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| {{about|the concept in physics and astronomy|the term in mechanical engineering|nutation (engineering)|other uses}}
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| [[File:Praezession.svg|thumb|170px|Rotation (green), precession (blue) and nutation in obliquity (red) of a planet]]
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| '''Nutation''' (from [[Latin]]: ''nūtāre'', to nod) 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 behavior 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 ''[[Leonard Euler|Euler]] nutation''.<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}} 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|year=2010|publisher=Princeton University Press|location=Princeton, N.J.|isbn=9780691135373|pages=526–527|coauthors=Paley, Derek A.}}</ref> | |
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| ==Rigid body==
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| {{further|rigid body dynamics}}
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| If a [[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|'''τ'''}} about the point of contact with the surface; the top rotates in the direction of this torque with an angular velocity {{math|'''Ω'''}} such that at any moment
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| :<math> \mathbf{\tau} = \mathbf{\Omega} \times \mathbf{L},</math>
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| where {{math|'''L'''}} is the instantaneous angular momentum of the top.<ref name=Feynman>{{harvnb|Feynman|Leighton|Sands|2011|pp=20–7{{clarification needed|date=December 2013}}}}</ref> | |
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| Initially, however, there is no precession, and the top falls straight downward. This gives rise to an imbalance in torques that starts the precession. In falling, the top overshoots the level 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>
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| The physics of nutation in tops and [[gyroscope]]s can be explored using the model of a ''heavy symmetrical top'' with its tip fixed. Initially, the effect of friction is ignored. The motion of the top can be described by three [[Euler angles]]: the tilt angle {{math|''θ''}} between the symmetry axis of the top and the vertical; the [[azimuth]] {{math|''φ''}} of the top about the vertical; and the rotation angle {{math|''ψ''}} of the top about its own axis. Thus, precession is the change in {{math|''φ''}} and nutation is the change in {{math|''θ''}}.<ref name=Goldstein217>{{harvnb|Goldstein|1980|p=217}}</ref>
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| 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]] is
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| :<math>V = Mgl\cos\theta.</math>
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| In a coordinate system where the {{math|''z''}} axis is the axis of symmetry, the top has [[angular velocity|angular velocities]] {{math|''ω''<sub>1</sub>, ''ω''<sub>2</sub>, ''ω''<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. The [[kinetic energy]] is
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| :<math>T = \frac{1}{2}I_1\left(\omega_1^2+\omega_2^2\right) + \frac{1}{2}I_3\omega_3^2.</math>
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| In terms of the Euler angles, this is
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| :<math>T = \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>
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| 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
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| :<math> \dot{\phi} = \frac{b - a\cos\theta}{\sin^2\theta}.</math>
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| The tilt is determined by a differential equation for {{math|''u'' {{=}} cos ''θ''}} of the form
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| :<math> \dot{u}^2 = f(u)</math>
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| 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|''θ''}} 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>
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| ==Astronomy==
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| The nutation of a planet happens because of gravitational attraction of other bodies that cause the [[precession of the equinoxes]] to vary over time so that the speed of precession is not constant. The nutation of the [[Earth's rotation|axis of the Earth]] was discovered in 1728 by the British astronomer [[James Bradley]], but this nutation was not explained in detail until 20 years later.<ref>{{cite web|author=Robert E. Bradley |title=The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler |work=The MAA Mathematical Sciences Digital Library |publisher=Mathematical Association of America |url=http://mathdl.maa.org/mathDL/?pa=content&sa=viewDocument&nodeId=962&bodyId=1147 |accessdate=17 April 2013}}</ref>
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| Because the dynamic motions of the planets are so well known, their nutations can be calculated to within [[arcsecond]]s over periods of many decades. There is another disturbance of the Earth's rotation called [[polar motion]] that can be estimated for only a few months into the future because it is influenced by rapidly and unpredictably varying things such as [[ocean current]]s, wind systems, and hypothesised motions in the liquid [[nickel-iron]] [[outer core|outer core of the Earth]].
