Editing Angular velocity (section)

=== 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}}