**Nutation** (from Latin *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 in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame it can be defined as a change in the second Euler angle. If it is not caused by forces external to the body, it is called *free nutation* or * Euler nutation*.^{ [1] } A *pure nutation* is a movement of a rotational axis such that the first Euler angle is constant.^{[ citation needed ]} In spacecraft dynamics, precession (a change in the first Euler angle) is sometimes referred to as nutation.^{ [2] }

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 **τ** about the point of contact with the surface; the top rotates in the direction of this torque with an angular velocity **Ω** such that at any moment

where **L** is the instantaneous angular momentum of the top.^{ [3] }

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.^{ [3] }^{ [4] }

The physics of nutation in tops and gyroscopes 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 *θ* between the symmetry axis of the top and the vertical; the azimuth *φ* of the top about the vertical; and the rotation angle *ψ* of the top about its own axis. Thus, precession is the change in *φ* and nutation is the change in *θ*.^{ [5] }

If the top has mass *M* and its center of mass is at a distance *l* from the pivot point, its gravitational potential relative to the plane of the support is

In a coordinate system where the *z* axis is the axis of symmetry, the top has angular velocities *ω*_{1}, *ω*_{2}, *ω*_{3} and moments of inertia *I*_{1}, *I*_{2}, *I*_{3} about the *x*, *y*, and *z* axes. Since we are taking a symmetric top, we have *I*_{1}=*I*_{2}. The kinetic energy is

In terms of the Euler angles, this is

If the Euler–Lagrange equations are solved for this system, it is found that the motion depends on two constants *a* and *b* (each related to a constant of motion). The rate of precession is related to the tilt by

The tilt is determined by a differential equation for *u* = cos(*θ*) of the form

where *f* is a cubic polynomial that depends on parameters *a* and *b* as well as constants that are related to the energy and the gravitational torque. The roots of *f* are cosines of the angles at which the rate of change of *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.^{ [6] }

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 axis in 1728.

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

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 Moon's orbital nodes.^{ [1] } 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 *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 mantle inelasticity and changes in the core–mantle boundary.^{ [7] }

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 9.2″ in obliquity.^{ [8] } 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^{[ who? ]} 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 Julian day according to IAU 2000B methodology.^{ [9] }

In the 1961 disaster film * The Day the Earth Caught Fire *, the near-simultaneous detonation of two super-hydrogen bombs 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.

**Precession** is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. 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 *nutation*. In physics, there are two types of precession: torque-free and torque-induced.

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the *pole*, and the ray from the pole in the reference direction is the *polar axis*. The distance from the pole is called the *radial coordinate*, *radial distance* or simply *radius*, and the angle is called the *angular coordinate*, *polar angle*, or *azimuth*. The radial coordinate is often denoted by *r* or *ρ*, and the angular coordinate by *φ*, *θ*, or *t*. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the *radial distance* of that point from a fixed origin, its *polar angle* measured from a fixed zenith direction, and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

**Tidal acceleration** is an effect of the tidal forces between an orbiting natural satellite, and the primary planet that it orbits. The acceleration causes a gradual recession of a satellite in a prograde orbit away from the primary, and a corresponding slowdown of the primary's rotation. The process eventually leads to tidal locking, usually of the smaller first, and later the larger body. The Earth–Moon system is the best-studied case.

A **rotation** is a circular movement of an object around a center of rotation. A three-dimensional object can always be rotated about an infinite number of imaginary lines called *rotation axes*. If the axis passes through the body's center of mass, the body is said to rotate upon itself, or spin. A rotation around an external point, e.g. the planet Earth around the Sun, is called a *revolution* or *orbital revolution*, typically when it is produced by gravity. The axis is called a **pole**.

**Kinematics** is a subfield of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

**Angular displacement** of a body is the angle in radians through which a point revolves around a centre or line has been rotated in a specified sense about a specified axis. When a body rotates about its axis, the motion cannot simply be analyzed as a particle, as in circular motion it undergoes a changing velocity and acceleration at any time (*t*). When dealing with the rotation of a body, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the body's motion, so for example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal and negligible. Thus the rotation of a rigid body over a fixed axis is referred to as rotational motion.

In physics, **angular velocity** refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time. There are two types of angular velocity: orbital angular velocity and spin angular velocity. Spin angular velocity refers to how fast a rigid body rotates with respect to its centre of rotation. Orbital angular velocity refers to how fast a point object revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. Spin angular velocity is independent of the choice of origin, in contrast to orbital angular velocity which depends on the choice of origin.

