  We formulate the concept of dominant interaction Hamiltonians to obtain an
integrable approximation to the dynamics of an electron exposed to a strong
laser field and an atomic potential leading to high harmonic generation. The
concept relies on local information in phase space to switch between the
interactions. This information is provided by classical integrable trajectories
from which we construct a semiclassical wave function. The high harmonic
spectrum obtained is in excellent agreement with the accurate quantum spectrum.
The separation in the atomic potential and laser coupling interactions should
facilitate the calculation of high harmonic spectra in complex systems.
