# [Physics] the difference between precession and spin angles

precessionreference framesrigid-body-dynamicsrotational-dynamics

I was recently introduced to Euler Angles in a Dynamics course, but I am confused on the difference between precession and spin angles. Both precession and spin consist in rotating a coordinate system about the $z$-axis, which means that they have the same transformation matrices.

Both precession and spin consist in rotating a coordinate system about the $z$-axis, which means that they have the same transformation matrices.
Yes, but in between the two there is a rotation about $x$. (Note: Some use $y$ for the middle rotation). This makes all the difference in the world for describing the orientation of a world with respect to the ecliptic, the original usage of Euler rotation sequences.
A bit overly simplified, the initial rotation about the original $z$-axis is the precession angle. Precession for a planet is very slow. The second rotation about the once-rotated $x$-axis is the nutation angle. For the Earth, this closely corresponds to the nearly constant obliquity. The final rotation about the twice-rotated $z$ axis is the daily rotation angle.
With two exceptions, given any orientation, the Euler sequence that rotates the initial $x$, $y$, and $z$ axes to the $X$, $Y$, and $Z$ axes that correspond to the desired orientation is unique with the constraint that the rotations about $z$ are between 0° (inclusive) and 360° degrees (exclusive) and the intermediate rotation about the once-rotated$x$-axis is between 0° and 180° (exclusive of both).
The two exceptions occur when the intermediate rotation about the once-rotated $x$-axis is 0° or 180°. These situations are called "gimbal lock." Here the distinction between precession and rotation is completely arbitrary; the standard approach is to arbitrarily set one of the two to zero.