# Rotation number of integrable symplectic mappings of the plane

## Abstract

Symplectic mappings are discrete-time analogs of Hamiltonian systems. They appear in many areas of physics, including, for example, accelerators, plasma, and fluids. Integrable mappings, a subclass of symplectic mappings, are equivalent to a Twist map, with a rotation number, constant along the phase trajectory. In this letter, we propose a succinct expression to determine the rotation number and present two examples. Similar to the period of the bounded motion in Hamiltonian systems, the rotation number is the most fundamental property of integrable maps and it provides a way to analyze the phase-space dynamics.

- Authors:

- Fermilab
- Oak Ridge

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1418136

- Report Number(s):
- FERMILAB-PUB-17-305-AD-APC-CD

1649868

- DOE Contract Number:
- AC02-07CH11359

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: TBD

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Arnold-Liouville theorem; KAM theory; discrete dynamical systems; integrability; McMillan map; symplectic topology; Poincare rotation number

### Citation Formats

```
Zolkin, Timofey, Nagaitsev, Sergei, and Danilov, Viatcheslav.
```*Rotation number of integrable symplectic mappings of the plane*. United States: N. p., 2017.
Web.

```
Zolkin, Timofey, Nagaitsev, Sergei, & Danilov, Viatcheslav.
```*Rotation number of integrable symplectic mappings of the plane*. United States.

```
Zolkin, Timofey, Nagaitsev, Sergei, and Danilov, Viatcheslav. Tue .
"Rotation number of integrable symplectic mappings of the plane". United States.
doi:. https://www.osti.gov/servlets/purl/1418136.
```

```
@article{osti_1418136,
```

title = {Rotation number of integrable symplectic mappings of the plane},

author = {Zolkin, Timofey and Nagaitsev, Sergei and Danilov, Viatcheslav},

abstractNote = {Symplectic mappings are discrete-time analogs of Hamiltonian systems. They appear in many areas of physics, including, for example, accelerators, plasma, and fluids. Integrable mappings, a subclass of symplectic mappings, are equivalent to a Twist map, with a rotation number, constant along the phase trajectory. In this letter, we propose a succinct expression to determine the rotation number and present two examples. Similar to the period of the bounded motion in Hamiltonian systems, the rotation number is the most fundamental property of integrable maps and it provides a way to analyze the phase-space dynamics.},

doi = {},

journal = {TBD},

number = ,

volume = ,

place = {United States},

year = {Tue Apr 11 00:00:00 EDT 2017},

month = {Tue Apr 11 00:00:00 EDT 2017}

}

Other availability

Save to My Library

You must Sign In or Create an Account in order to save documents to your library.