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reorients the spinning top with respect to the torque that is being exerted. The result is that the torque exerted by gravity – via the pitching motion – elicits gyroscopic precession (which in turn yields a counter torque against the gravity torque) rather than causing the spinning top to fall to its side.
1527:
1828:
In the adjunct image, Earth's apsidal precession is illustrated. As the Earth travels around the Sun, its elliptical orbit rotates gradually over time. The eccentricity of its ellipse and the precession rate of its orbit are exaggerated for visualization. Most orbits in the Solar System have a much
1820:
do not really follow an identical ellipse each time, but actually trace out a flower-petal shape because the major axis of each planet's elliptical orbit also precesses within its orbital plane, partly in response to perturbations in the form of the changing gravitational forces exerted by other
1154:
Due to the way the torque vectors are defined, it is a vector that is perpendicular to the plane of the forces that create it. Thus it may be seen that the angular momentum vector will change perpendicular to those forces. Depending on how the forces are created, they will often rotate with the
998:
In the discussion above, the setup was kept unchanging by preventing pitching around the gimbal axis. In the case of a spinning toy top, when the spinning top starts tilting, gravity exerts a torque. However, instead of rolling over, the spinning top just pitches a little. This pitching motion
454:
192:
1164:
1690:(c. 190–120 BC) is generally accepted to be the earliest known astronomer to recognize and assess the precession of the equinoxes at about 1° per century (which is not far from the actual value for antiquity, 1.38°), although there is some minor dispute about whether he was. In
991:
The same reasoning applies for the bottom half of the wheel, but there the arrows point in the opposite direction to that of the top arrows. Combined over the entire wheel, there is a torque around the gimbal axis when some spinning is added to rotation around a vertical axis.
662:
905:
First, imagine that the entire device is rotating around the (vertical) pivot axis. Then, spinning of the wheel (around the wheelhub) is added. Imagine the gimbal axis to be locked, so that the wheel cannot pitch. The gimbal axis has sensors, that measure whether there is a
38:
586:
1374:
379:
810:{\displaystyle E\left(\exp \left(\left_{\times }\right){\boldsymbol {R}}\right)\approx E\left({\boldsymbol {R}}\right)+\left({\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\times {\boldsymbol {L}}\right)\cdot {\boldsymbol {v}}}
1777:
Precessional movement of the axis (left), precession of the equinox in relation to the distant stars (middle), and the path of the north celestial pole among the stars due to the precession. Vega is the bright star near the bottom
1105:
1618:
In astronomy, precession refers to any of several gravity-induced, slow and continuous changes in an astronomical body's rotational axis or orbital path. Precession of the equinoxes, perihelion precession, changes in the
600:
182:). If an object is asymmetric about its principal axis of rotation, the moment of inertia with respect to each coordinate direction will change with time, while preserving angular momentum. The result is that the
1287:{\displaystyle {\boldsymbol {\omega }}_{\mathrm {p} }={\frac {\ mgr}{I_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }}}={\frac {\tau }{I_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }\sin(\theta )}}}
107:. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called
483:
283:{\displaystyle {\boldsymbol {\omega }}_{\mathrm {p} }={\frac {{\boldsymbol {I}}_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }}{{\boldsymbol {I}}_{\mathrm {p} }\cos({\boldsymbol {\alpha }})}}}
1652:
Axial precession is the movement of the rotational axis of an astronomical body, whereby the axis slowly traces out a cone. In the case of Earth, this type of precession is also known as the
128:
refers to any of several slow changes in an astronomical body's rotational or orbital parameters. An important example is the steady change in the orientation of the axis of rotation of the
1664:. Earth goes through one such complete precessional cycle in a period of approximately 26,000 years or 1° every 72 years, during which the positions of stars will slowly change in both
1127:
189:
The torque-free precession rate of an object with an axis of symmetry, such as a disk, spinning about an axis not aligned with that axis of symmetry can be calculated as follows:
2293:
1059:
1149:
1522:{\displaystyle T_{\mathrm {p} }={\frac {4\pi ^{2}I_{\mathrm {s} }}{\ mgrT_{\mathrm {s} }}}={\frac {4\pi ^{2}I_{\mathrm {s} }\sin(\theta )}{\ \tau T_{\mathrm {s} }}}}
872:(reaction) of the ground is pushing up on it at the point of contact with the support. These two opposite forces produce a torque which causes the top to precess.
