Exploring Infinitesimal Rotations in Classical Mechanics

[Kashmir]

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.

 

[PeroK]

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.

 

[Kashmir]

That link doesn't work.

Thank you. Edited.

 

[PeroK]

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.

 

[Kashmir]

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 ?

 

[PeroK]

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.

 

[Kashmir]

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 .

 

[PeroK]

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.

 

[Kashmir]

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.