4.3.2 Semi-discretized LLG equation properties

By using the above notations, the spatially semi-discretized LLG equation consists in a system of $3N$ ODEs which, for the generic $l-$th cell, can be written in the following form :

$$\frac{d}{dt}\textbf{{m}}_l=-\textbf{{m}}_l\times\left[ \textbf{h}... ...l}(\underline{\textbf{m}},t) - \alpha\frac{d}{dt}\textbf{{m}}_l \right] \quad,$$ (4.22)

and for the whole collection of cells as:

$$\frac{d}{dt}\underline{\textbf{m}}=-{\underline{\Lambda}}(\underl... ...erline{\textbf{m}},t) - \alpha\frac{d}{dt}\underline{\textbf{m}}\right] \quad,$$ (4.23)

where ${\underline{\Lambda}}(\underline{\textbf{m}})$ is a block-diagonal matrix

$${\underline{\Lambda}}(\underline{\textbf{m}})= (\Lambda(\textbf{{m}}_1),\ldots,\Lambda(\textbf{{m}}_N))$$ (4.24)

with blocks $\Lambda( \cdot ) \in \mathbb{R}^{3\times3}$ such that $\Lambda(\textbf{v})\cdot \textbf{w}= \textbf{v}\times \textbf{w}$, $\forall \textbf{v}, \textbf{w}\in \mathbb{R}^3$. We also observe that ${\underline{\Lambda}}(\underline{\textbf{m}})$ is linearly dependent on $\underline{\textbf{m}}$ through an appropriate third order tensor ${\underline{\Gamma}}$ as

$${\underline{\Lambda}}= {\underline{\Gamma}}\cdot \underline{\textbf{m}}$$ (4.25)

where ${\underline{\Gamma}}$ is block diagonal with N diagonal $3 \times 3 \times 3$ blocks constituted by third order permutation tensors and the dot product in Eq. (4.25) represent an index contraction. The meaning of this contraction can be inferred by the considering that the component of the vector $\underline{\textbf{v}}\cdot ({\underline{\Gamma}} \cdot \underline{\textbf{w}})$ corresponding to the $l$-th cell is given by

$$\left( \underline{\textbf{v}}\cdot ({\underline{\Gamma}}\cdot \underline{\textbf{w}}) \right)_l = \textbf{v}_l \times \textbf{w}_l ,$$ (4.26)

where we have used the notation introduced above for mesh vectors ( $\underline{\textbf{v}}$, $\underline{\textbf{w}}$) and cell vectors ( $\textbf{v}_l$, $\textbf{w}_l$). Now, we briefly summarize the properties of the semi-discretized LLG. These properties are completely analogous to the properties (4.13)-(4.17) and the demonstration can be obtained by following the very same line of reasoning. Indeed, we can easily prove that

$$\vert\textbf{{m}}_l(t)\vert =\vert\textbf{{m}}_l (t_0 )\vert \qquad \forall t \ge t_0 , l=1,\ldots,N ,$$ (4.27)

$$\frac{d }{d t}\underline{{\text{g}}}(t) = -\alpha \left\vert \fra... ...\vert^2 -\sum_{l=1}^{N} \textbf{{m}}_l \cdot \frac{d\textbf{h}_{a,l}}{dt} ,$$ (4.28)

and, in the case of constant applied field, that:

$$\frac{d }{d t}\underline{{\text{g}}}(t) = -\alpha \left\vert \fra... ... \sum_{l=1}^{N} \alpha \left\vert\frac{d \textbf{{m}}_l}{dt}\right\vert^2 .$$ (4.29)

Finally, in the case $\alpha=0$, the semi-discretized LLG (4.23) takes the form

$$\frac{d \underline{\textbf{m}}}{d t}= {\underline{\Lambda}}(\underline{\textbf{m}}) \cdot \frac{\partial g}{\partial \underline{\textbf{m}}} ,$$ (4.30)

which is related to the semi-discretized version of Poisson bracket (4.17)

$$\{ f(\underline{\textbf{m}}) , h(\underline{\textbf{m}}) \}= \fra... ...m}}) \cdot \left( \frac{\partial h}{\partial \underline{\textbf{m}}} \right)^T$$ (4.31)

where $T$ indicate the matrix transpose, $f(\underline{\textbf{m}})$ and $h(\underline{\textbf{m}})$ are two generic functions of $\underline{\textbf{m}}$, and ${\partial f}/{\partial \underline{\textbf{m}}}$, ${\partial h}/{\partial \underline{\textbf{m}}}$ are the corresponding gradients. In connection with the hamiltonian structure (4.30), it is interesting to mention that when the matrix ${\underline{\Lambda}}(\underline{\textbf{m}})$ has the linear form (4.25), the related hamiltonian system (4.30) is said to have a Lie-Poisson structure [88]. As we will discuss in the sequel, this structure affects the nature of mid-point approximation of LLG equation.