Derivation of LLG equation in polar coordinates

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Discussion Overview

The discussion revolves around the derivation of the Landau-Lifshitz-Gilbert (LLG) equation in polar coordinates, specifically addressing the torque contribution due to uniaxial anisotropy and its representation in the LLG framework. The focus is on the transition from Cartesian to polar coordinates and the challenges associated with this derivation.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant presents the torque contribution due to uniaxial anisotropy and its relation to the LLG equation in polar coordinates.
  • Another participant asks for clarification on the specific difficulties faced, suggesting that mentioning certain parameters could simplify the discussion.
  • A participant acknowledges the need to derive the equation in polar angles while noting that the initial equation is in Cartesian coordinates, expressing uncertainty about how to achieve a clean final expression.
  • A later reply references additional papers that provide more explicit theoretical work, although they may be considered excessive for the current problem.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and uncertainty regarding the derivation process. There is no consensus on how to proceed with the derivation, and multiple viewpoints on the complexity of the task are present.

Contextual Notes

Participants mention the need to consider specific conditions, such as setting certain parameters to zero, which may affect the derivation process. The transition from Cartesian to polar coordinates is highlighted as a critical step that introduces complexity.

apervaiz
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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 \ &#039; \\<br /> \phi \ &#039; \\<br /> \end{bmatrix}<br /> = \begin{bmatrix}<br /> \theta \\<br /> \phi \\<br /> \end{bmatrix}<br />

Now the result of this derivation is already given as

\begin{bmatrix}<br /> \theta \ &#039; \\<br /> \phi \ &#039; \\<br /> \end{bmatrix}<br /> = \begin{bmatrix}<br /> \alpha \sin \theta \cos \theta \\<br /> \cos \theta \\<br /> \end{bmatrix}<br />

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
 
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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.
 
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.
 

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