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Describing r21=r2-r1 in spherical harmonics

  1. May 20, 2008 #1
    Describing the a model
    Two hands of an analog clock: r1 (hand of the minutes) and r2 (hand of the hours),
    and a relative vector r21 between them.

    The question:
    In spherical harmonics representation how can I describe the motion of the vector r21 by the rotation of r1 relative to r2 (r2 is fixed).

    I would like to describe the first rank spherical harmonics of r21 -> [tex]\Upsilon^{1}_{k(m,n)}[/tex](r21) by the product of the two spherical harmonics r1 -> [tex]\Upsilon^{1}_{m}[/tex]([tex]\theta[/tex][tex]_{1}[/tex],[tex]\varphi[/tex][tex]_{1}[/tex])
    and r2 -> [tex]\Upsilon^{1}_{n}[/tex]([tex]\theta[/tex][tex]_{2}[/tex],[tex]\varphi[/tex][tex]_{2}[/tex])

    somthing like ~ [tex]\Upsilon^{1}_{m}[/tex]([tex]\theta[/tex][tex]_{1}[/tex],[tex]\varphi[/tex][tex]_{1}[/tex])[tex]\otimes[/tex][tex]\Upsilon^{1}_{n}[/tex]([tex]\theta[/tex][tex]_{2}[/tex],[tex]\varphi[/tex][tex]_{2}[/tex]).

    Should I use the bipolar harmonics ? I don't realy understand this formula.

  2. jcsd
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