Betatron frequency and the Poincare rotation number

TL;DR

This paper introduces an analytic method to determine the betatron frequency (rotation number) for integrable symplectic maps in accelerators, aiding in the analysis of accelerator dynamics and Hamiltonian topology.

Contribution

It provides a new analytic expression for the rotation number of integrable symplectic maps, enhancing understanding of nonlinear accelerator dynamics.

Findings

Derived an explicit formula for the rotation number in integrable symplectic maps.

Validated the formula with examples relevant to accelerator physics.

Facilitates analysis of Hamiltonian topology and perturbation theories.

Abstract

Symplectic maps are routinely used to describe single-particle dynamics in circular accelerators. In the case of a linear accelerator map, the rotation number (the betatron frequency) can be easily calculated from the map itself. In the case of a nonlinear map, the rotation number is normally obtained numerically, by iterating the map for given initial conditions, or through a normal form analysis, a type of a perturbation theory for maps. Integrable maps, a subclass of symplectic maps, allow for an analytic evaluation of their rotation numbers. In this paper we propose an analytic expression to determine the rotation number for integrable symplectic maps of the plane and present several examples, relevant to accelerators. These new results can be used to analyze the topology of the accelerator Hamiltonians as well as to serve as the starting point for a perturbation theory for maps.