Mathematical and Numerical Methods for Non-linear Beam Dynamics

TL;DR

This paper reviews mathematical and numerical techniques for analyzing non-linear beam dynamics in particle accelerators, emphasizing recent frameworks that improve understanding and stability analysis of particle beams.

Contribution

It introduces a unified, superior framework for handling complex non-linear problems in accelerator physics, integrating methods from other physics and mathematics fields.

Findings

New framework simplifies non-linear beam dynamics analysis

Enhanced stability understanding of particle beams

Improved numerical methods for accelerator design

Abstract

Non-linear effects in accelerator physics are important for both successful operation of accelerators and during the design stage. Since both of these aspects are closely related, they will be treated together in this overview. Some of the most important aspects are well described by methods established in other areas of physics and mathematics. The treatment will be focused on the problems in accelerators used for particle physics experiments. Although the main emphasis will be on accelerator physics issues, some of the aspects of more general interest will be discussed. In particular, we demonstrate that in recent years a framework has been built to handle the complex problems in a consistent form, technically superior and conceptually simpler than the traditional techniques. The need to understand the stability of particle beams has substantially contributed to the development of new techniques and is an important source of examples which can be verified experimentally. Unfortunately, the documentation of these developments is often poor or even unpublished, in many cases only available as lectures or conference proceedings.