High-order classical adiabatic reaction forces: slow manifold for a spin model

The influence of a fast system on the hamiltonian dynamics of a slow system coupled to it is explored by calculating, in a model, high-order smooth (nonoscillating) adiabatic reaction forces (i.e. beyond Born–Oppenheimer and geometric magnetism). The model is a spin (fast) driven by, and reacting on, the position vector (slow) of a particle coupled to it. The search for smooth solutions is equivalent to determining the slow manifold in the full phase space, on which, in the model system, the spin would not precess. The series of reactions for the nonlinear coupled system diverges factorially, as in the simpler linear case of a spin being driven passively by a position vector changing in a prescribed manner. When the particle is closest to the origin, all terms in the divergent series have the same sign, indicating a Stokes phenomenon and suggesting that a solution of the slow manifold equation exists but contains exponentially weak precession oscillations. The predicted oscillations are observed numerically, and shown to be inevitable for the exactly solvable linearized slow manifold which is equivalent to the Landau–Majorana–Zener model of quantum mechanics.