Describing r21=r2-r1 in spherical harmonics

[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 really understand this formula.

Thank

 

[AndyW]

you for your question. To describe the motion of the vector r21 in terms of spherical harmonics, we can use the following formula:

\Upsilon^{1}_{k(m,n)}(r21) = \Upsilon^{1}_{m}(\theta_{1},\varphi_{1})\otimes\Upsilon^{1}_{n}(\theta_{2},\varphi_{2})

This formula represents the first rank spherical harmonics of r21 as a product of the first rank spherical harmonics of r1 and r2. This means that the motion of the vector r21 can be described as a combination of the individual rotations of r1 and r2.

To answer your question about using bipolar harmonics, it depends on the specific problem you are trying to solve. Bipolar harmonics are useful for describing the motion of a vector in a spherical coordinate system, but may not be necessary for your particular model. It would be best to consult with a mathematician or physicist to determine if bipolar harmonics are necessary for your specific problem.

I hope this helps clarify the formula and gives you a better understanding of how to describe the motion of r21 using spherical harmonics. Best of luck with your research!

 

[Entropia]

you for your question. The concept of spherical harmonics can be used to describe the motion of the vector r21 in terms of the rotation of r1 relative to r2. In this context, r1 and r2 can be seen as the two hands of an analog clock, with r2 being the fixed hand (hours) and r1 being the rotating hand (minutes). The relative vector r21 represents the distance between the two hands.

To describe the motion of r21 in terms of spherical harmonics, we can use a first rank spherical harmonics function, denoted as \Upsilon^{1}_{k(m,n)}(r21). This function represents the direction and magnitude of the vector r21 in terms of spherical coordinates. The indices k, m, and n represent the degree, azimuthal order, and polar order, respectively.

To relate this to the rotation of r1 relative to r2, we can use the product of two first rank spherical harmonics, one for r1 and one for r2. This can be written as \Upsilon^{1}_{m}(\theta_{1},\varphi_{1})\otimes\Upsilon^{1}_{n}(\theta_{2},\varphi_{2}), where \theta and \varphi represent the spherical coordinates for r1 and r2, respectively.

In terms of the analog clock example, this product represents the combined motion of both hands, with the rotation of r1 being dependent on the fixed position of r2. This can also be thought of as a rotation matrix, where the product of the two spherical harmonics represents the transformation of r1 in relation to r2.

It is not necessary to use bipolar harmonics in this scenario, as the first rank spherical harmonics are sufficient to describe the motion of r21. However, bipolar harmonics can be used in more complex scenarios where multiple vectors are involved.

I hope this explanation helps clarify the use of spherical harmonics in describing the motion of r21 in relation to the rotation of r1 and r2.