# Eulerian angle understanding

I have alot of confusion with Eulerian angle so first of all I would like to address something I don't understand from the book and maybe that would shed some light on the intuition of eulerian angles.

Here is the book Explanation for Eulerian angles:

The coordinate system labeled O-123 is defined by the three principal axes fixed to the rigid body and rotates with it. The coordinate system fixed in space is labeled Oxyz. A third,rotating system, O$x'y'z'$ providing a connection between the principal axes attached to the body and axes fixed in space, is also defined as follows: The $z'$-axis coincides with the 3-axis of the body --- its symmetry axis The $x'$-axis is defined by the intersection of the body 1-2 plane with the fixed xy plane ? This is called the line of nodes.

So I don't understand is the plane O-xyz` changing always with time or is it fixed?

• I edited to make the primes visible (e.g. in $Ox'y'z'$). But I don't understand your question in the last paragraph. Can you fix up the grammar/punctuation so it is clear? Aug 14, 2015 at 5:10
• alright. I will fix it now to make it more clear.
– Dude
Aug 14, 2015 at 5:16

If the rigid body is rotating then in general the primed axes will be changing with time. An easier way to see this is to look at the Euler angles themselves as in this diagram. If, for instance, $\alpha$ is changing, then both the line of nodes (the $N$-axis in the diagram) and the $z'$-axis (the $Z$-axis in the diagram) are changing.