Need help on an exercise from Gravitation(MTW)

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The discussion focuses on deriving the equation related to infinitesimal and finite rotations in the context of exercise 9.13 from the book "Gravitation" by Misner, Thorne, and Wheeler (MTW). The equation involves rotations represented by R_x(t), R_z(ψ), R_x(θ), and R_z(φ), where R_x(t) denotes an infinitesimal rotation about the x-axis. The key to solving this problem lies in applying approximations due to the condition t << 1 and analyzing both sides of the equation for consistency.

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

[itex]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)[/itex]

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

Can anyone help me on this? Thanks!

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

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shichao116 said:
Hi, I'm working on exercise 9.13 of the "bible" Gravitation. The problem I have is how to derive the following equation:

[itex]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)[/itex]

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

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