Editing Nutation

{{Short description|Wobble of the axis of rotation}}
{{about|the concept in physics|the term in astronomy|Astronomical nutation|the term in mechanical engineering|Nutation (engineering)|other uses}}
{{distinguish|nunation}}
[[File:Nutation.gif|thumb|Nutation of a sphere animation]]
[[File:Praezession.svg|thumb|{{legend-line|green solid 2px|Rotation}} {{legend-line|blue solid 2px|Precession}}{{legend-line|red solid 2px|Nutation}}in obliquity of a planet]]

'''Nutation''' ({{etymology|la|{{wikt-lang|la|nūtātiō}}|nodding, swaying}}) is a rocking, swaying, or nodding motion in the [[axis of rotation]] of a largely axially symmetric object, such as a [[gyroscope]], [[planet]], or [[bullet]] [[external ballistics|in flight]], or as an intended behaviour of a mechanism. In an appropriate [[frame of reference|reference frame]] it can be defined as a change in the second [[Euler angles#Euler rotations|Euler angle]]. If it is not caused by forces external to the body, it is called free nutation or Euler nutation (after [[Leonhard Euler]]).<ref name=Lowrie/> A pure nutation is a movement of a rotational axis such that the first Euler angle is constant.{{citation needed|date=August 2012}} 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 Euler angle). However, in spacecraft dynamics, [[precession]] (a change in the first Euler angle) is sometimes referred to as nutation.<ref>{{cite book|last=Kasdin|first=N. Jeremy|title=Engineering dynamics : a comprehensive introduction|date=2010|publisher=[[Princeton University Press]]|location=Princeton, N.J.|isbn=9780691135373|pages=526–527|author2=Paley, Derek A. }}</ref>

==In a rigid body==
{{further|Rigid body dynamics}}
If a [[Spinning top|top]] 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 {{math|'''&tau;'''}} about the point of contact with the surface; the top rotates in the direction of this torque with an angular velocity {{math|'''&Omega;'''}} such that at any moment
:<math> \boldsymbol{\tau} = \mathbf{\Omega} \times \mathbf{L},</math>       (vector [[cross product]])

where {{math|'''L'''}} is the instantaneous angular momentum of the top.<ref name=Feynman>{{harvnb|Feynman|Leighton|Sands|2011|pp=20–7{{clarify|date=December 2013}}}}</ref>

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 this level. This oscillation is called ''nutation''. If the motion is damped, the oscillations will die down until the motion is a steady precession.<ref name=Feynman/><ref name=Goldstein220>{{harvnb|Goldstein|1980|p=220}}</ref>

The physics of nutation in tops and [[gyroscope]]s can be explored using the model of a ''heavy [[symmetrical top]]'' 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 [[Euler angles]]: the tilt angle {{math|''&theta;''}} between the symmetry axis of the top and the vertical (second Euler angle); the [[azimuth]] {{math|''&phi;''}} of the top about the vertical (first Euler angle); and the rotation angle {{math|''&psi;''}} of the top about its own axis (third Euler angle). Thus, precession is the change in {{math|''&phi;''}} and nutation is the change in {{math|''&theta;''}}.<ref name=Goldstein217>{{harvnb|Goldstein|1980|p=217}}</ref>

If the top has mass {{math|''M''}} and its [[center of mass]] is at a distance {{math|''l''}} from the pivot point, its [[gravitational potential]] relative to the plane of the support is
:<math>V = Mgl\cos(\theta).</math>

In a coordinate system where the {{math|''z''}} axis is the axis of symmetry, the top has [[angular velocity|angular velocities]] {{math|''&omega;''<sub>1</sub>, ''&omega;''<sub>2</sub>, ''&omega;''<sub>3</sub>}} and [[moments of inertia]] {{math|''I''<sub>1</sub>, ''I''<sub>2</sub>, ''I''<sub>3</sub>}} about the {{math|''x'', ''y''}}, and {{math|''z''}} axes. Since we are taking a symmetric top, we have {{math|''I''<sub>1</sub>}}={{math|''I''<sub>2</sub>}}. The [[kinetic energy]] is

:<math>E_\text{r} = \frac{1}{2}I_1\left(\omega_1^2 + \omega_2^2\right) + \frac{1}{2}I_3\omega_3^2.</math>

In terms of the Euler angles, this is
:<math>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.</math>

If the [[Lagrangian mechanics|Euler–Lagrange equations]] are solved for this system, it is found that the motion depends on two constants {{math|''a''}} and {{math|''b''}} (each related to a [[constant of motion]]). The rate of precession is related to the tilt by
:<math>\dot{\phi} = \frac{b - a\cos(\theta)}{\sin^2(\theta)}.</math>

The tilt is determined by a differential equation for {{math|''u'' {{=}} cos(''&theta;'')}} of the form
:<math>\dot{u}^2 = f(u)</math>

where {{math|''f''}} is a [[cubic function|cubic polynomial]] that depends on parameters {{math|''a''}} and {{math|''b''}} as well as constants that are related to the energy and the gravitational torque. The roots of {{math|''f''}} are [[cosine]]s of the angles at which the [[time derivative|rate of change]] of {{math|''&theta;''}} 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.<ref>{{harvnb|Goldstein|1980|pp=213–217}}</ref>

==Astronomy==
{{main|Astronomical nutation|Perturbation (astronomy)}}

The nutation of a planet occurs because the gravitational effects of other bodies cause the speed of its [[axial precession]] to vary over time, so that the speed is not constant. English astronomer [[James Bradley]] discovered the nutation of [[Earth's rotation|Earth's axis]] in 1728.

