In solid-state physics, the Landau–Lifshitz equation (LLE), named for Lev Landau and Evgeny Lifshitz, is a partial differential equation describing time evolution of magnetism in solids, depending on 1 time variable and 1, 2, or 3 space variables.
The LLE describes an anisotropic magnet. The equation is described in (Faddeev & Takhtajan 2007, chapter 8) as follows: it is an equation for a vector field S, in other words a function on R1+n taking values in R3. The equation depends on a fixed symmetric 3-by-3 matrix J, usually assumed to be diagonal; that is, J = diag --> ( J 1 , J 2 , J 3 ) {\displaystyle J=\operatorname {diag} (J_{1},J_{2},J_{3})} . The LLE is then given by Hamilton's equation of motion for the Hamiltonian
(where J(S) is the quadratic form of J applied to the vector S) which is
In 1+1 dimensions, this equation is
In 2+1 dimensions, this equation takes the form
which is the (2+1)-dimensional LLE. For the (3+1)-dimensional case, the LLE looks like
In the general case LLE (2) is nonintegrable, but it admits two integrable reductions:
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