Euler equation of the optimal trajectory for the fastest magnetization reversal of nano-magnetic structures

The Euler equation of the optimal reversal trajectory for the fastest magnetization reversal is obtained for an arbitrary nano-magnetic structure. The Euler equation is useful in designing a magnetic field pulse and/or a polarized electric current pulse in magnetization reversal for two reasons. 1) It is straightforward to obtain the solution of the Euler equation, at least numerically, for a given magnetic nano-structure characterized by its magnetic anisotropy energy. 2) After obtaining the optimal reversal trajectory for a given magnetic nano-structure, finding a proper field/current pulse is an algebraic problem instead of the original non-linear differential equation.