  In atomic force microscopy (AFM) tip-surface interactions are usually
considered as functions of the tip position only, so-called force curves.
However, tip-surface interactions often depend on the tip velocity and the past
tip trajectory. Here, we introduce a compact and general description of these
interactions appropriate to dynamic AFM where the measurement of force is
restricted to a narrow frequency band. We represent the tip-surface interaction
in terms of a force disk in the phase space of position and velocity.
Determination of the amplitude dependence of tip-surface forces at a fixed
static probe height allows for a comprehensive treatment of conservative and
dissipative interactions. We illuminate the fundamental limitations of force
reconstruction with narrow band dynamic AFM and we show how the amplitude
dependence of the Fourier component of the force at the tip oscillation
frequency, gives qualitative insight into the detailed nature of the
tip-surface interaction. With minimal assumptions this amplitude dependence
force spectroscopy allows for a quantitative reconstruction of the effective
conservative tip-surface force as well as a position-dependent damping factor.
We demonstrate this reconstruction on simulated intermodulation AFM data.
