:
776:
773:
772:
771:
770:
757:
740:
739:
735:
734:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
654:
649:
644:
639:
634:
629:
624:
619:
613:
612:
609:
608:
605:
604:
601:
600:
581:
580:
575:
570:
565:
563:Coriolis force
560:
559:
558:
548:
543:
538:
532:
531:
526:
525:
522:
521:
517:
516:
511:
506:
505:
504:
499:
489:
484:
479:
472:
461:
460:
459:
454:
441:
440:
435:
430:
425:
418:
413:
408:
402:
401:
398:
397:
394:
393:
390:
389:
386:
385:
380:
375:
370:
365:
360:
354:
348:
346:
339:
336:
333:
332:
329:
328:
324:
323:
318:
313:
308:
303:
298:
293:
288:
282:
277:
271:
266:
255:
250:
245:
240:
235:
234:
233:
228:
218:
213:
208:
203:
197:
196:
193:
192:
189:
188:
184:
183:
178:
173:
168:
163:
158:
153:
148:
142:
141:
138:
137:
134:
133:
131:
130:
125:
120:
114:
111:
110:
105:
91:
88:
82:
78:
72:
54:
53:
47:
46:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3891:
3880:
3877:
3875:
3872:
3870:
3867:
3866:
3864:
3854:
3849:
3845:
3844:
3840:
3836:
3833:
3830:
3825:
3820:
3817:
3814:
3811:
3808:
3804:
3800:
3794:
3792:0-89838-230-0
3788:
3784:
3779:
3778:
3774:
3766:
3759:
3757:
3753:
3748:
3741:
3739:
3737:
3733:
3728:
3721:
3719:
3715:
3710:
3703:
3700:
3696:
3692:
3688:
3681:
3677:
3673:
3670:
3666:
3662:
3659:
3658:
3656:
3652:
3648:
3644:
3640:
3636:
3632:
3628:
3624:
3620:
3616:
3612:
3608:
3604:
3598:
3595:
3591:
3587:
3583:
3579:
3575:
3569:
3566:
3562:
3557:
3550:
3547:
3542:
3540:1-85233-221-2
3536:
3532:
3528:
3521:
3518:
3512:
3508:
3506:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3484:Born rigidity
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3455:
3452:
3450:
3447:
3446:
3442:
3440:
3438:
3434:
3430:
3426:
3422:
3420:
3415:
3411:
3407:
3399:
3394:
3391:
3390:
3389:
3387:
3378:
3374:
3373:
3372:
3368:
3366:
3362:
3361:
3352:
3348:
3344:
3340:
3336:
3332:
3324:
3319:
3315:
3311:
3308:
3304:
3301:
3297:
3294:
3290:
3286:
3282:
3278:
3274:
3271:
3268:The (linear)
3267:
3266:
3265:
3259:
3255:
3251:
3247:
3244:
3240:
3239:
3238:
3235:
3233:
3227:
3219:
3217:
3215:
3211:
3207:
3203:
3201:
3193:
3189:
3170:
3162:
3148:
3145:
3124:
3114:
3111:
3096:
3093:
3074:
3059:
3039:
3029:
3026:
3011:
2991:
2981:
2978:
2963:
2960:
2941:
2926:
2923:
2919:
2915:
2911:
2907:
2903:
2899:
2879:
2873:
2866:
2863:
2845:
2831:
2830:
2829:
2813:
2800:
2794:
2791:
2785:
2774:
2768:
2760:
2750:
2742:
2732:
2729:
2718:
2712:
2701:
2693:
2683:
2680:
2669:
2661:
2651:
2648:
2632:
2628:
2624:
2621:
2620:
2615:
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2608:
2604:
2596:
2594:
2592:
2548:
2520:
2517:
2489:
2461:
2429:
2428:
2427:
2421:
2419:
2417:
2373:
2345:
2313:
2312:
2311:
2305:
2303:
2290:
2215:
2212:As mentioned
2207:
2205:
2203:
2120:
2092:
2088:
2067:
2038:
2034:
2006:
1978:
1946:
1945:
1944:
1942:
1938:
1930:
1928:
1876:
1856:
1828:
1799:
1767:
1766:
1765:
1728:
1713:
1697:
1660:
1638:
1631:
1599:
1598:
1597:
1595:
1587:
1585:
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1538:
1509:
1477:
1476:
