Linear time-dependent Hamiltonian systems beyond the adiabatic limit

A classical version of the Magnus expansion well suited to studying adiabatic time-evolution is built up. The method improves the adiabatic approximation while being symplectic in character. It is shown that the first-order approximation is already accurate enough even far from the adiabatic limit. An analysis of the changes suffered by the adiabatic invariant of a linear Hamiltonian system along its time-evolution illustrates part of the above results. Asymptotic formulae for such changes are also obtained with explicit computation of pre-exponential factors.