Two successive rotation (Goldstein problem 4.13)

[arpon]

Homework Statement

Suppose two successive coordinate rotations through angles $\Phi_1$ and $\Phi_2$ are carried out, equivalent to a single rotation through an angle $\Phi$. Show that $\Phi_1$, $\Phi_2$ and $\Phi$ can be considered as the sides of a spherical triangle with the angle opposite to $\Phi$ given by the angle between the two axes of rotation.

(Source: Classical Mechanics, 3rd edition, Goldstein, Problem 13, Chapter 4)

Homework Equations

If $A$ is a rotation matrix,
$$Tr~A=1+ 2\cos{\theta}$$
,where $\theta$ is the rotation angle.

The Attempt at a Solution

Let $R_1$ and $A_1$ be the rotation axis (unit vector) and the rotation matrix respectively for the $\Phi_1$ rotation. So we get,
$$A_1R_1=R_1$$
and $$Tr~ A_1 = 1+2 \cos{\Phi_1}$$
In the same way, for $\Phi_2$,
$$A_2R_2=R_2$$
and $$Tr~ A_2 = 1+2 \cos{\Phi_2}$$
The rotation matrix corresponding to $\Phi$ rotation will be $A_2A_1$. So we get
$$Tr ~A_2A_1 = 1+2 \cos{\Phi}$$
If the angle between $R_1$ and $R_2$ is $\psi$, then
$$\cos{\psi}=R_1^TR_2$$
Now I need to show that $\psi$ is the angle opposite to $\Phi$ in the spherical triangle with $\Phi_1$, $\Phi_2$ and $\Phi$ considered as the sides.
I am not sure if the problem can be solved in this approach.
Any help or suggestion will be appreciated.

 

[TSny]

I'm not sure how rigorous the derivation needs to be. But if you construct a diagram of the spherical triangle, then the result appears to follow immediately by inspection of the diagram.

 

[arpon]

I'm not sure how rigorous the derivation needs to be. But if you construct a diagram of the spherical triangle, then the result appears to follow immediately by inspection of the diagram.

I was looking for a rigorous derivation.

 

[TSny]

OK. I don't see a rigorous derivation at the moment. For what it's worth, here's the diagram that seems to me to show the result.

Start with a spherical apple. Let a radial line sweep out the arc Φ₁ from a to b, slicing the apple along the yellow plane. Continue with two more slicings Φ₂ and Φ along the blue and green planes, respectively. Remove the wedge of apple that has been sliced out. The picture above shows peering down inside the apple. The red angle is the angle opposite Φ. This angle is clearly the angle between the yellow and blue planes. The result follows by considering how the angle between the yellow and blue planes is related to the angle between the rotation axes corresponding to Φ₁ and Φ₂.

Hopefully, someone can provide some hints on constructing a rigorous argument.