4.2.2 Constraints for magnetization dynamics
and hamiltonian structure of conservative dynamics

Let us now summarize the fundamental properties of LLG dynamics in the normalized quantities introduced above. The first property a) is expressed as

$$\vert\textbf{{m}}(\textbf{r},t)\vert= \vert\textbf{{m}}(\textbf{r... ...0)\vert \qquad \forall t \ge t_0 , \forall \textbf{r}\in \Omega ,$$ (4.13)

which can be easily derived as explained in section 1.3.5. In the typical micromagnetic problem it is assumed initially $\vert\textbf{{m}}(\textbf{r},t_0)\vert=1$ which is the normalized version of the micromagnetic constraint $\vert\textbf{M}\vert=$M$_$s. The second fundamental property b) is related to the nature of the energy balance in LLG dynamics:

$$\frac{d }{d t}g(t) = - \int_{\Omega} \alpha \left\vert \frac{\par... ...nt_{\Omega} \textbf{m}\cdot \frac{\partial \textbf{h}_a }{\partial t} dv$$ (4.14)

which has very important implications. For constant applied field, Eq. (4.14) reduces to

$$\frac{d }{d t}g(t) = - \int_{\Omega} \alpha \left\vert \frac{\partial \textbf{{m}}}{\partial t}\right\vert^2 dv ,$$ (4.15)

This equation reveals the role of the damping and shows that LLG dynamics has a Lyapunov structure, namely, for constant external field, the free energy is always a decreasing function of time. In addition, it also demonstrates the nature of the Gilbert phenomenological damping: the dissipation is proportional to the square of the velocity of variation of magnetization. This is connected to the fact that the Gilbert damping term can be introduced by using the Rayleigh approach described in section 1.3.5. This property is very important because it guarantees that the system tends toward minima of free energy (i.e. meta-stable equilibrium points). Third, for $\alpha=0$ the LLG equation becomes an Hamiltonian dynamical system for the vector field $\textbf{{m}}$ defined as

$$\frac{\partial \textbf{{m}}}{\partial t} = \textbf{{m}}\times \frac{\delta g}{\delta \textbf{{m}}} .$$ (4.16)

The LLG hamiltonian form (4.16) is related to the following LLG Poisson bracket

$$\{ f(\textbf{{m}}) , h(\textbf{{m}}) \}= -\textbf{{m}}\cdot \frac{\delta f}{\delta \textbf{{m}}}\times \frac{\delta h}{\delta \textbf{{m}}}$$ (4.17)

where $f(\textbf{{m}})$ and $h(\textbf{{m}})$ are two generic functionals of $\textbf{{m}}$. In Eq. (4.16), the role of the hamiltonian is played by $g(\textbf{{m}};\textbf{h}_$a$)$. It should be underlined that, although the LLG dynamics is always dissipative, it is interesting to consider the conservative case as in most experimental situations the dissipative effect are quite small (typically $\alpha\sim 10^{-2}$). In other terms, the LLG dynamics, on relatively short time scale, is a perturbation of the conservative (hamiltonian) precessional dynamics. In this respect, many interesting conclusions on the nature of dynamics can be also derived from the conservative^4.1 Eq. (4.16).