- #1
IanBerkman
- 54
- 1
I have posted this question earlier but I think it was ill-stated. I try to give it in a simpler fashion in this thread.
In the problem it is stated that there a rotation around the x-axis of a (stereographic) sphere is given by
$$\delta \phi = \cot \phi \cot \theta \delta \theta$$
where ##\theta## is the angle from the z-axis and ##\phi## the angle from the x-axis.
The rotation maps the sphere on itself.
I need to prove that this equation indeed represents a rotation around the x-axis.
________________________________________________________________________________________________________
In spherical coordinates we have
$$ x = \rho\sin\theta\cos\phi\\
y = \rho\sin\theta\sin\phi\\
z = \rho\cos\theta$$
I changed the ##\delta## in the equation for the infinitesimally small d. (It is nowhere stated what ##\delta## is, it could also be that I have to substitute this in ##\delta S = \delta\int L dt = 0## but this will give a huge integral. I can give this integral in a comment).
This gives
$$
\tan\phi d\phi = \cot\theta d \theta$$
Integration gives
$$sin\phi = const\cdot\csc\theta\tan\phi$$
Where the constant follows from the integration.
Substituting this equation in the spherical coordinates will give a constant ##x##, but no constant for ##y^2+z^2## which must follow from a rotation around the x-axis.
I am stuck with this problem for over a week, I am really thankful if someone can provide some insight.
Thanks in advance,
Ian
In the problem it is stated that there a rotation around the x-axis of a (stereographic) sphere is given by
$$\delta \phi = \cot \phi \cot \theta \delta \theta$$
where ##\theta## is the angle from the z-axis and ##\phi## the angle from the x-axis.
The rotation maps the sphere on itself.
I need to prove that this equation indeed represents a rotation around the x-axis.
________________________________________________________________________________________________________
In spherical coordinates we have
$$ x = \rho\sin\theta\cos\phi\\
y = \rho\sin\theta\sin\phi\\
z = \rho\cos\theta$$
I changed the ##\delta## in the equation for the infinitesimally small d. (It is nowhere stated what ##\delta## is, it could also be that I have to substitute this in ##\delta S = \delta\int L dt = 0## but this will give a huge integral. I can give this integral in a comment).
This gives
$$
\tan\phi d\phi = \cot\theta d \theta$$
Integration gives
$$sin\phi = const\cdot\csc\theta\tan\phi$$
Where the constant follows from the integration.
Substituting this equation in the spherical coordinates will give a constant ##x##, but no constant for ##y^2+z^2## which must follow from a rotation around the x-axis.
I am stuck with this problem for over a week, I am really thankful if someone can provide some insight.
Thanks in advance,
Ian