LL方程推导LLG方程
LL: L.Landau and Lifshitz $$\frac{\partial \vec{M}}{\partial t} = \gamma M \times \vec{H} - \frac{\lambda}{M^2} \times \vec{M} \times (\vec{M} \times \vec{H}) \label{eq:1} \tag{1}$$
$\vec{H}$ refers to effective magnetic fields, $\vec{H} =-\frac{\partial \vec{U}}{\partial \vec{M}}$
$\vec{U} = \vec{U_e} + \vec{U_d} + \vec{U_{ex}} + \vec{U_e} + \vec{U_a} + \vec{U_{me}}$
So: $\vec{H} = \vec{H_e} + \vec{H_d} + \vec{H_{ex}} + \vec{H_e} + \vec{H_a} + \vec{H_{me}}$
where $\vec{U_e}$ is external fields energy, $\vec{U_d}$ is demagnetization energy, $\vec{U_{ex}}$ is exchage energy, $\vec{U_a}$ is magnetoelastic energy.
LL equation can be written as:
$$ \vec{M} \times \vec{H} = \left[ \frac{\partial \vec{M}}{\partial t } + \frac{\lambda}{M^2} \times \vec{M} \times (\vec{M} \times \vec{H}) \right] \gamma^{-1} \label{eq:2} \tag{2} $$
Substitute $\eqref{eq:2}$ to $\eqref{eq:1}$:
$$ \begin{aligned}
\frac{\partial \vec{M}}{\partial t}&=\gamma \vec{M}\times \vec{H}-\frac{\lambda}{M^2}\vec{M}\times \gamma ^{-1}\left[ \frac{\partial \vec{M}}{\partial t}+\frac{\lambda}{M^2}\vec{M}\times \left( \vec{M}\times \vec{H} \right) \right] \\
&=\gamma \vec{M}\times \vec{H}-\frac{\lambda}{\gamma M^2}\times \frac{\partial \vec{M}}{\partial t}-\frac{\lambda ^2}{\gamma M^4}\vec{M}\times \left[ \vec{M}\times \left( \vec{M}\times \vec{H} \right) \right]\ \end{aligned} \label{eq:3} \tag{3} $$
because $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \cdot \vec{b} - (\vec{a} \cdot \vec{b}) \cdot \vec{c}$
so $\vec{M} \times [\vec{M} \times (\vec{M} \times \vec{H})] = [\vec{M} \cdot (\vec{M} \times \vec{H})] \cdot \vec{M} - (\vec{M} \cdot \vec{M}) \cdot (\vec{M} \times \vec{H}) \ = -M^2 \cdot (\vec{M} \times \vec{H})$
so $\eqref{eq:3}$ can be simplified as:
$$\begin{aligned}
\frac{\partial \vec{M}}{\partial t}&=\gamma \vec{M}\times \vec{H}-\frac{\lambda}{\gamma M^2}\vec{M}\times \frac{\partial \vec{M}}{\partial t}+\frac{\lambda ^2M^2}{\lambda M^4}\vec{M}\times \vec{H} \\
&=\left( \gamma +\frac{\lambda ^2M^2}{\gamma M^4} \right) \vec{M}\times \vec{H}-\frac{\lambda}{\gamma M^2}\vec{M}\times \frac{\partial \vec{M}}{\partial t} \\
&=\left( \frac{\gamma ^2M^4+\gamma ^2M^4\alpha ^2}{\gamma M^4} \right) \vec{M}\times \vec{H}-\frac{\alpha}{M}\vec{M}\times \frac{\partial \vec{M}}{\partial t} \\
&=\gamma \left( 1+\alpha ^2 \right) \vec{M}\times \vec{H}-\frac{\alpha}{M}\vec{M}\times \frac{\partial \vec{M}}{\partial t} \\
&=\gamma ^* \vec{M}\times \vec{H}-\frac{\alpha}{M}\vec{M}\times \frac{\partial \vec{M}}{\partial t} \\
\end{aligned}$$
where $\alpha = \eta \cdot \gamma \cdot M = \lambda/\gamma M$
$\gamma^* = \gamma(1+\alpha^2)$