===Earth===
{{split section|Earth's nutation|date=October 2020|discuss=Talk:Astronomical nutation#Split Earth's nutation}}
{{further|Geodynamics}}
[[File:Trópico de Cáncer en México - Carretera 83 (Vía Corta) Zaragoza-Victoria, Km 27+800.jpg|thumb|240px|Yearly changes in the location of the [[Tropic of Cancer]] near a highway in Mexico]]

Nutation subtly changes the [[axial tilt]] of Earth with respect to the [[ecliptic]] plane, shifting the [[Circle of latitude#Major circles of latitude|major circles of latitude]] that are defined by the Earth's tilt (the [[tropical circle]]s and the [[polar circle]]s).

In the case of Earth, the principal sources of tidal force are the [[Sun]] and [[Moon]], 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 [[Lunar node|Moon's orbital nodes]].<ref name=Lowrie>{{cite book |last=Lowrie |first=William |title=Fundamentals of Geophysics |url=https://archive.org/details/fundamentalsgeop00lowr |url-access=limited |date=2007 |publisher=[[Cambridge University Press]] |location=Cambridge [u.a.] |isbn=9780521675963 |pages=[https://archive.org/details/fundamentalsgeop00lowr/page/n69 58]–59 |edition=2nd}}</ref> 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{{by whom|date=February 2022}} a "theory of nutation".{{citation needed|date=February 2022}} In the theory, parameters are adjusted in a more or less ''ad hoc'' method to obtain the best fit to data. Simple [[rigid body dynamics]] do not give the best theory; one has to account for deformations of the Earth, including [[asthenosphere|mantle inelasticity]] and changes in the [[core–mantle boundary]].<ref>{{cite web |url=http://www.iers.org/nn_10382/IERS/EN/Science/Recommendations/resolutionB3.html |title=Resolution 83 on non-rigid Earth nutation theory |work=[[International Earth Rotation and Reference Systems Service]] |publisher=Federal Agency for Cartography and Geodesy |date=2 April 2009 |access-date=2012-08-06}}</ref>

The principal term of nutation is due to the regression of the Moon's [[nodal line]] and has the same period of 6798 days (18.61 years). It reaches plus or minus 17&Prime; in [[longitude]] and 9.2&Prime; in [[axial tilt|obliquity]].<ref>{{cite web |url=http://www2.jpl.nasa.gov/basics/bsf2-1.php#nutation |title=Basics of Space Flight, Chapter 2 |date=28 August 2013 |access-date=2015-03-26 |publisher=[[Jet Propulsion Laboratory]]/NASA}}</ref> All other terms are much smaller; the next-largest, with a period of 183 days (0.5 year), has amplitudes 1.3&Prime; and 0.6&Prime; respectively. The periods of all terms larger than 0.0001&Prime; (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{{according to whom|date=February 2022}} 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 [[Julian day]] according to IAU 2000B methodology.<ref>{{Cite web | url=http://www.neoprogrammics.com/nutations/ |title = NeoProgrammics - Science Computations}}</ref>

==In popular culture==
{{Unreferenced section|date=November 2022}}
In the 1961 disaster film ''[[The Day the Earth Caught Fire]]'', the near-simultaneous detonation of two super-[[hydrogen bomb]]s 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.

In ''[[Star Trek: The Next Generation]]'', 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.

==See also==
* [[Libration]]
* [[Teetotum]]

==Notes==
{{Reflist}}

==References==
{{refbegin}}
*[https://feynmanlectures.caltech.edu/I_20.html The Feynman Lectures on Physics Vol. I Ch. 20: Rotation in space]
*{{cite book|last=Feynman|first=Richard P.|first2= Robert B. |last2=Leighton |first3=Matthew |last3=Sands |title=The Feynman lectures on physics |year=2011|publisher=BasicBooks|location=New York|isbn=978-0465024933|edition=New millennium}}
*{{cite book|last=Goldstein|first=Herbert|title=Classical mechanics|date=1980|publisher=Addison-Wesley Pub. Co.|location=Reading, Mass.|isbn=0201029189|edition=2d }}
*{{cite book|last=Lambeck|first=Kurt|title=The earth's variable rotation : geophysical causes and consequences|date=2005|publisher=[[Cambridge University Press]]|location=Cambridge|isbn=9780521673303|edition=Digitally printed 1st pbk.}}
*{{cite book|last=Munk|first=Walter H.|title=The rotation of the earth : a geophysical discussion|date=1975|publisher=Cambridge University Press|location=Cambridge, Eng.|isbn=9780521207782|others=Reprint. with corr.|author2=MacDonald, Gordon J.F.|url-access=registration|url=https://archive.org/details/rotationofearthg0000munk}}
{{refend}}

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[[Category:Rotation in three dimensions]]
[[Category:Astrometry]]
[[Category:Geodynamics]]