2262:(The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and Earth.)
2140:
2101:
2273:"An even larger value for a precession has been found, for a black hole in orbit around a much more massive black hole, amounting to 39 degrees each orbit"
1710:
had drifted roughly one degree over the course of fifty years relative to the position of the stars. The precession of Earth's axis was later explained by
449:{\displaystyle {\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)={\boldsymbol {R}}{\boldsymbol {I}}_{0}^{-1}{\boldsymbol {R}}^{T}{\boldsymbol {L}}}
876:
347:, inelastic dissipation will tend to damp torque-free precession, and the rotation axis will align itself with one of the inertia axes of the body.
350:
For a generic solid object without any axis of symmetry, the evolution of the object's orientation, represented (for example) by a rotation matrix
970:
tends to move in the direction of the top-left arrow in the diagram (shown at 45°) in the direction of rotation around the pivot axis. Section
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2346:
2240:
2202:
2163:
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463:
1703:
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give three types of corrections to the
Newtonian precession, of a gyroscope near a large mass such as Earth, described above. They are:
979:
of the wheel is moving away from the pivot axis, and so a force (again, a
Coriolis force) acts in the same direction as in the case of
657:{\displaystyle E\left({\boldsymbol {R}}\right)={\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\cdot {\frac {\boldsymbol {L}}{2}}}
1852:), which accurately predicted the anomalies. Deviating from Newton's law, Einstein's theory of gravitation predicts an extra term of
2214:
1969:
1594:, a general-relativistic correction accounting for the frame dragging by the Kerr metric of curved space near a large rotating mass.
879:
The response of a rotating system to an applied torque. When the device swivels, and some roll is added, the wheel tends to pitch.
887:
mounted. From inside to outside there are three axes of rotation: the hub of the wheel, the gimbal axis, and the vertical pivot.
1807:
1603:
861:
2116:
2077:
1591:
1588:, a general-relativistic correction accounting for the Schwarzschild metric of curved space near a large non-rotating mass.
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864:
that would intuitively result from the external torque. In the case of a toy top, its weight is acting downwards from its
2337:
1582:, a special-relativistic correction accounting for an object (such as a gyroscope) being accelerated along a curved path.
995:
It is important to note that the torque around the gimbal axis arises without any delay; the response is instantaneous.
104:
1789:
144:
Torque-free precession implies that no external moment (torque) is applied to the body. In torque-free precession, the
31:
1341:
is the distance between the center of mass and the pivot. The torque vector originates at the center of mass. Using
2232:
1695:
356:
that transforms internal to external coordinates, may be numerically simulated. Given the object's fixed internal
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produced by a torque. The general equation that relates the torque to the rate of change of angular momentum is:
581:{\displaystyle {\boldsymbol {R}}_{\text{new}}=\exp \left(\left_{\times }dt\right){\boldsymbol {R}}_{\text{old}}}
357:
186:
of the angular velocities of the body about each axis will vary inversely with each axis' moment of inertia.
1930:
1599:
853:
1829:
smaller eccentricity and precess at a much slower rate, making them nearly circular and nearly stationary.
