Generalized Panofsky-Wenzel theorem in curvilinear coordinate systems applicable to non-ultrarelativistic beams

TL;DR

This paper extends the classical impedance and wake theories in particle accelerators to curved beam trajectories, assuming constant velocity, broadening their applicability beyond straight-line motion.

Contribution

It introduces a generalized formulation of impedances and wakes for particles moving along curved paths with constant speed, expanding the theoretical framework.

Findings

Extended impedance and wake theories to curved orbits

Applicable to non-ultrarelativistic beams

Maintains constant velocity assumption

Abstract

This note gives an introduction to the theories of impedances and wakes in particle accelerators. The standard formulation assumes that the beam is traveling along a straight orbit with constant velocity $\vec{v}=v\vec{e}_z$. On this note, we show the possibility of extending the formulation for beams traveling along a curved orbit but assuming $|\vec{v}|=v$ to be constant.