Editing Angular velocity (section)

== Spin angular velocity of a rigid body or reference frame {{anchor|Spin}} ==

Given a rotating frame of three linearly independent unit coordinate vectors, at each instant in time, there always exists a common axis (called the axis of rotation) around which all three vectors rotate with the same angular speed and in the same angular direction (clockwise or counterclockwise). In such a frame, each vector may be considered as a moving particle with constant scalar radius. A collection of such particles is called a rigid body.

[[Euler's rotation theorem]] says that in a rotating frame, the axis of rotation one obtains from one choice of three linearly independent unit vectors is the same as that for any other choice; that is, there is one ''single'' [[instantaneous axis of rotation]] to the frame, around which all points rotate at the same angular speed and in the same angular direction (clockwise or counterclockwise). The spin angular velocity of a frame or rigid body is defined to be the pseudovector whose magnitude is this common angular speed, and whose direction is along the common axis of rotation in accordance with the right-hand rule (that is, for counterclockise rotation, it points "upward" along the axis, while for clockwise rotation, it points "downward").

In larger than 3 spatial dimensions, the interpretation of spin angular velocity as a pseudovector is not valid; however, it may be characterized by a more general type of object known as an antisymmetric rank-2 [[tensor]].

The addition of angular velocity vectors for frames is also defined by the usual vector addition (composition of linear movements), and can be useful to decompose the rotation as in a [[gimbal]]. All components of the vector can be calculated as derivatives of the parameters defining the moving frames (Euler angles or rotation matrices). As in the general case, addition is commutative: <math>\omega_1 + \omega_2 = \omega_2 + \omega_1</math>.

If we choose a reference point <math>{\boldsymbol r_0}</math> fixed in a rotating frame, the velocity <math> \dot {\boldsymbol r}</math> of any point in the frame is given by

:<math> \dot {\boldsymbol r}= \dot {\boldsymbol r_0}+ {\boldsymbol\omega}\times({\boldsymbol r}-{\boldsymbol r_0})
</math>

=== Components from the basis vectors of a body-fixed frame ===

Consider a rigid body rotating about a fixed point O. Construct a reference frame in the body consisting of an orthonormal set of vectors <math>\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 </math> fixed to the body and with their common origin at O. The spin angular velocity vector of both frame and body about O is then
: <math>\boldsymbol\omega = \left(\dot \mathbf{e}_1\cdot\mathbf{e}_2\right) \mathbf{e}_3 + \left(\dot \mathbf{e}_2\cdot\mathbf{e}_3\right) \mathbf{e}_1 + \left(\dot \mathbf{e}_3\cdot\mathbf{e}_1\right) \mathbf{e}_2,
</math>
where <math> \dot \mathbf{e}_i= \frac{d  \mathbf{e}_i}{dt} </math> is the time rate of change of the frame vector <math> \mathbf{e}_i, i=1,2,3,</math>  due to the rotation.

This formula is incompatible with the expression for ''orbital'' angular velocity

: <math>\boldsymbol\omega
=\frac{\boldsymbol{r}\times\boldsymbol{v}}{r^2},</math>

as that formula defines angular velocity for a ''single point'' about O, while the formula in this section applies to a frame or rigid body. In the case of a rigid body a ''single'' <math> \boldsymbol\omega</math> has to account for the motion of ''all'' particles in the body.

=== Components from Euler angles ===

[[Image:Eulerframe.svg|class=skin-invert-image|thumb|Diagram showing Euler frame in green]]

The components of the spin angular velocity pseudovector were first calculated by [[Leonhard Euler]] using his [[Euler angles]] and the use of an intermediate frame:
* One axis of the reference frame (the precession axis)
* The line of nodes of the moving frame with respect to the reference frame (nutation axis)
* One axis of the moving frame (the intrinsic rotation axis)

Euler proved that the projections of the angular velocity pseudovector on each of these three axes is the derivative of its associated angle (which is equivalent to decomposing the instantaneous rotation into three instantaneous [[Euler rotations]]). Therefore:<ref>[http://www.vti.mod.gov.rs/ntp/rad2007/3-07/hedr/hedr.pdf K.S.HEDRIH: Leonhard Euler (1707–1783) and rigid body dynamics]</ref>

: <math>\boldsymbol\omega = \dot\alpha\mathbf u_1+\dot\beta\mathbf u_2+\dot\gamma \mathbf u_3</math>

This basis is not orthonormal and it is difficult to use, but now the velocity vector can be changed to the fixed frame or to the moving frame with just a change of bases. For example, changing to the mobile frame:

: <math>\boldsymbol\omega =
(\dot\alpha \sin\beta \sin\gamma + \dot\beta\cos\gamma) \hat\mathbf i+
(\dot\alpha \sin\beta \cos\gamma - \dot\beta\sin\gamma) \hat\mathbf j +
(\dot\alpha \cos\beta + \dot\gamma) \hat\mathbf k</math>

where <math>\hat\mathbf i, \hat\mathbf j, \hat\mathbf k</math> are unit vectors for the frame fixed in the moving body. This example has been made using the Z-X-Z convention for Euler angles.{{Citation needed|date=June 2020}}