1110:
1706:
307–345 AD) made a similar discovery centuries later, noting that the position of the Sun during the
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161:
88:
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160:. The inertia matrix is composed of the moments of inertia of a body calculated with respect to separate
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2021:
1845:
1624:
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of its orbit over tens of thousands of years are all important parts of the astronomical theory of
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is forced closer to the pivot axis of the rotation (by the wheel spinning further), because of the
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Mechanical
Systems, Classical Models: Volume II: Mechanics of Discrete and Continuous Systems
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To distinguish between the two horizontal axes, rotation around the wheel hub will be called
2375:
2207:
Science and
Civilization in China: Mathematics and the Sciences of the Heavens and the Earth
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1955:
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this unphysical tendency can be counteracted by repeatedly applying a small rotation vector
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133:
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is moment of inertia about either of the other two equal perpendicular principal axes, and
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1715:
1707:
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958:
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597:. The errors induced by finite time steps tend to increase the rotational kinetic energy:
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Our Place in the
Universe: Understanding Fundamental Astronomy from Ancient Discoveries
1100:{\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}}
865:
2322:
17:
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2034:
2009:
1840:
were prominent among the forms of experimental evidence leading to the acceptance of
1738:, but instead causing it to precess. The torque exerted by the planets, particularly
1718:, Earth has a non-spherical shape, bulging outward at the equator. The gravitational
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1677:
1672:. Over this cycle, Earth's north axial pole moves from where it is now, within 1° of
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37:
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vector changes orientation with time. What makes this possible is a time-varying
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Discrepancies between the observed perihelion precession rate of the planet
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is the angle between the moment of inertia direction and the symmetry axis.
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Under these circumstances the angular velocity of precession is given by:
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11.4 Precession of a
Gyroscope - University Physics Volume 1 | OpenStax
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11.4 Precession of a
Gyroscope - University Physics Volume 1 | OpenStax
1739:
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114:
1602:(sometimes Schwarzschild precession) is used in the prediction of the
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1023:
907:
884:
841:
118:
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1006:
performance at high speed. Precession is also the mechanism behind
2209:, vol. 3, reprint edition. Cambridge: Cambridge University Press.
1813:
1788:
1699:
1563:. In general, the problem is more complicated than this, however.
1335:
is the angle between the spin axis and the axis of precession and
1155:
angular momentum vector, and then circular precession is created.
1017:
874:
849:
129:
47:
36:
935:
of the rotating motion around the (vertical) pivot axis. Section
1874:, which accurately gives the observed excess turning rate of 43
1723:
1606:
of the planets, most notably for the accurate prediction of the
1044:
in the direction of that torque. This causes the top to precess.
856:
of the external torque are constant, the spin axis will move at
1817:
1727:
2323:
Explanation and derivation of formula for precession of a top
2115:
Moebs, William; Ling, Samuel J.; Sanny, Jeff (Sep 19, 2016).
2076:
Moebs, William; Ling, Samuel J.; Sanny, Jeff (Sep 19, 2016).
948:
with respect to the rotation around the pivot axis, and as
1151:
are the torque and angular momentum vectors respectively.
1002:
Precession or gyroscopic considerations have an effect on
848:, but all rotating objects can undergo precession. If the
2275:. 18 March 2008. Archived from the original on 2018-08-07
2205:, p. 120; see also Needham, Joseph; Wang, Ling. (1995) .
844:
is applied to it. The phenomenon is commonly seen in a
2055:. Springer Science & Business Media. p. 420.
913:
In the picture, a section of the wheel has been named
2335:
2183:. Classical Philology, 86, (1991), 48–63. p. 59.
1986:"Lecture 26 – Torque-free rotation – body-fixed axes"
1377:
1321:
is the angular velocity of spin about the spin axis,
1167:
1135:
1113:
1062:
988:. Note that both arrows point in the same direction.
898:. Rotation around the vertical pivot axis is called
894:, and rotation around the gimbal axis will be called
688:
603:
486:
382:
323:
is the moment of inertia about the axis of symmetry,
195:
2008:
Sharma, Ishan; Burns, Joseph A.; Hui, C.-H. (2005).
1730:
apply torque to the equator, attempting to pull the
27:
Periodic change in the direction of a rotation axis
2158:. New York: Taylor and Francis Group. p. 71.
2010:"Nutational damping times in solids of revolution"
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2014:Monthly Notices of the Royal Astronomical Society
1821:planets. This is called perihelion precession or
1035:and the weight of the top causes a change in the
2292:: CS1 maint: bot: original URL status unknown (
2197:, second edition. Cham, Switzerland: Springer.
