1135:
1212:. All other terms are much smaller; the next-largest, with a period of 183 days (0.5 year), has amplitudes 1.3″ and 0.6″ respectively. The periods of all terms larger than 0.0001″ (about as accurately as available technology can measure) lie between 5.5 and 6798 days; for some reason (as with ocean tidal periods) they seem to avoid the range from 34.8 to 91 days, so it is customary to split the nutation into long-period and short-period terms. The long-period terms are calculated and mentioned in the almanacs, while the additional correction due to the short-period terms is usually taken from a table. They can also be calculated from the
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61:
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240:
Initially, however, there is no precession, and the upper part of the top falls sideways and downward, thereby tilting. This gives rise to an imbalance in torques that starts the precession. In falling, the top overshoots the amount of tilt at which it would precess steadily and then oscillates about
143:
is a movement of a rotational axis such that the first Euler angle is constant. Therefore it can be seen that the circular red arrow in the diagram indicates the combined effects of precession and nutation, while nutation in the absence of precession would only change the tilt from vertical (second
166:
is set at a tilt on a horizontal surface and spun rapidly, its rotational axis starts precessing about the vertical. After a short interval, the top settles into a motion in which each point on its rotation axis follows a circular path. The vertical force of gravity produces a horizontal torque
1181:. However, there are other significant periodic terms that must be accounted for depending upon the desired accuracy of the result. A mathematical description (set of equations) that represents nutation is called a "theory of nutation". In the theory, parameters are adjusted in a more or less
607:
618:
259:
with its tip fixed. (A symmetrical top is one with rotational symmetry, or more generally one in which two of the three principal moments of inertia are equal.) Initially, the effect of friction is ignored. The motion of the top can be described by three
936:
1309:, rapidly 'cycling' or 'changing' the 'shield nutation' is frequently mentioned as a means by which to delay the antagonist in their efforts to break through the defences and pillage the Enterprise or other spacecraft.
221:
1177:, which continuously change location relative to each other and thus cause nutation in Earth's axis. The largest component of Earth's nutation has a period of 18.6 years, the same as that of the precession of the
816:{\displaystyle E_{\text{r}}={\frac {1}{2}}I_{1}\left({\dot {\theta }}^{2}+{\dot {\phi }}^{2}\sin ^{2}(\theta )\right)+{\frac {1}{2}}I_{3}\left({\dot {\psi }}+{\dot {\phi }}\cos(\theta )\right)^{2}.}
478:
1004:
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is zero. One of these is not related to a physical angle; the other two determine the upper and lower bounds on the tilt angle, between which the gyroscope oscillates.
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852:
185:
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near the poles causes a change in Earth's nutation, as well as an 11° shift in the axial tilt and a change in Earth's orbit around the Sun.
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602:{\displaystyle E_{\text{r}}={\frac {1}{2}}I_{1}\left(\omega _{1}^{2}+\omega _{2}^{2}\right)+{\frac {1}{2}}I_{3}\omega _{3}^{2}.}
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about the point of contact with the surface; the top rotates in the direction of this torque with an angular velocity
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The nutation of a planet occurs because the gravitational effects of other bodies cause the speed of its
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245:. If the motion is damped, the oscillations will die down until the motion is a steady precession.
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as well as constants that are related to the energy and the gravitational torque. The roots of
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and has the same period of 6798 days (18.61 years). It reaches plus or minus 17″ in
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do not give the best theory; one has to account for deformations of the Earth, including
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of the top about its own axis (third Euler angle). Thus, precession is the change in
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are solved for this system, it is found that the motion depends on two constants
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between the symmetry axis of the top and the vertical (second Euler angle); the
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931:{\displaystyle {\dot {\phi }}={\frac {b-a\cos(\theta )}{\sin ^{2}(\theta )}}.}
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This article is about the concept in physics. For the term in astronomy, see
216:{\displaystyle {\boldsymbol {\tau }}=\mathbf {\Omega } \times \mathbf {L} ,}
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of the top about the vertical (first Euler angle); and the rotation angle
148:(a change in the first Euler angle) is sometimes referred to as nutation.
