Editing Rotation (section)

== Physics ==
The [[speed of rotation]] is given by the [[angular frequency]] (rad/s) or [[frequency]] ([[turn (geometry)|turns]] per time), or [[Frequency|period]] (seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s<sup>2</sup>), caused by [[torque]]. The ratio of torque {{mvar|τ}} to the angular acceleration {{mvar|α}} is given by the [[moment of inertia]]:
<math display="block"> I = \frac{\tau}{\alpha}.</math>

The [[angular velocity]] vector (an ''[[axial vector]]'') also describes the direction of the axis of rotation. Similarly,  the torque is an axial vector.

The physics of the [[rotation around a fixed axis]] is mathematically described with the [[axis–angle representation]] of rotations. According to the [[right-hand rule]], the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like a [[screw]].

=== Circular motion ===
{{See also|Circular motion|Rotation around a fixed axis}}[[File:Circular_motion_vs_rotation.svg|thumb|The motion on the left, an example of curvilinear translation, cannot be treated as rotation since there is no change in orientation, whereas the right can be treated as rotation. |264x264px]]
It is possible for [[Rigid body|objects]] to have periodic [[Circular motion|circular trajectories]] without changing their [[Orientation (geometry)|orientation]]. These types of motion are treated under [[circular motion]] instead of rotation, more specifically as a curvilinear translation. Since translation involves [[Displacement (geometry)|displacement]] of [[Rigid body|rigid bodies]] while preserving the [[Orientation (geometry)|orientation]] of the body, in the case of curvilinear translation, all the points have the same instantaneous velocity whereas relative motion can only be observed in motions involving rotation.<ref name=":0">{{Cite book |last1=Harrison |first1=H. |url=https://books.google.com/books?id=IBkD6pSdjl0C |title=Advanced Engineering Dynamics |last2=Nettleton |first2=T. |date=1997-08-01 |publisher=Butterworth-Heinemann |isbn=978-0-08-052335-4 |pages=55 |language=en |chapter=Rigid body motion in three dimensions |chapter-url=https://books.google.com/books?id=IBkD6pSdjl0C&pg=PA55}}</ref>

In rotation, the [[Orientation (geometry)|orientation]] of the object changes and the change in [[Orientation (geometry)|orientation]] is independent of the observers whose [[Frame of reference|frames of reference]] have constant relative orientation over time. By [[Euler's rotation theorem|Euler's theorem]], any change in orientation can be described by rotation about an axis through a chosen reference point.<ref name=":0" /> Hence, the distinction between rotation and circular motion can be made by requiring an instantaneous axis for rotation, a line passing through [[Instant centre of rotation|instantaneous center of circle]] and perpendicular to the [[Plane of rotation|plane of motion]]. In the example depicting curvilinear translation, the center of circles for the motion lie on a straight line but it is parallel to the plane of motion and hence does not resolve to an axis of rotation. In contrast, a rotating body will always have its instantaneous axis of zero velocity, perpendicular to the plane of motion.<ref>{{Cite book |last=Hibbeler |first=R. C. |url=https://books.google.com/books?id=PxjGxmLhSBwC |title=Engineering Mechanics: Statics & dynamics |date=2007 |publisher=Prentice-Hall |isbn=978-0-13-221509-1 |language=en |chapter=Planar kinematics of a rigid body: Instantaneous center of zero velocity |chapter-url=https://books.google.com/books?id=PxjGxmLhSBwC&pg=PA680}}</ref>

More generally, due to [[Chasles' theorem (kinematics)|Chasles' theorem]], any motion of [[Rigid body|rigid bodies]] can be treated as a composition of '''rotation''' and [[Translation (geometry)|translation]], called general plane motion.<ref name=":0" /> A simple example of pure rotation is considered in [[rotation around a fixed axis]].

=== Cosmological principle ===
The [[laws of physics]] are currently believed to be [[rotational invariance#Application to quantum mechanic|invariant under any fixed rotation]]. (Although they do appear to change when viewed from a rotating viewpoint: see [[rotating frame of reference]].)

In modern physical cosmology, the [[cosmological principle]] is the notion that the distribution of matter in the universe is [[homogeneous]] and [[isotropic]] when viewed on a large enough scale, since the forces are expected to act uniformly throughout the universe and have no preferred direction, and should, therefore, produce no observable irregularities in the large scale structuring over the course of evolution of the matter field that was initially laid down by the Big Bang.

In particular, for a system which behaves the same regardless of how it is oriented in space, its [[Lagrangian mechanics|Lagrangian]] is [[Rotational invariance|rotationally invariant]]. According to [[Noether's theorem]], if the [[Action (physics)|action]] (the [[integral over time]] of its Lagrangian) of a physical system is invariant under rotation, then [[Conservation of angular momentum|angular momentum is conserved]].

=== Euler rotations ===
{{Main|Euler angles}}
[[Image:Praezession.svg|thumb|right|upright=0.8|Euler rotations of the Earth. Intrinsic (green), Precession (blue) and Nutation (red)]]
Euler rotations provide an alternative description of a rotation. It is a composition of three rotations defined as the movement obtained by changing one of the [[Euler angles]] while leaving the other two constant. Euler rotations are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation system, where the first angle moves the [[line of nodes]] around the external axis ''z'', the second rotates around the [[line of nodes]] and the third one is an intrinsic rotation around an axis fixed in the body that moves.

These rotations are called [[precession]], [[nutation]], and ''intrinsic rotation''.