# Describing r21=r2-r1 in spherical harmonics

1. May 20, 2008

### borish

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).

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

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

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

Thank