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arpon
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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.