Orbital Angular Momentum as Generator for Circular Translations

[unscientific]

Homework Statement

Taken from Binney's Text, pg 143.

Homework Equations

The Attempt at a Solution

From equation (7.36): we see that $\delta a$ is in the direction of the angle rotated, $\vec{x}$ is the position vector, and $\vec{n}$ is the unit normal to the plane of rotation.

But at the last line of (7.37), we see that $\vec{\alpha} = \alpha \vec{n}$. Why is the angle rotated now the same direction as the unit normal to the plane?

 

[TSny]

I'm not sure if I'm understanding your question. But, I think it's just a matter of definition. The symbol $\vec{\alpha}$ is defined to be $\alpha \hat{n}$. Note how Binney uses the definition symbol (or equivalence symbol) \equiv. This definition allows the rotation operator $U$ to be written more compactly. An angle of rotation has a certain magnitude about a certain axis. In this case $\alpha$ is the angle of rotation and $\hat{n}$ is the direction of the axis of rotation. The notation $\vec{\alpha} = \alpha \hat{n}$ is just a way to express that.

 

[unscientific]

I'm not sure if I'm understanding your question. But, I think it's just a matter of definition. The symbol $\vec{\alpha}$ is defined to be $\alpha \hat{n}$. Note how Binney uses the definition symbol (or equivalence symbol) \equiv. This definition allows the rotation operator $U$ to be written more compactly. An angle of rotation has a certain magnitude about a certain axis. In this case $\alpha$ is the angle of rotation and $\hat{n}$ is the direction of the axis of rotation. The notation $\vec{\alpha} = \alpha \hat{n}$ is just a way to express that.

If $\alpha$ is the angle of rotation and $\hat{n}$ is the direction vector of rotation, then wouldn't $\delta a = \delta \alpha$ x $x$ be into the plane, perpendicular to the plane of rotation?

 

[TSny]

$\hat{n}$ is a unit vector perpendicular to the plane of rotation. It points along the axis of rotation. So, $\hat{n} \times \vec{x}$ lies in the plane of rotation.

Since $\vec{\alpha} = \alpha \hat{n}$, $\vec{\alpha}$ also lies along the axis of rotation (perpendicular to the plane of rotation). For example, a rotation in the x-y plane of $\pi$ radians could be written $\vec{\alpha} = \pi \hat{z}$

 

[unscientific]

For example, a rotation in the x-y plane of $\pi$ radians could be written $\vec{\alpha} = \pi \hat{z}$

That doesn't make sense at all..Usually $\vec{\alpha}$ is the direction vector in which the system moves - I would say it's more like $\alpha_x \vec{i} + \alpha_y \vec{j}$

 

[TSny]

Watch Prof. Binney himself explain the notation here . The relevant part is at 36 min 30 sec into the video.

Or see his https://www.amazon.com/dp/B00GGJR9Y4/?tag=pfamazon01-20 just after equation (4.13), page 71.