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Rotations around the x and y axes of stereographic sphere

  1. Jul 15, 2016 #1
    1. The problem statement, all variables and given/known data
    Show that the equations
    $$ \delta \phi = \cot \theta \cot \phi \delta \theta, \quad \delta \phi =- \cot \theta \tan \phi \delta \theta$$
    represent rotations around the x and y axes respectively of a stereographic sphere.
    Both these two rotations map the sphere on itself and map any geodesic line on another geodesic line.

    2. Relevant equations
    The sphere is a momentum sphere with radius ##p_0##. Assuming ##\theta## is the angle from the z-axis and ##\phi## the angle from the x-axis. We obtain the stereographic projection in the ##p_x,p_y##-plane by
    $$
    \textbf{p} = p_0\cot(\theta/2)(\cos\phi, \sin\phi, 0)$$

    3. The attempt at a solution
    The meaning of ##\delta## is not clearly stated and I used it as a deviation (similar to ##\Delta##).
    An infinitesimal small deviation gives for the x-rotation
    $$d \phi = \cot \theta \cot \phi d\theta$$
    Integrating gives
    $$\cos\phi = a\csc\theta$$
    Where ##a## is the constant of the integration.
    Intuitively I think the answer of it being a rotation around the x-axis is hidden in this equation, but I fail to see it.

    Furthermore, I need to express the new stereographic project of the momentum under a rotation.
    Differentiating w.r.t. ##\phi## gives
    $$ \sin \phi = a \cot\theta\csc\theta \frac{d\theta}{d\phi} =
    a\csc\theta\tan\phi$$
    Substituting these equations of ##\sin\phi## and ##\cos\phi## into the expression ##\textbf{p}## at the top gives
    $$
    \textbf{p}=ap_0\cot(\theta/2)\csc\theta(1,\tan\phi,0)$$

    The problem is, I do not know if I do this correct or entirely wrong. I want to hear your opinion about this.

    Thanks in advance,

    Ian
     
    Last edited: Jul 15, 2016
  2. jcsd
  3. Jul 15, 2016 #2
    Substituting the ##\cos\phi## term in the spherical coordinates gives an constant for ##x##. Furthermore, I obtain ##y=rc\tan\phi## and ##z=rc\tan\phi\sec\phi##. This gives ##y^2+z^2\neq constant##, which is not possible.
     
    Last edited: Jul 15, 2016
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