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Scaling the parameter of the SO(2) rotation matrix

  1. Apr 18, 2015 #1
    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?
     
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  3. Apr 18, 2015 #2

    Orodruin

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    SO(2) is a one dimensional group. You should not have two parameters.
     
  4. Apr 19, 2015 #3

    Fredrik

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    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.
     
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