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) [itex]\equiv[/itex]. 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) [itex]\equiv[/itex]. 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.