1553:is the period of spin about the spin axis, and
1642:Axial precession (precession of the equinoxes)
456:Precession occurs by repeatedly recalculating
312:is the spin rate about the axis of symmetry,
8:
1902:For the precession of the Moon's orbit, see
103:, whereas the third Euler angle defines the
961:, with respect to the vertical pivot axis,
944:, therefore, has a lot of angular rotating
99:it can be defined as a change in the first
2139:: CS1 maint: location missing publisher (
2100:: CS1 maint: location missing publisher (
1936:Precession as a form of parallel transport
922:. At the depicted moment in time, section
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2328:Precession and the Milankovich theory
1961:Analytical Mechanics of Space Systems
1122:{\displaystyle {\boldsymbol {\tau }}}
117:, there are two types of precession:
7:
2181:On the cosmical mysteries of Mithras
1571:The special and general theories of
1331:is the acceleration due to gravity,
883:The device depicted on the right is
370:and fixed external angular momentum
156:, or more precisely, a time-varying
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2035:10.1111/j.1365-2966.2005.08864.x
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1080:
343:When an object is not perfectly
2260:Einstein's Theory of Relativity
1808:Anomalous perihelion precession
1604:anomalous perihelion precession
140:Torque-free or Torque neglected
1493:
1487:
1278:
1272:
832:of a spinning object (e.g., a
274:
266:
1:
1026:caused by the normal force –
2331:From Stargazers to Starships
2049:Teodorescu, Petre P (2002).
1850:General Theory of Relativity
1144:{\displaystyle \mathbf {L} }
1048:Precession is the change of
91:of the rotational axis of a
1654:precession of the equinoxes
1371:of precession is given by:
824:Torque-induced precession (
134:precession of the equinoxes
32:Precession (disambiguation)
2478:
2233:Cambridge University Press
1964:, AIAA, pp. 149–150,
1901:
1885:
1805:
1799:
1645:
1567:Relativistic (Einsteinian)
840:in space when an external
121:-free and torque-induced.
29:
2156:Fundamentals of Astronomy
2154:Barbieri, Cesare (2007).
1676:, in a circle around the
1662:precession of the equator
1592:Lense–Thirring precession
1685:ancient Greek astronomer
910:around the gimbal axis.
852:of the rotation and the
358:moment of inertia tensor
301:is the precession rate,
95:body. In an appropriate
2179:Swerdlow, Noel (1991).
1931:Precession (mechanical)
1600:Schwarzschild geodesics
148:is a constant, but the
1836:and that predicted by
1816:of planets around the
1797:
1734:into the plane of the
1666:equatorial coordinates
1623:to its orbit, and the
1523:
1288:
1145:
1123:
1101:
1045:
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670:perpendicular to both
658:
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18:Precession (astronomy)
1792:
1742:, also plays a role.
1524:
1289:
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1102:
1021:
1014:Classical (Newtonian)
878:
826:gyroscopic precession
812:
659:
590:skew-symmetric matrix
583:
462:and applying a small
451:
285:
51:
40:
2462:Dynamics (mechanics)
2319:at Wikimedia Commons
2227:Bradt, Hale (2007).
1984:Boal, David (2001).
1848:(in particular, his
1846:Theory of Relativity
1658:lunisolar precession
1621:tilt of Earth's axis
1586:de Sitter precession
1375:
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1111:
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193:
30:For other uses, see
2193:Sun, Kwok. (2017).
2026:2005MNRAS.359...79S
1838:classical mechanics
1635:Milankovitch cycles
1367:, we find that the
474:for the short time
428:
87:is a change in the
1823:apsidal precession
1802:Apsidal precession
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1794:Apsidal precession
1785:Apsidal precession
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2229:Astronomy Methods
2203:978-3-319-54171-6
2165:978-0-7503-0886-1
2062:978-1-4020-8988-6
1956:Schaub, Hanspeter
1916:Larmor precession
1878:every 100 years.