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| Values of nutations are usually divided into components parallel and [[perpendicular]] to the [[ecliptic]]. The component that works along the ecliptic is known as the ''nutation in longitude''. The component perpendicular to the [[ecliptic]] is known as the ''nutation in obliquity''. Celestial coordinate systems are based on an "equator" and "equinox", which means a great circle in the sky that is the projection of the Earth's equator outwards, and a line, the [[Vernal equinox]] intersecting that circle, which determines the starting point for measurement of [[right ascension]]. These items are affected both by [[precession]] of the equinoxes and nutation, and thus depend on the theories applied to precession and nutation, and on the date used as a reference date for the coordinate system. In simpler terms, nutation (and precession) values are important in observation from [[Earth]] for calculating the apparent positions of astronomical objects.
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| ==Earth==
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| {{further|geodynamics}}
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| [[File:Trópico de Cáncer en México - Carretera 83 (Vía Corta) Zaragoza-Victoria, Km 27+800.jpg|thumb|240px|Evolution of the position where the [[Tropic of Cancer]] crosses Carretera 83 (Vía Corta) Zaragoza-Victoria, Km 27+800 (Mexico). Nutation makes a small change to the angle at which the Earth tilts with respect to the Sun, thereby moving the location of the [[Tropic of Cancer]], the most northerly latitude which the Sun can reach directly overhead. All four [[Circle_of_latitude#Major_circles_of_latitude|major circles of latitude]] that are defined by the Earth's tilt (both [[tropical circle|Tropical Circles]] and both [[polar circle|Polar Circles]]) will shift correspondingly.]]
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| In the case of the 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|year=2007|publisher=Cambridge Univ. Press|location=Cambridge [u.a.]|isbn=9780521675963|pages=58–59|edition=2nd}}</ref> However, there are other significant periodical terms that must be calculated depending on 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 ''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 |accessdate=6 August 2012}}</ref>
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| 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″ in [[longitude]] and 9″ in [[Axial tilt|obliquity]]. 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 one can measure) lie between 5.5 and 6798 days; for some reason 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.
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| ==In popular culture==
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| In the movie [[The Day The Earth Caught Fire]], the near-simultaneous detonation of two super-hydrogen bombs near the poles causes a change in the Earth's nutation, as well as an 11 degree shift in the axis of rotation and a change in the Earth's orbit around the sun.
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| The verb ''[[wikt:nutate|to nutate]]'' was used by MIT physicist Peter Fisher on the television show [[Late Night with Conan O'Brien]] on February 8, 2008. Fisher used the term to describe the motion of a spinning ring as it began to slow down and wobble.
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| ==See also==
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| * [[Libration]]
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| ==References==
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| <references/>
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| ==Further reading==
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| {{refbegin}}
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| *{{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 |ref=harv}}
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| *{{cite book|last=Goldstein|first=Herbert|title=Classical mechanics|year=1980|publisher=Addison-Wesley Pub. Co.|location=Reading, Mass.|isbn=0201029189|edition=2d |ref=harv}}
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| *{{cite book|last=Lambeck|first=Kurt|title=The earth's variable rotation : geophysical causes and consequences|year=2005|publisher=Cambride University Press|location=Cambridge|isbn=9780521673303|edition=Digitally printed 1st pbk.}}
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| *{{cite book|last=Munk|first=Walter H.|title=The rotation of the earth : a geophysical discussion|year=1975|publisher=Cambridge University Press|location=Cambridge, Eng.|isbn=9780521207782|others=Reprint. with corr.|coauthors=MacDonald, Gordon J.F.}}
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| {{refend}}
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| [[Category:Rotation in three dimensions]]
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| [[Category:Astrometry]]
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| [[Category:Geodynamics]]
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Hello, I'm Garfield Murtagh. My residence has become in Nj-New Jersey. Distributing generation is where my main revenue arises from but I intend on changing it. Among the absolute best things in the entire world for me is acting and now I've occasion to battle new things.
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