  Based on the modified Landau-Lifshitz-Gilbert equation for an arbitrary
Stoner particle under an external magnetic field and a spin-polarized electric
current, differential equations for the optimal reversal trajectory, along
which the magnetization reversal is the fastest one among all possible reversal
routes, are obtained. We show that this is a Euler-Lagrange problem with
constrains. The Euler equation of the optimal trajectory 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 nonlinear differential equation.
