On the field Lagrangians in micromagnetics

Micromagnetic calculations are commonly carried out starting from an expression for the energy of a magnetic body which contains a six-fold integral representing the nonlocal dipole-dipole interactions. This term is responsible for the long running times in numerical applications. There exist two alternative field Lagrangians in which nonlocal terms do not appear; this locality is obtained at the cost of introducing new variables in the equations. Only one of the two Lagrangians has local minima in one-to-one correspondence with those of the energy. The interrelation of the two field Lagrangians with the energy is discussed. Special formulas are derived for the two-dimensional case, and possible numerical applications are discussed.