Longitudinal Force on a Moving Potential

We show a formal result of the longitudinal force acting on a moving potential. The potential can be velocity-dependent, which appears in various interesting physical systems, such as electrons in the presence of a magnetic flux-line, or phonons scattering off a moving vortex. By using the phase-shift analysis, we are able to show the equivalence between the adiabatic perturbation theory and the kinetic theory for the longitudinal force in the dilute gas limit.