# Preserving Simplecticity in the Numerical Integration of Linear Beam Optics

## Abstract

Presented are mathematical tools and methods for the development of numerical integration techniques that preserve the symplectic condition inherent to mechanics. The intended audience is for beam physicists with backgrounds in numerical modeling and simulation with particular attention to beam optics applications. The paper focuses on Lie methods that are inherently symplectic regardless of the integration accuracy order. Section 2 provides the mathematically tools used in the sequel and necessary for the reader to extend the covered techniques. Section 3 places those tools in the context of charged-particle beam optics; in particular linear beam optics is presented in terms of a Lie algebraic matrix representation. Section 4 presents numerical stepping techniques with particular emphasis on a third-order leapfrog method. Section 5 discusses the modeling of field imperfections with particular attention to the fringe fields of quadrupole focusing magnets. The direct computation of a third order transfer matrix for a fringe field is shown.

- Authors:

- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1408583

- Report Number(s):
- ORNL/TM-2017/358

- DOE Contract Number:
- AC05-00OR22725

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 43 PARTICLE ACCELERATORS

### Citation Formats

```
Allen, Christopher K.
```*Preserving Simplecticity in the Numerical Integration of Linear Beam Optics*. United States: N. p., 2017.
Web. doi:10.2172/1408583.

```
Allen, Christopher K.
```*Preserving Simplecticity in the Numerical Integration of Linear Beam Optics*. United States. doi:10.2172/1408583.

```
Allen, Christopher K. Sat .
"Preserving Simplecticity in the Numerical Integration of Linear Beam Optics". United States.
doi:10.2172/1408583. https://www.osti.gov/servlets/purl/1408583.
```

```
@article{osti_1408583,
```

title = {Preserving Simplecticity in the Numerical Integration of Linear Beam Optics},

author = {Allen, Christopher K.},

abstractNote = {Presented are mathematical tools and methods for the development of numerical integration techniques that preserve the symplectic condition inherent to mechanics. The intended audience is for beam physicists with backgrounds in numerical modeling and simulation with particular attention to beam optics applications. The paper focuses on Lie methods that are inherently symplectic regardless of the integration accuracy order. Section 2 provides the mathematically tools used in the sequel and necessary for the reader to extend the covered techniques. Section 3 places those tools in the context of charged-particle beam optics; in particular linear beam optics is presented in terms of a Lie algebraic matrix representation. Section 4 presents numerical stepping techniques with particular emphasis on a third-order leapfrog method. Section 5 discusses the modeling of field imperfections with particular attention to the fringe fields of quadrupole focusing magnets. The direct computation of a third order transfer matrix for a fringe field is shown.},

doi = {10.2172/1408583},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Sat Jul 01 00:00:00 EDT 2017},

month = {Sat Jul 01 00:00:00 EDT 2017}

}