I'm studying the following equation

$$\frac{d\mathbf{M}}{dt}=\mathbf{B} (t) \times \mathbf{M}(t) \Longleftrightarrow \frac{d}{dt} \left(\begin{array}{c} M_x\\ M_y\\ M_z \end{array}\right)=\left(\begin{array}{c} B_y M_z-B_z M_y \\ B_z M_x-B_x M_z \\ B_x M_y-B_y M_x \end{array}\right)$$

which is known in physics as the Landau–Lifshitz–Gilbert equation without damping.

I would like to know if this equation (which describes the precession of the magnetisation $M$ around the vector $B$) has a name in mathematics, and if its solutions are known in the general case for a smooth vector $B(t)$ and for dimension $3$ (as I write them in components).


1 Answer 1


The equation can be written in 'matrix form' as \begin{equation} \frac{\text{d} \mathbf{x}}{\text{d} t} = \mathbf{A}(t)\,\mathbf{x}, \tag{1} \end{equation} where \begin{equation} \mathbf{A}(t) = \begin{pmatrix} 0 & -B_z & B_y \\ B_z & 0 & -B_x \\ -B_y & B_x & 0\end{pmatrix}. \end{equation} For general smooth $B(t)$, equation $(1)$ is the general form of a nonautonomous linear (three-dimensional) system, with the only additional constraint that the matrix $\mathbf{A}(t)$ is antisymmetric, i.e. $\mathbf{A}^T = - \mathbf{A}$. Because this setting is so general, there are no explicit results for general $\mathbf{B}(t)$. If $\mathbf{B}(t)$ is periodic in time, however, it's worth looking into Floquet theory.

Regarding the name, as far as I know this equation is only known as the Landau-Lifshitz-Gilbert equation. There are some interesting results on the solutions of this equation when $\mathbf{B}$ is a function of $\mathbf{M}$, yielding a nonlinear but autonomous system of ODE's, see here.


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