  Scanning probe microscopy is often extended beyond topographic imaging to
study electrical forces and sample properties, with the most widely used
experiment being frequency-modulated Kelvin probe force microscopy. The
equations commonly used to interpret this experiment, however, rely on two
hidden assumptions: (1) the tip charge oscillates in phase with the cantilever
motion to keep the tip voltage constant, and (2) any changes in the tip-sample
interaction happen slowly. Starting from an electro-mechanical model of the
cantilever-sample interaction, we use Lagrangian mechanics to derive coupled
equations of motion for the cantilever position and charge. This general
approach rigorously describes scanned probe experiments even in the case when
the usual assumptions of fast tip charging and slowly changing samples
properties are violated. We develop a Magnus-expansion approximation to
illustrate how abrupt changes in the tip-sample interaction cause abrupt
changes in the cantilever amplitude and phase. We show that feedback-free
time-resolved electric force microscopy cannot uniquely determine sub-cycle
photocapacitance dynamics. We then use first-order perturbation theory to
relate cantilever frequency shift and dissipation to the sample impedance even
when the tip charge oscillates out of phase with the cantilever motion.
Analogous to the treatment of impedance spectroscopy in electrochemistry, we
apply this approximation to determine the cantilever frequency shift and
dissipation for an arbitrary sample impedance in (broadband) local dielectric
spectroscopy experiments. The general approaches we develop provide a path
forward for rigorously modeling the coupled motion of the cantilever position
and charge in the wide range of electrical scanned probe microscopy experiments
where the hidden assumptions of the conventional equations are violated or
inapplicable.