1712:Newtonian physics
1698:scholar-official
1580:Thomas precession
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181:
175:
169:
150:angular velocity
146:angular momentum
70:
65: Precession
64:
55:
41:Precession of a
21:
2477:
2476:
2472:
2471:
2470:
2468:
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2442:
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2177:
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2153:
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2131:
2124:
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2114:
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2109:
2092:
2085:
2083:
2075:
2074:
2070:
2063:
2048:
2047:
2043:
2007:
2006:
2002:
1993:
1991:
1988:
1983:
1982:
1978:
1972:
1954:
1953:
1949:
1944:
1912:
1907:
1890:
1884:
1865:
1862:
1857:
1856:
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1810:
1804:
1787:
1782:
1781:
1780:
1779:
1774:
1773:
1772:
1764:
1763:
1755:
1754:
1716:oblate spheroid
1708:winter solstice
1650:
1644:
1616:
1569:
1554:
1551:
1545:
1538:
1532:
1504:
1497:
1469:
1459:
1455:
1435:
1422:
1409:
1399:
1395:
1378:
1373:
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1355:
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1342:
1336:
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1168:
1163:
1162:
1131:
1130:
1109:
1108:
1085:
1073:
1058:
1057:
1039:
1033:
1027:
1016:
986:
980:
977:
971:
968:
962:
959:Coriolis effect
955:
949:
942:
936:
929:
923:
920:
914:
822:
773:
767:
763:
747:
708:
707:
703:
696:
692:
684:
683:
677:
671:
665:
628:
607:
599:
598:
595:
592:
566:
528:
524:
518:
514:
513:
512:
508:
487:
482:
481:
475:
466:
464:rotation vector
457:
429:
388:
378:
377:
371:
368:
360:
351:
335:
332:
324:
321:
313:
310:
302:
299:
291:
246:
245:
230:
216:
215:
196:
191:
190:
177:
171:
165:
162:coordinate axes
142:
132:, known as the
105:rotation itself
97:reference frame
75:
68:
66:
62:
60:
53:
35:
28:
23:
22:
15:
12:
11:
5:
2475:
2473:
2465:
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2459:
2454:
2444:
2443:
2439:
2438:
2426:
2414:
2402:
2390:
2378:
2366:
2354:
2334:
2333:
2325:
2320:
2306:
2305:External links
2303:
2300:
2299:
2264:
2248:
2241:
2235:. p. 66.
2219:
2186:
2171:
2164:
2146:
2107:
2068:
2061:
2041:
2000:
1976:
1970:
1946:
1945:
1943:
1940:
1939:
1938:
1933:
1928:
1923:
1918:
1911:
1908:
1886:Main article:
1883:
1880:
1800:Main article:
1786:
1783:
1776:
1775:
1766:
1765:
1757:
1756:
1748:
1747:
1746:
1745:
1744:
1646:Main article:
1643:
1640:
1615:
1612:
1596:
1595:
1589:
1583:
1568:
1565:
1549:
1536:
1512:
1507:
1503:
1495:
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1462:
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1381:
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1262:
1256:
1248:
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1210:
1205:
1199:
1196:
1193:
1184:
1178:
1172:
1139:
1117:
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1066:
1031:
1015:
1012:
984:
975:
966:
953:
940:
927:
918:
866:center of mass
836:) describes a
821:
820:Torque-induced
818:
805:
801:
797:
792:
788:
784:
780:
776:
771:
766:
762:
758:
754:
750:
746:
743:
739:
734:
729:
724:
719:
715:
711:
706:
702:
699:
695:
691:
682:, noting that
651:
648:
643:
639:
635:
631:
626:
622:
618:
614:
610:
606:
593:
570:
564:
560:
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532:
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522:
517:
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507:
504:
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491:
444:
438:
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423:
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407:
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297:
276:
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268:
265:
262:
256:
250:
240:
234:
226:
220:
212:
206:
200:
158:inertia matrix
141:
138:
124:In astronomy,
67:
61:
52:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2474:
2463:
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2348:
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2321:
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2238:
2234:
2230:
2223:
2220:
2216:
2215:0-521-05801-5
2212:
2208:
2204:
2200:
2196:
2190:
2187:
2182:
2175:
2172:
2167:
2161:
2157:
2150:
2147:
2142:
2136:
2120:
2119:
2111:
2108:
2103:
2097:
2081:
2080:
2072:
2069:
2064:
2058:
2054:
2053:
2045:
2042:
2036:
2031:
2027:
2023:
2019:
2015:
2011:
2004:
2001:
1987:
1980:
1977:
1973:
1971:9781600860270
1967:
1963:
1962:
1957:
1951:
1948:
1941:
1937:
1934:
1932:
1929:
1927:
1924:
1922:
1919:
1917:
1914:
1913:
1909:
1905:
1900:
1898:
1894:
1893:Orbital nodes
1889:
1881:
1879:
1877:
1868:
1860:
1851:
1847:
1843:
1839:
1835:
1830:
1826:
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1725:
1721:
1717:
1713:
1709:
1701:
1697:
1693:
1692:ancient China
1689:
1686:
1681:
1679:
1678:ecliptic pole
1675:
1671:
1667:
1663:
1659:
1655:
1649:
1641:
1639:
1638:
1636:
1630:
1626:
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1611:
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1593:
1590:
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1562:
1557:
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1535:
1529:
1505:
1501:
1490:
1484:
1481:
1470:
1464:
1460:
1456:
1450:
1436:
1432:
1429:
1426:
1410:
1404:
1400:
1396:
1390:
1379:
1370:
1361:
1347:
1346:
1339:
1329:
1325:is the mass,
1316:
1315:
1309:
1301:
1275:
1269:
1266:
1241:
1236:
1231:
1203:
1197:
1194:
1191:
1182:
1161:
1160:
1159:
1156:
1152:
1091:
1068:
1055:
1051:
1042:
1038:
1030:
1025:
1020:
1013:
1011:
1009:
1008:gyrocompasses
1005:
1000:
996:
993:
989:
983:
974:
965:
960:
952:
947:
939:
934:
926:
917:
911:
909:
903:
901:
897:
893:
888:
886:
877:
873:
871:
867:
863:
859:
855:
851:
847:
843:
839:
835:
831:
827:
819:
817:
799:
795:
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782:
774:
764:
760:
756:
748:
744:
741:
737:
727:
722:
717:
709:
704:
700:
697:
693:
689:
680:
674:
668:
649:
641:
637:
629:
620:
616:
608:
604:
591:
562:
558:
555:
550:
545:
540:
525:
515:
509:
505:
502:
499:
478:
472:
470:
465:
460:
436:
424:
421:
416:
401:
397:
389:
374:
365:
364:
359:
354:
348:
346:
341:
338:
329:
328:
318:
317:
307:
306:
296:
295:
263:
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210:
187:
185:
180:
174:
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163:
159:
155:
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147:
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116:
112:
111:
106:
102:
98:
94:
90:
86:
79:
74:
59:
50:
44:
39:
33:
19:
2424:Solar System
2277:. Retrieved
2267:
2259:
2251:
2228:
2222:
2206:
2194:
2189:
2180:
2174:
2155:
2149:
2123:. Retrieved
2117:
2110:
2084:. Retrieved
2078:
2071:
2051:
2044:
2017:
2013:
2003:
1992:. Retrieved
1979:
1960:
1950:
1926:Polar motion
1891:
1866:
1858:
1831:
1827:
1811:
1720:tidal forces
1682:
1661:
1657:
1653:
1651:
1632:
1625:eccentricity
1617:
1597:
1570:
1555:
1546:
1533:
1530:
1359:
1344:
1343:
1337:
1327:
1313:
1312:
1299:
1296:
1157:
1153:
1047:
1040:
1028:
1001:
997:
994:
990:
981:
972:
963:
950:
937:
924:
915:
912:
904:
899:
895:
891:
889:
882:
870:normal force
858:right angles
825:
823:
678:
672:
666:
476:
468:
467:
458:
372:
362:
361:
352:
349:
342:
336:
326:
325:
315:
314:
304:
303:
293:
292:
188:
178:
172:
166:
143:
125:
123:
108:
84:
83:
2412:Outer space
2400:Spaceflight
2364:Mathematics
1899:over time.