1475:
1473:
1465:
1463:
1446:
1426:
1406:
1382:
1381:
1380:
1374:
1372:
1355:
1326:
1297:
1264:
1263:
1262:
1256:
1251:
1249:
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1222:angular speed
1219:
1218:
1213:
1211:
1206:
1202:
1198:
1194:
1189:
1187:
1183:
1182:
1177:
1174:(also called
1173:
1172:
1164:
1162:
1160:
1159:
1154:
1153:
1147:
1131:
1127:
1123:
1118:
1114:
1110:
1105:
1101:
1089:
1082:
1075:
1068:
1061:
1054:
1051:
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1043:
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1035:
1031:
1027:
1023:
1018:
1016:
1012:
1008:
1004:
1000:
996:
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987:
985:
981:
977:
973:
969:
965:
961:
957:
953:
945:
941:
937:
936:
931:
928:
924:
920:
919:
914:
910:
909:
908:
905:
901:
896:
888:
883:
881:
879:
875:
871:
867:
863:
859:
854:
852:
848:
844:
840:
836:
832:
828:
824:
821:
817:
813:
809:
797:
792:
790:
785:
783:
778:
777:
775:
774:
768:
758:
755:
750:
744:
743:
742:
741:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
614:
607:
606:
599:
595:
591:
587:
584:
583:
579:
576:
574:
571:
569:
566:
564:
561:
557:
554:
553:
552:
549:
547:
544:
542:
539:
537:
534:
533:
529:
524:
523:
515:
512:
510:
507:
503:
500:
498:
495:
494:
493:
490:
488:
485:
483:
480:
478:
473:
470:
466:
463:
462:
458:
455:
452:
448:
444:
443:
439:
436:
434:
431:
429:
426:
424:
419:
417:
414:
412:
409:
407:
404:
403:
396:
395:
384:
381:
379:
376:
374:
371:
369:
366:
364:
361:
359:
356:
355:
353:
352:
347:
344:
343:
338:
337:
331:
330:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
283:
281:
278:
276:
272:
270:
267:
264:
260:
256:
254:
251:
249:
246:
244:
241:
239:
236:
232:
229:
227:
224:
223:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
198:
191:
190:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
147:
144:
143:
136:
135:
129:
126:
124:
121:
119:
116:
115:
112:
108:
89:
86:
76:
70:
56:
55:
52:
48:
44:
43:
37:
32:
19:
3869:Rigid bodies
3853:Rigid bodies
3823:
3799:screw theory
3785:. Springer.
3782:
3764:
3746:
3726:
3708:
3702:
3695:contour line
3679:
3675:
3668:
3664:
3650:
3646:
3642:
3638:
3606:
3597:
3577:
3573:
3568:
3555:
3549:
3530:
3520:
3504:
3479:Euler's laws
3433:translations
3418:
3403:
3383:
3369:
3364:
3356:
3343:mirror image
3328:
3263:
3236:
3231:
3229:
3204:
3199:
3197:
3191:
2921:
2917:
2913:
2909:
2861:
2630:
2623:acceleration
2622:
2617:
2613:
2609:
2602:
2600:
2590:
2588:
2425:
2415:
2413:
2309:
2211:
2152:
1940:
1934:
1891:
1717:
1591:
1581:
1579:
1469:
1461:
1378:
1370:
1260:
1215:
1214:
1190:
1179:
1175:
1169:
1168:
1156:
1150:
1148:
1087:
1080:
1073:
1072:, such that
1066:
1059:
1052:
1050:unit vectors
1026:Euler angles
1019:
988:
972:acceleration
955:
951:
949:
943:
939:
933:
916:
912:
892:
870:point masses
855:
816:rigid object
815:
811:
805:
596: /
592: /
590:displacement
588: /
491:
449: /
411:Displacement
349:
340:
334:Formulations
321:Virtual work
261: /
201:Acceleration
194:Fundamentals
3697:is a curve.
