Scaling the parameter of the SO(2) rotation matrix

[spaghetti3451]

For the distance function $(\Delta s)^2 = (\Delta r)^2 + (r \Delta \theta)^2$, the rotation matrix is $R(\theta) = \begin{pmatrix} cos\ \theta & - sin\ \theta \\ sin\ \theta & cos\ \theta \end{pmatrix}$.

That means that for the distance function $(\Delta s)^2 = (\Delta r)^2 + ((1-\frac{\phi}{2 \pi})r \Delta \theta)^2$, the rotation matrix is $R(\theta) = \begin{pmatrix} cos\ [(1-\frac{\phi}{2 \pi})\ \theta] & - sin\ [(1-\frac{\phi}{2 \pi})\ \theta] \\ sin\ [(1-\frac{\phi}{2 \pi})\ \theta] & cos\ [(1-\frac{\phi}{2 \pi})\ \theta] \end{pmatrix}$?

The generator for the original rotation matrix is $X = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$. That means that the new rotation matrix has the generator $X = \begin{pmatrix} 0 & -i(1-\frac{\phi}{2 \pi}) \\ i(1-\frac{\phi}{2 \pi}) & 0 \end{pmatrix}$?

The problem with this is that because $R(\theta) = \mathbb{1} - i \theta X + ...$, I can only see one generator $X$ when in fact there should be two generators because there are two parameters $\theta$ and $\phi$.

Any thoughts on this?

 

[Orodruin]

when in fact there should be two generators because there are two parameters

SO(2) is a one dimensional group. You should not have two parameters.

 

[Fredrik]

A map from $\mathbb R^2$ to $\mathbb R^2$ that preserves (Euclidean) distances and angles is always an orthogonal linear operator. The determinant of an orthogonal linear operator is always 1 or -1. SO(2) consists of the ones with determinant 1.

I'm not sure what happens when you replace the Euclidean distance function on $\mathbb R^2$ with another one. Are you sure that your new function even satisfies the definition of a metric? I don't think it can be a metric, since there's a second parameter in there. Did you intend for it to be a metric on $\mathbb R^3$ rather than $\mathbb R^2$? Or is it supposed to be a metric on a plane through the origin in $\mathbb R^3$ that isn't the xy-plane? (If $\phi$ is the angle that the plane makes with the z axis, or something like that, then it's not really a parameter).

I think you will have to explain that distance function and how you got from the first rotation matrix to the second. The second is just a standard rotation matrix for the angle $\big(1-\frac{\phi}{2\pi}\big)\theta$. The group has only one generator, not one generator for each angle.