# Betatron motion with coupling of horizontal and vertical degrees of freedom

## Abstract

The Courant-Snyder parameterization of one-dimensional linear betatron motion is generalized to two-dimensional coupled linear motion. To represent the 4 x 4 symplectic transfer matrix the following ten parameters were chosen: four beta-function, four alpha-functions and two betatron phase advances. The beta-functions have a meaning similar to the Courant-Snyder parameterization, and the definition of alpha-functions coincides with the standard one at regions with zero longitudinal magnetic field, where they are equal to negative half-derivatives of the beta-functions. Such a parameterization can be useful for analysis of coupled betatron motion in circular machines and transfer lines.

- Authors:

- Publication Date:

- Research Org.:
- Thomas Jefferson National Accelerator Facility, Newport News, VA (US)

- Sponsoring Org.:
- USDOE Office of Energy Research (ER) (US)

- OSTI Identifier:
- 754650

- Report Number(s):
- DOE/ER/40150-1275; JLAB-ACC-99-19

TRN: US0002259

- DOE Contract Number:
- AC05-84ER40150

- Resource Type:
- Journal Article

- Resource Relation:
- Other Information: PBD: 1 Oct 1999

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 43 PARTICLE ACCELERATORS; BETATRON OSCILLATIONS; ACCELERATORS; BEAM TRANSPORT; TRANSFER FUNCTIONS; MATRICES; ACCELERATOR

### Citation Formats

```
Lebedev, V A, and Bogacz, S A.
```*Betatron motion with coupling of horizontal and vertical degrees of freedom*. United States: N. p., 1999.
Web.

```
Lebedev, V A, & Bogacz, S A.
```*Betatron motion with coupling of horizontal and vertical degrees of freedom*. United States.

```
Lebedev, V A, and Bogacz, S A. Fri .
"Betatron motion with coupling of horizontal and vertical degrees of freedom". United States. https://www.osti.gov/servlets/purl/754650.
```

```
@article{osti_754650,
```

title = {Betatron motion with coupling of horizontal and vertical degrees of freedom},

author = {Lebedev, V A and Bogacz, S A},

abstractNote = {The Courant-Snyder parameterization of one-dimensional linear betatron motion is generalized to two-dimensional coupled linear motion. To represent the 4 x 4 symplectic transfer matrix the following ten parameters were chosen: four beta-function, four alpha-functions and two betatron phase advances. The beta-functions have a meaning similar to the Courant-Snyder parameterization, and the definition of alpha-functions coincides with the standard one at regions with zero longitudinal magnetic field, where they are equal to negative half-derivatives of the beta-functions. Such a parameterization can be useful for analysis of coupled betatron motion in circular machines and transfer lines.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1999},

month = {10}

}

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