Jacobi equations and particle accelerator beam dynamics

TL;DR

This paper presents a geometric approach to linear beam dynamics in accelerators, interpreting it as an approximation to Jacobi equations of an affine Lorentz connection, including statistical perturbations.

Contribution

It introduces a geometric formulation of beam dynamics using Jacobi equations and accounts for statistical perturbations in particle bunches.

Findings

Linear transverse and longitudinal dynamics modeled as Jacobi equations

Geometric interpretation of beam trajectories as integral curves

Inclusion of statistical effects as perturbations

Abstract

A geometric formulation of the linear beam dynamics in accelerator physics is presented. In particular, it is proved that the linear transverse and longitudinal dynamics can be interpret geometrically as an approximation to the Jacobi equation of an affine averaged Lorentz connection. We introduce a specific notion reference trajectory as integral curves of the main velocity vector field. A perturbation caused by the statistical nature of the bunch of particles is considered.