General formulation of impedance in Frenet-Serret coordinate system

Abstract

In accelerator physics, the concept of impedance is popularly used to describe the interactions of charged particles inside a bunch or between bunches in a train. Standard formulations of impedance assume that the driving charge has a constant velocity $\vec{v}=v\vec{i}_z$ in the $z$ direction of the Cartesian coordinate system. For the case of driving charge moving along a curved orbit, impedance can be formulated in the Frenet-Serret coordinate system, but there seems to be a lack of systematic formulations. This note presents an effort in this direction.