Post-adiabatic forces and Lagrangians with higher-order derivatives
We study a slow classical system [particle] coupled to a fast quantum system with discrete energy spectrum. We adiabatically exclude the quantum system and construct an autonomous dynamics for the classical particle in successive orders of the small ratio $ε$ of the characteristic times. It is known that in the order $ε^0$ the particle gets an additional [Born-Oppenheimer] potential, while in the order $\eps^1$ it feels an effective magnetic field related to the Berry phase. In the order $ε^2$ the motion of the classical particle can be reduced to a free [geodesic] motion on a curved Riemannian manifold, with the metric generated by the excluded quantum system. This motion has a number of unusual features, e.g., it combines subspaces of different (Riemannian and pseudo-Riemannian) signature for the metric tensor. In the order $ε^3$ the motion of the classical particle is still described by a Lagrangian, but the latter linearly depends on the particle's acceleration. This implies the existence of a spin tensor [non-orbital angular momentum] for the particle. This spin tensor is related to the momentum via an analogue of the zitterbewegung effect. The Hamiltonian structure of the system is non-trivial and is defined via non-linear Poisson brackets. The linear dependence of the effective classical Lagrangian on higher-order derivatives is seen as well in the higher orders $ε^n$.