1.3.5.1 Magnetization magnitude conservation

Let us now briefly recall the fundamental properties of Landau-Lifshitz-Gilbert (LLG) dynamics. By scalar multiplying both sides of the LLG equation (1.95) by $\textbf{{m}}$ one can easily obtain:

$$\frac{d}{dt}\left(\frac{1}{2} \vert\textbf{{m}}\vert^2 \right) = 0 \quad,$$ (1.96)

which implies that, for any $t_0,t$ and $\textbf{r}\in \Omega$, it happens that:

$$\vert\textbf{{m}}(t,\textbf{r})\vert= \vert\textbf{{m}}(t_0,\textbf{r})\vert \quad .$$ (1.97)

Thus, any magnetization motion, at a given location $\textbf{r}$, will occur on the unit sphere.