- #1
IanBerkman
- 54
- 1
Homework Statement
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.
Homework 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
$$
\textbf{p} = p_0\cot(\theta/2)(\cos\phi, \sin\phi, 0)$$
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: