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Orbital Angular Momentum as Generator for Circular Translations

  1. Mar 21, 2014 #1
    1. The problem statement, all variables and given/known data

    Taken from Binney's Text, pg 143.

    21e509d.png

    2. Relevant equations



    3. 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.

    51dctx.png

    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?
     
  2. jcsd
  3. Mar 21, 2014 #2

    TSny

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    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.
     
  4. Mar 21, 2014 #3
    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?
     
  5. Mar 21, 2014 #4

    TSny

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    ##\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}##
     
  6. Mar 22, 2014 #5
    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}##
     
  7. Mar 22, 2014 #6

    TSny

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    Last edited by a moderator: May 6, 2017
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