101: 'nodding, swaying') is a rocking, swaying, or nodding motion in the
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to vary over time, so that the speed is not constant. English astronomer
1567:
The earth's variable rotation : geophysical causes and consequences
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The principal term of nutation is due to the regression of the Moon's
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1596:. Reprint. with corr. Cambridge, Eng.: Cambridge University Press.
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In the case of Earth, the principal sources of tidal force are the
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128:. If it is not caused by forces external to the body, it is called
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The
Feynman Lectures on Physics Vol. I Ch. 20: Rotation in space
120:, or as an intended behaviour of a mechanism. In an appropriate
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60:
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52:
1587:
1479:. Federal Agency for Cartography and Geodesy. 2 April 2009
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International Earth
Rotation and Reference Systems Service
1419:
harvnb error: no target: CITEREFFeynmanLeightonSands2011 (
1592:
The rotation of the earth : a geophysical discussion
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1385:
Engineering dynamics : a comprehensive introduction
1550:(2d ed.). Reading, Mass.: Addison-Wesley Pub. Co.
941:
The tilt is determined by a differential equation for
1382:
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846:). The rate of precession is related to the tilt by
450:axes. Since we are taking a symmetric top, we have
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1569:(Digitally printed 1st pbk. ed.). Cambridge:
1471:"Resolution 83 on non-rigid Earth nutation theory"
998:
930:
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237:is the instantaneous angular momentum of the top.
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105:of a largely axially symmetric object, such as a
1613:
1588:Munk, Walter H.; MacDonald, Gordon J.F. (1975).
1296:, the near-simultaneous detonation of two super-
144:Euler angle). However, in spacecraft dynamics,
1185:method to obtain the best fit to data. Simple
34:. For the term in mechanical engineering, see
124:it can be defined as a change in the second
1439:
1427:
1383:Kasdin, N. Jeremy; Paley, Derek A. (2010).
1258:. Unsourced material may be challenged and
1102:It has been suggested that this section be
1158:that are defined by the Earth's tilt (the
382:axis is the axis of symmetry, the top has
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1278:Learn how and when to remove this message
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320:relative to the plane of the support is
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1523:"NeoProgrammics - Science Computations"
241:this level. This oscillation is called
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14:
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1138:Yearly changes in the location of the
612:In terms of the Euler angles, this is
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1219:
1106:out into another article titled
252:can be explored using the model of a
1256:adding citations to reliable sources
1223:
1216:according to IAU 2000B methodology.
1088:
248:The physics of nutation in tops and
1498:"Basics of Space Flight, Chapter 2"
999:{\displaystyle {\dot {u}}^{2}=f(u)}
366:{\displaystyle V=Mgl\cos(\theta ).}
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1415:Feynman, Leighton & Sands 2011
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376:In a coordinate system where the
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291:and nutation is the change in
56:Nutation of a sphere animation
27:Wobble of the axis of rotation
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1293:The Day the Earth Caught Fire
1150:of Earth with respect to the
1622:Rotation in three dimensions
1354:(2nd ed.). Cambridge :
1146:Nutation subtly changes the
1054:
7:
1546:Goldstein, Herbert (1980).
1312:
1077:discovered the nutation of
1041:of the angles at which the
1019:that depends on parameters
10:
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1571:Cambridge University Press
1389:Princeton University Press
1356:Cambridge University Press
1350:Fundamentals of Geophysics
1290:In the 1961 disaster film
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316:from the pivot point, its
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1502:Jet Propulsion Laboratory
1156:major circles of latitude
40:Nutation (disambiguation)
1346:Lowrie, William (2007).
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1142:near a highway in Mexico
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1065:Perturbation (astronomy)
828:Euler–Lagrange equations
179:such that at any moment
82:in obliquity of a planet
45:Not to be confused with
318:gravitational potential
1565:Lambeck, Kurt (2005).
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38:. For other uses, see
36:Nutation (engineering)
1504:/NASA. 28 August 2013
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1128:Further information:
1061:Astronomical nutation
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156:Further information:
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32:Astronomical nutation
1391:. pp. 526–527.
