Approximate Invariant Analysis: An Efficient Framework for Nonlinear Beam Dynamics, Part I: Geometric Approaches of the Poincar\'e Rotation Number

TL;DR

This paper introduces Approximate Invariant Analysis (AIA), a new efficient framework for nonlinear beam dynamics that uses geometric methods to analyze invariants and betatron frequencies, demonstrated on NSLS-II.

Contribution

It presents the first part of AIA, combining approximate invariants with Poincaré rotation number geometry for nonlinear beam analysis.

Findings

AIA effectively analyzes nonlinear beam dynamics.

The method accurately extracts betatron frequencies.

Demonstrated successfully on NSLS-II storage ring.

Abstract

We present the first part of an efficient framework for nonlinear beam dynamics, termed Approximate Invariant Analysis (AIA). The framework is based on the construction of approximate invariants~[Y.~Li, D.~Xu, and Y.~Hao, Phys.\ Rev.\ Accel.\ Beams \textbf{28}, 074001 (2025)] and on the extraction of the betatron frequency with the geometric foundations of Poincar\'e rotation number~[S.~Nagaitsev and T.~Zolkin, Phys.\ Rev.\ Accel.\ Beams \textbf{23}, 054001 (2020)]. The method is demonstrated using the National Synchrotron Light Source~II (NSLS-II) storage ring as an illustrative example.