4.4.1.3 Preservation of Hamiltonian structure

Finally, it is also important to address the issue of the preservation of the hamiltonian structure [89] (in the case $\alpha=0$) given by Eq. (4.30). Let us indicate by $\underline{\textbf{m}}(t)=\phi(t,\underline{\textbf{m}}_0)$ the flow of Eq. (4.30), namely the solution of the Cauchy problem for the system of ODEs (4.30) with the initial condition $\underline{\textbf{m}}(t=t_0)=\underline{\textbf{m}}_0$. It is well known that the map $\phi(t,\underline{\textbf{m}}_0)$ mapping $\underline{\textbf{m}}_0$ into $\underline{\textbf{m}}(t)$ satisfies the following symplecticity condition

$$\frac{\partial \phi(t,\underline{\textbf{m}})}{\partial \underlin... ...bf{m}}} \right)^T = {\underline{\Lambda}}(\phi(t,\underline{\textbf{m}})) .$$ (4.41)

A numerical scheme is said to preserve the hamiltonian structure if the associated map, which connects one step to the following (in the case of mid-point rule the map $\Phi(\underline{\textbf{m}}^{n},\Delta t)$ introduced in Eq. (4.34)), fulfills the condition (4.41). In this respect, by using the fact that the LLG equation has a Lie-Poisson structure (i.e. the matrix ${\underline{\Lambda}}(\underline{\textbf{m}})$ is linear with respect to $\underline{\textbf{m}}$ as expressed in Eq. (4.25)), it is possible to prove the following error formula [83]

$$\frac{\partial \Phi(\Delta t,\underline{\textbf{m}})}{\partial \u... ...ine{\Lambda}}(\Phi(\Delta t,\underline{\textbf{m}})) = \mathcal{O}(\Delta t ^3)$$ (4.42)

which means that, the mid-point rule applied to LLG equation preserves hamiltonian structure up to the third order term in $\Delta t$. It is also interesting to underline that the preservation of hamiltonian structure would be exact for an hamiltonian system defined by a Poisson bracket of the type $\{ f(\underline{\textbf{m}}) , h(\underline{\textbf{m}}) \}= ({\partial f}/{\... ...derline{\textrm{J}}} \cdot ( {\partial h}/{\partial \underline{\textbf{m}}})^T$ where the matrix ${\underline{\textrm{J}}}$ does not depend on $\underline{\textbf{m}}$. In LLG studies this situation is encountered in all those problems in which LLG equation is linearized around a given magnetization state as it is generally done in Spin-wave analysis [4] and nucleation problems [5]. In this respect, it must be underlined that, although these problems are linear in nature, analytical solutions are obtainable only under quite restrictive assumptions about the geometry of the magnetic body. General geometries can be treated only by numerical techniques.