1252:improve this section
1195:core–mantle boundary
1179:Moon's orbital nodes
1154:plane, shifting the
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300:If the top has mass
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1548:Classical mechanics
1387:. Princeton, N.J.:
1193:and changes in the
1191:mantle inelasticity
1187:rigid body dynamics
842:(each related to a
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158:Rigid body dynamics
1460:, pp. 213–217
1220:In popular culture
1208:and 9.2″ in
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80: Nutation
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68: Rotation
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1506:. Retrieved
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1481:. Retrieved
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1632:Geodynamics
1358:. pp.
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126:Euler angle
1627:Astrometry
1616:Categories
1557:0201029189
1534:References
1508:2015-03-26
1483:2012-08-06
1214:Julian day
1202:nodal line
1148:axial tilt
434:about the
250:gyroscopes
146:precession
89:(from
1319:Libration
1239:does not
1210:obliquity
1206:longitude
1081:in 1728.
1055:Astronomy
970:˙
917:θ
911:
893:θ
887:
878:−
863:˙
860:ϕ
794:θ
788:
779:˙
776:ϕ
764:˙
761:ψ
718:θ
712:
690:˙
687:ϕ
668:˙
665:θ
583:ω
540:ω
522:ω
355:θ
349:
203:×
199:Ω
191:τ
118:in flight
107:gyroscope
1324:Teetotum
1313:See also
1162:and the
1152:ecliptic
306:and its
243:nutation
223:(vector
137:nutation
87:Nutation
47:nunation
1260:removed
1245:sources
1114:Discuss
1039:cosines
826:If the
272:azimuth
97:nūtātiō
1600:
1577:
1554:
1395:
1366:
1183:ad hoc
1048:θ
1009:where
948:θ
946:= cos(
468:. The
444:, and
402:ω
395:ω
388:ω
294:θ
288:φ
282:ψ
276:φ
267:θ
254:heavy
231:where
176:Ω
170:τ
115:bullet
111:planet
78:
72:
66:
1362:–59.
1330:Notes
1104:split
1085:Earth
1015:is a
162:If a
135:Euler
113:, or
93:
91:Latin
1598:ISBN
1575:ISBN
1552:ISBN
1421:help
1393:ISBN
1364:ISBN
1243:any
1241:cite
1175:Moon
1173:and
1063:and
1037:are
1025:and
836:and
408:and
139:. A
1303:In
1254:by
1171:Sun
1166:).
1112:. (
1045:of
902:sin
884:cos
785:cos
703:sin
472:is
346:cos
164:top
132:or
1618::
1573:.
1500:.
1473:.
1407:^
1360:58
1338:^
1197:.
1116:)
439:,
426:,
419:,
400:,
393:,
297:.
109:,
1606:.
1583:.
1560:.
1525:.
1511:.
1486:.
1423:)
1401:.
1372:.
1281:)
1275:(
1270:)
1266:(
1262:.
1248:.
1034:f
1028:b
1022:a
1012:f
994:)
991:u
988:(
985:f
982:=
977:2
967:u
950:)
944:u
926:.
920:)
914:(
906:2
896:)
890:(
881:a
875:b
869:=
839:b
833:a
811:.
806:2
801:)
797:)
791:(
770:+
754:(
747:3
743:I
737:2
734:1
729:+
725:)
721:)
715:(
707:2
697:2
680:+
675:2
657:(
651:1
647:I
641:2
638:1
633:=
628:r
624:E
597:.
592:2
587:3
577:3
573:I
567:2
564:1
559:+
555:)
549:2
544:2
536:+
531:2
526:1
517:(
511:1
507:I
501:2
498:1
493:=
488:r
484:E
465:2
462:I
459:=
456:1
453:I
447:z
441:y
437:x
431:3
428:I
424:2
421:I
417:1
414:I
405:3
398:2
391:1
379:z
361:.
358:)
352:(
343:l
340:g
337:M
334:=
331:V
313:l
303:M
234:L
227:)
211:,
207:L
195:=
49:.
42:.
20:)
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