3691:meteorology
3637:, the term
3617:, however,
3615:mathematics
3507:(Goldstein)
3489:Rigid rotor
2906:orientation
2900:matrix, an
2898:orientation
1152:translation
1022:orientation
989:The linear
940:orientation
827:deformation
732:von Neumann
399:Core topics
3863:Categories
3775:References
3682:= spread).
3671:= spread),
3643:trajectory
3635:kinematics
3603:kinematics
3312:The total
3300:precession
1030:quaternion
898:three non-
884:Kinematics
812:rigid body
667:d'Alembert
647:Maupertuis
610:Scientists
492:Rigid body
166:Kinematics
3586:basis set
3437:rotations
3331:different
3158:ψ
3105:ψ
3068:α
2935:ω
2792:×
2779:α
2756:ψ
2719:×
2706:ω
2702:−
2642:ψ
2549:×
2537:ω
2279:α
2237:ω
2177:α
2121:×
2109:α
2068:×
2056:ω
2035:×
2023:ω
1857:×
1845:ω
1741:ω
1344:ω
1315:ω
1286:ω
1124:×
1042:basis set
960:kinematic
900:collinear
874:molecules
825:in which
712:Liouville
594:frequency
514:Vibration
231:potential
156:Continuum
151:Celestial
128:Textbooks
3803:dynamics
3687:topology
3655:dynamics
3443:See also
3410:manifold
3347:symmetry
3325:Geometry
3270:momentum
3220:Kinetics
3206:Vehicles
1201:velocity
1171:Velocity
1158:rotation
1011:centroid
991:position
976:momentum
968:velocity
944:attitude
927:centroid
918:position
895:position
831:distance
767:Category
692:Hamilton
677:Lagrange
672:Clairaut
637:Horrocks
598:velocity
568:Pendulum
556:reactive
528:Rotation
497:dynamics
447:Inertial
433:Friction
316:Velocity
291:Momentum
171:Kinetics
161:Dynamics
139:Branches
123:Timeline
36:attitude
3828:(link:
3560:(link:
3427:of the
3412:of the
3349:or its
3341:if its
2896:is the
2614:spatial
2200:is the
1032:, or a
980:impulse
964:kinetic
956:angular
843:moments
818:, is a
808:physics
727:Koopman
687:Poisson
682:Laplace
627:Huygens
622:Galileo
467: (
406:Damping
259:Inertia
253:Impulse
226:kinetic
176:Statics
146:Applied
118:History
3789:
3680:linere
3676:curvus
3669:linere
3665:rectus
3619:linear
3613:). In
3607:linear
3537:
3380:plane.
3376:plane.
3339:chiral
3289:torque
3260:, etc.
2828:where
2153:where
1892:where
1205:motion
1193:vector
1178:) and
995:vector
982:, and
952:linear
839:forces
835:points
765:
717:Appell
702:Cauchy
697:Jacobi
642:Halley
632:Newton
617:Kepler
469:linear
465:Motion
311:Torque
286:Moment
221:Energy
211:Couple
3651:curve
3645:, or
3631:curve
3627:curve
3611:space
3588:, or
3513:Notes
3254:hinge
2619:twist
2214:above
999:space
942:, or
864:. In
820:solid
722:Gibbs
707:Routh
662:Euler
301:Speed
296:Space
238:Force
3787:ISBN
3689:and
3647:path
3639:line
3623:line
3535:ISBN
3435:and
3404:The
3275:The
3250:axle
3241:the
1226:axis
1155:and
1044:(or
1028:, a
962:and
954:and
932:the
911:the
851:mass
823:body
810:, a
306:Time
269:Mass
3685:In
3601:In
3439:).
