# Rotation number of integrable symplectic mappings of the plane

## TL;DR

This paper introduces a concise method to calculate the rotation number of integrable symplectic mappings, which are key in understanding phase-space dynamics in physics, with practical examples demonstrating its application.

## Contribution

It provides a new succinct expression for the rotation number of integrable symplectic maps, enhancing analysis of their phase-space behavior.

## Key findings

- Derived a simple formula for the rotation number
- Presented two illustrative examples
- Highlighted the importance of rotation number in phase-space analysis

## 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.