An **inverted pendulum** is a pendulum that has its center of mass above its pivot point. It is unstable and without additional help will fall over. It can be suspended stably in this inverted position by using a control system to monitor the angle of the pole and move the pivot point horizontally back under the center of mass when it starts to fall over, keeping it balanced. The inverted pendulum is a classic problem in dynamics and control theory and is used as a benchmark for testing control strategies. It is often implemented with the pivot point mounted on a cart that can move horizontally under control of an electronic servo system as shown in the photo; this is called a **cart and pole** apparatus. Most applications limit the pendulum to 1 degree of freedom by affixing the pole to an axis of rotation. Whereas a normal pendulum is stable when hanging downwards, an inverted pendulum is inherently unstable, and must be actively balanced in order to remain upright; this can be done either by applying a torque at the pivot point, by moving the pivot point horizontally as part of a feedback system, changing the rate of rotation of a mass mounted on the pendulum on an axis parallel to the pivot axis and thereby generating a net torque on the pendulum, or by oscillating the pivot point vertically. A simple demonstration of moving the pivot point in a feedback system is achieved by balancing an upturned broomstick on the end of one's finger.

**Rigid-body dynamics** studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid, which means that they do not deform under the action of applied forces, simplifies the analysis by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. This excludes bodies that display fluid, highly elastic, and plastic behavior.

In physics, **circular motion** is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.

**Rolling** is a type of motion that combines rotation and translation of that object with respect to a surface, such that, if ideal conditions exist, the two are in contact with each other without sliding.

A **rotating frame of reference** is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth.

The **rigid rotor** is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the *linear rotor* requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water, ammonia, or methane.

**Rotation around a fixed axis** or **about a fixed axis of revolution** or **motion with respect to a fixed axis of rotation** is a special case of rotational motion. The fixed axis hypothesis excludes the possibility of an axis changing its orientation, and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible. If two rotations are forced at the same time, a new axis of rotation will appear.

In classical mechanics, **Routh's procedure** or **Routhian mechanics** is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the **Routhian** is the function which replaces both the Lagrangian and Hamiltonian functions.

In geometry, various **formalisms** exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

In mathematics, the **axis–angle representation** of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector **e** indicating the direction of an axis of rotation, and an angle *θ* describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector **e** rooted at the origin because the magnitude of **e** is constrained. For example, the elevation and azimuth angles of **e** suffice to locate it in any particular Cartesian coordinate frame.

**Astronomical nutation** is a phenomenon which causes the orientation of the axis of rotation of a spinning astronomical object to vary over time. It is caused by the gravitational forces of other nearby bodies acting upon the spinning object. Although they are caused by the same effect operating over different timescales, astronomers usually make a distinction between *precession*, which is a steady long-term change in the axis of rotation, and *nutation*, which is the combined effect of similar shorter-term variations.

- 1 2 Lowrie, William (2007).
*Fundamentals of Geophysics*(2nd ed.). Cambridge [u.a.]: Cambridge University Press. pp. 58–59. ISBN 9780521675963. - ↑ Kasdin, N. Jeremy; Paley, Derek A. (2010).
*Engineering dynamics : a comprehensive introduction*. Princeton, N.J.: Princeton University Press. pp. 526–527. ISBN 9780691135373. - 1 2 Feynman, Leighton & Sands 2011 , pp. 20–7
^{[ clarification needed ]} - ↑ Goldstein 1980 , p. 220
- ↑ Goldstein 1980 , p. 217
- ↑ Goldstein 1980 , pp. 213–217
- ↑ "Resolution 83 on non-rigid Earth nutation theory".
*International Earth Rotation and Reference Systems Service*. Federal Agency for Cartography and Geodesy. 2 April 2009. Retrieved 2012-08-06. - ↑ "Basics of Space Flight, Chapter 2". Jet Propulsion Laboratory/NASA. 28 August 2013. Retrieved 2015-03-26.
- ↑ "NeoProgrammics - Science Computations".

- Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (2011).
*The Feynman lectures on physics*(New millennium ed.). New York: BasicBooks. ISBN 978-0465024933.CS1 maint: ref=harv (link) - Goldstein, Herbert (1980).
*Classical mechanics*(2d ed.). Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0201029189.CS1 maint: ref=harv (link) - Lambeck, Kurt (2005).
*The earth's variable rotation : geophysical causes and consequences*(Digitally printed 1st pbk. ed.). Cambridge: Cambridge University Press. ISBN 9780521673303. - Munk, Walter H.; MacDonald, Gordon J.F. (1975).
*The rotation of the earth : a geophysical discussion*. Reprint. with corr. Cambridge, Eng.: Cambridge University Press. ISBN 9780521207782.

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