3421:(3)
3252:or
2629:of
2616:or
2601:If
1935:By
1009:or
925:or
915:or
880:).
849:of
841:or
806:In
3865::
3831:).
3818:).
3755:^
3735:^
3717:^
3605:,
3529:.
3359:2n
3216:.
3200:xy
3194:).
2593:.
2418:.
1212:.
1188:.
1146:.
1065:,
1058:,
986:.
978:,
974:,
970:,
853:.
3795:.
3592:.
3563:)
3543:.
3419:E
3365:n
3357:S
3320:.
3302:.
3295:.
3232:L
3192:L
3174:)
3171:t
3168:(
3163:c
3130:)
3125:0
3120:r
3115:,
3112:t
3109:(
3078:)
3075:t
3072:(
3045:)
3040:0
3035:r
3030:,
3027:t
3024:(
3020:a
2997:)
2992:0
2987:r
2982:,
2979:t
2976:(
2972:v
2945:)
2942:t
2939:(
2924:.
2922:G
2918:L
2914:G
2910:L
2883:)
2880:t
2877:(
2874:A
2862:L
2846:0
2841:r
2814:0
2809:r
2804:)
2801:t
2798:(
2795:A
2789:)
2786:t
2783:(
2775:+
2772:)
2769:t
2766:(
2761:c
2751:=
2748:)
2743:0
2738:r
2733:,
2730:t
2727:(
2723:v
2716:)
2713:t
2710:(
2699:)
2694:0
2689:r
2684:,
2681:t
2678:(
2674:a
2670:=
2667:)
2662:0
2657:r
2652:,
2649:t
2646:(
2631:C
2610:L
2603:C
2571:R
2565:v
2557:B
2543:B
2529:N
2521:2
2518:+
2512:R
2506:a
2498:B
2490:+
2484:Q
2478:a
2470:N
2462:=
2456:R
2450:a
2442:N
2396:R
2390:v
2382:B
2374:+
2368:Q
2362:v
2354:N
2346:=
2340:R
2334:v
2326:N
2291:.
2285:B
2271:N
2243:B
2229:N
2183:B
2168:N
2135:Q
2132:P
2126:r
2115:B
2101:N
2093:+
2089:)
2082:Q
2079:P
2073:r
2062:B
2048:N
2039:(
2029:B
2015:N
2007:+
2001:P
1995:a
1987:N
1979:=
1973:Q
1967:a
1959:N
1911:Q
1908:P
1902:r
1877:.
1871:Q
1868:P
1862:r
1851:B
1837:N
1829:+
1823:P
1817:v
1808:N
1800:=
1794:Q
1788:v
1780:N
1747:B
1733:N
1698:.
1695:)
1689:P
1683:v
1675:N
1667:(
1661:t
1657:d
1650:d
1643:N
1632:=
1626:P
1620:a
1612:N
1582:t
1565:)
1559:P
1556:O
1550:r
1545:(
1539:t
1535:d
1528:d
1521:N
1510:=
1504:P
1498:v
1490:N
1447:.
1441:R
1438:Q
1432:r
1427:+
1421:Q
1418:P
1412:r
1407:=
1401:R
1398:P
1392:r
1356:.
1350:B
1335:D
1327:+
1321:D
1306:N
1298:=
1292:B
1277:N
1132:2
1128:b
1119:1
1115:b
1111:=
1106:3
1102:b
1091:3
1088:b
1084:2
1081:b
1077:1
1074:b
1070:3
1067:b
1063:2
1060:b
1056:1
1053:b
795:e
788:t
781:v
471:)
90:t
87:d
81:p
77:d
71:=
66:F
20:)
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