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Rotational symmetry

  1. Jun 6, 2009 #1
    I've been following along with Lenny Susskinds lectures on modern classical mechanics on youtube.

    at 34:30 he writes a few translation formulas on the board:
    delta X = - epsilon Y
    delta Y = epsilon X


    It's not obvious to me why these equations are true. I can't seem to find a derivation anywhere, nor can I work one out myself. Any help?
     
    Last edited by a moderator: Sep 25, 2014
  2. jcsd
  3. Jul 9, 2009 #2
    I haven't watched the video, but if you perform a http://mathworld.wolfram.com/RotationMatrix.html" [Broken] by an angle [itex]\theta[/itex] about the [itex]z[/itex] axis on the vector [itex] {\bf r} = \left(x,y,z\right) [/itex], you get [itex] r' = r + \Delta r = \left(\cos \theta x - \sin \theta y, \sin \theta x + \cos \theta y ,z\right) [/itex]. For [itex]\theta = \epsilon[/itex] infinitesimal, this becomes [itex] r' = r + \delta r = \left(x - \epsilon y, \epsilon x + y ,z\right) = r + \left(- \epsilon y, \epsilon x ,0\right) [/itex], so that [itex] \delta x = - \epsilon y[/itex] and [itex] \delta y = - \epsilon x[/itex].
     
    Last edited by a moderator: May 4, 2017
  4. Jul 14, 2009 #3
    thanks, that's what I was looking for.
     
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