Need help on an exercise from Gravitation(MTW)

[shichao116]

Hi, I'm working on exercise 9.13 of the "bible" Gravitation. The problem I have is how to derive the following equation:

$R_x(t)R_z(\psi)R_x(\theta)R_z(\phi) = R_z(\psi-tsin\psi cot\theta)R_x(\theta+tcos\psi)R_z(\phi+tsin\psi/sin\theta)$

Where $R_x(t)$ denotes an infinitesimal rotation about x-axis, i.e. t<<1. $R_z(\psi)R_x(\theta)R_z(\phi)$ denote three consecutive FINITE angle rotations about z, x, and z axis, with Euler angles $\psi$, $\theta$ and $\phi$ respectively.

Can anyone help me on this? Thanks!

In case the latex doesn't work, I attach the equation below

 

[qinglong.1397]

Hi, I'm working on exercise 9.13 of the "bible" Gravitation. The problem I have is how to derive the following equation:

$R_x(t)R_z(\psi)R_x(\theta)R_z(\phi) = R_z(\psi-tsin\psi cot\theta)R_x(\theta+tcos\psi)R_z(\phi+tsin\psi/sin\theta)$

Where $R_x(t)$ denotes an infinitesimal rotation about x-axis, i.e. t<<1. $R_z(\psi)R_x(\theta)R_z(\phi)$ denote three consecutive FINITE angle rotations about z, x, and z axis, with Euler angles $\psi$, $\theta$ and $\phi$ respectively.

Can anyone help me on this? Thanks!

In case the latex doesn't work, I attach the equation below

Since t<<1, you should solve this problem by doing approximation. And also, go from both sides of the equation.

(I don't want to type more words only if you're still searching for help )