Derivation of LLG equation in polar coordinates

[apervaiz]

The torque contribution due to the uniaxial anisotropy is given by the equation below

$$\frac{\Gamma}{l_m K} = (2 \sin\theta \cos\theta)[\sin\phi e_x - \cos\phi e_y] (3)$$

This contribution can be taken in the LLG equation to derive the LLG equation in polar coordinates

$$\frac{\partial n_m}{\partial t} + ( n_m \times \frac{\partial n_m}{\partial t})=\frac{1}{2}\Omega_K \frac{\Gamma}{l_m K} (9)$$
where $n_m= [r,\theta,\phi]$. Since r is unity for magnetization a differential equation in the two angles should be possible which should have the form
$$\begin{bmatrix}<br /> \theta \ ' \\<br /> \phi \ ' \\<br /> \end{bmatrix}<br /> = \begin{bmatrix}<br /> \theta \\<br /> \phi \\<br /> \end{bmatrix}$$

Now the result of this derivation is already given as

$$\begin{bmatrix}<br /> \theta \ ' \\<br /> \phi \ ' \\<br /> \end{bmatrix}<br /> = \begin{bmatrix}<br /> \alpha \sin \theta \cos \theta \\<br /> \cos \theta \\<br /> \end{bmatrix}$$

I'm having a hard time deriving this result from (3) using (9). Could anyone help me with this?
here is the link of this paper
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.62.570

 

[Hypersphere]

Can you be more specific about where you are having a hard time? Is it the change of coordinate system?

Also, when asking a question like this, it's quite reasonable to mention that you're setting $h_p=h=h_s=0$. Makes it easier to relate your equations to those in the paper.

 

[apervaiz]

Thanks for the reply. You are right this equation only caters for uniaxial anisotropy. the equation has to be derived in polar angles while (3) is in cartesian. I think going from (3) to polar is not hard, but how to get a clean expression like the final one.
I'm not sure how to proceed for the derivation at all.

 

[rigetFrog]

these papers works out theory a little more explicitly, but they are over kill. (One is for a coupled system)

http://iopscience.iop.org/0034-4885/61/7/001/pdf/0034-4885_61_7_001.pdf

http://journals.aps.org/prb/pdf/10.1103/PhysRevB.74.064409