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

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.