A Exploring Infinitesimal Rotations in Classical Mechanics

AI Thread Summary
The discussion focuses on understanding the transformation of coordinates under infinitesimal rotations as presented in Goldstein's Classical Mechanics. Participants suggest deriving the rotation matrix for small angles and applying trigonometric approximations to simplify it. They emphasize that the infinitesimal rotation can be expressed in a standard form using Euler angles, leading to a specific matrix representation. The conversation highlights the importance of understanding the limits of the rotation matrix as the angles approach zero. Overall, the thread aims to clarify the mathematical foundations of infinitesimal rotations in rigid body motion.
Kashmir
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Can anybody please help me to understand that why under infinitesimal rotation ##x1## transforms in the way as shown in equation 4-100?

This is from Goldstein's Classical Mechanics page chapter 5 and page 168 on the Kinematics of Rigid body motion.
 
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Kashmir said:
https://www.physicsforums.com/attachments/292210

Can anybody please help me to understand that why under infinitesimal rotation ##x'_1## transforms in the way as shown ?

This is from Goldstein's Classical Mechanics page chapter 5 and page 168 on the Kinematics of Rigid body motion.
That link doesn't work.
 
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PeroK said:
That link doesn't work.
Thank you. Edited.
 
The way I would do it is to write out the rotation matrix for a rotation of ##\theta## about the ##x_1## axis. And apply this matrix to an arbitrary vector ##(x_1, x_2,x_3)##.

When ##\theta## is small, we have ##\cos \theta \approx 1##, ##\sin \theta \approx \theta##. If you apply those approximations you should get the limit for an infinitesimal rotation.

I think Goldstein is just using generic matric entries, rather than ##\cos \theta## and ##\sin \theta## explicity.
 
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Wouldn't it be more general to apply these trigonometric limits for small angles directly into the rotation matrix here ##A=\left[\begin{array}{ccc}\cos \psi \cos \phi-\cos \theta \sin \phi \sin \psi & \cos \psi \sin \phi+\cos \theta \cos \phi \sin \psi & \sin \psi \sin \theta \\ -\sin \psi \cos \phi-\cos \theta \sin \phi \cos \psi & -\sin \psi \sin \phi+\cos \theta \cos \phi \cos \psi & \cos \psi \cdot \sin \theta \\ \sin \theta \sin \phi & -\sin \theta \cos \phi & \cos \theta\end{array}\right]## reducing to ##A=\left[\begin{array}{ccc}1-\phi \psi & \phi+\psi & \psi \theta \\ -\psi-\phi & -\psi \phi+1 & \theta \\\theta \phi & -\theta & 1\end{array}\right]## hence the required equation A-400 ?
 
Kashmir said:
Yes I got the idea. Thank you. Wouldn't it be more general to apply these trigonometric limits for small angles directly into the rotation matrix here ##A=\left[\begin{array}{ccc}\cos \psi \cos \phi-\cos \theta \sin \phi \sin \psi & \cos \psi \sin \phi+\cos \theta \cos \phi \sin \psi & \sin \psi \sin \theta \\ -\sin \psi \cos \phi-\cos \theta \sin \phi \cos \psi & -\sin \psi \sin \phi+\cos \theta \cos \phi \cos \psi & \cos \psi \cdot \sin \theta \\ \sin \theta \sin \phi & -\sin \theta \cos \phi & \cos \theta\end{array}\right]## reducing to ##A=\left[\begin{array}{ccc}1-\phi \psi & \phi+\psi & \phi \theta \\ 4 \beta-\phi & -\psi \phi+1 & \theta \\ -\psi & -\theta & 1\end{array}\right## hence the required equation A-400 ?
We simply want to analyse infinitesimal rotations about the three coordinate axes. Those have a well-known simple form.
 
PeroK said:
We simply want to analyse infinitesimal rotations about the three coordinate axes. Those have a well-known simple form.
I agree. We can also think like this:
Given a general rotation matrix A defined with the three Euler angles phi, theta and psi we can find the limit by letting all three go to zero and find that the infinitesimal rotation is
##A=\left[\begin{array}{ccc}1-\phi \psi & \phi+\psi & \psi \theta \\ -\psi-\phi & -\psi \phi+1 & \theta \\\theta \phi & -\theta & 1\end{array}\right] =I+e## exactly as the author writes at equation A-102 .
 
Kashmir said:
I agree. We can also think like this:
Given a general rotation matrix A defined with the three Euler angles phi, theta and psi we can find the limit by letting all three go to zero and find that the infinitesimal rotation is
##A=\left[\begin{array}{ccc}1-\phi \psi & \phi+\psi & \psi \theta \\ -\psi-\phi & -\psi \phi+1 & \theta \\\theta \phi & -\theta & 1\end{array}\right] =I+e## exactly as the author writes at equation A-102 .
Possibly. I don't have Goldstein, so I don't know where he's going with this. I might have misunderstood what he's trying to do.
 
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PeroK said:
Possibly. I don't have Goldstein, so I don't know where he's going with this. I might have misunderstood what he's trying to do.
Thank you for your help.
 
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