1714:. Being an
1696:Jin-dynasty
1610:of Mercury
101:Euler angle
89:orientation
80:of a planet
2452:Precession
2446:Categories
2317:Precession
2279:2023-11-15
2125:23 October
2086:23 October
1994:2008-09-17
1942:References
1876:arcseconds
1806:See also:
1688:Hipparchus
1573:relativity
931:is at the
126:precession
85:Precession
2376:Astronomy
2217:, p. 220.
2135:cite book
2096:cite book
2020:(1): 79.
1614:Astronomy
1502:τ
1491:θ
1485:
1461:π
1401:π
1276:θ
1270:
1255:ω
1237:τ
1217:ω
1171:ω
1116:τ
1065:τ
933:perimeter
862:direction
854:magnitude
834:gyroscope
800:⋅
787:×
770:ω
742:≈
723:×
701:
642:⋅
625:ω
551:×
521:ω
506:
422:−
385:ω
271:α
264:
233:ω
199:ω
184:component
78:obliquity
43:gyroscope
2288:cite web
2258:(1924),
2256:Max Born
1958:(2003),
1921:Nutation
1910:See also
1842:Einstein
1778:(right).
1736:ecliptic
1629:ice ages
946:velocity
900:rotation
896:pitching
892:spinning
868:and the
588:for the
480:; e.g.:
110:nutation
93:rotating
73:Nutation
58:Rotation
2436:Science
2352:Physics
2338:Portals
2022:Bibcode
1897:precess
1871:
1855:
1834:Mercury
1740:Jupiter
1722:of the
1674:Polaris
1559:is the
1540:is the
1364:
1350:
1306:is the
1004:bicycle
860:to the
115:physics
2239:
2213:
2201:
2162:
2059:
1968:
1814:orbits
1694:, the
1561:torque
1531:Where
1499:
1424:
1369:period
1297:where
1189:
1107:where
1024:torque
908:torque
885:gimbal
842:torque
290:where
164:(e.g.
119:torque
71:
69:
63:
56:
54:
2457:Earth
2388:Stars
1989:(PDF)
1895:also
1700:Yu Xi
1660:, or
1633:(See
850:speed
345:rigid
130:Earth
113:. In
2294:link
2237:ISBN
2211:ISBN
2199:ISBN
2160:ISBN
2141:link
2127:2020
2102:link
2088:2020
2057:ISBN
1966:ISBN
1812:The
1726:and
1724:Moon
1683:The
1668:and
1598:The
1129:and
1052:and
1022:The
838:cone
830:axis
676:and
2030:doi
2018:359
1844:'s
1818:Sun
1728:Sun
1704:fl.
1482:sin
1267:sin
698:exp
574:old
536:old
503:exp
495:new
261:cos
76:in
2448::
2290:}}
2286:{{
2231:.
2137:}}
2133:{{
2098:}}
2094:{{
2028:.
2016:.
2012:.
1825:.
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1631:.
1544:,
1353:2π
1348:=
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1010:.
982:dm
973:dm
964:dm
951:dm
938:dm
925:dm
916:dm
902:.
477:dt
471:dt
176:,
170:,
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2340::
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2282:.
2245:.
2168:.
2143:)
2129:.
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2090:.
2065:.
2038:.
2032::
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1997:.
1906:.
1867:r
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1488:(
1476:s
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1385:p
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1360:T
1356:/
1345:ω
1338:r
1333:θ
1328:g
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1314:ω
1303:s
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1273:(
1261:s
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1183:=
1177:p
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1076:d
1069:=
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919:1
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779:R
775:(
765:(
761:+
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749:(
745:E
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718:]
714:v
710:[
705:(
694:(
690:E
679:L
673:ω
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634:R
630:(
621:=
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613:R
609:(
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569:R
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531:R
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510:(
500:=
490:R
469:ω
459:ω
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417:0
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406:R
402:=
398:)
394:R
390:(
373:L
367:0
363:I
353:R
337:α
331:p
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320:s
316:I
309:s
305:ω
298:p
294:ω
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267:(
255:p
249:I
239:s
225:s
219:I
211:=
205:p
179:z
173:y
167:x
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20:)
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