Dynamics of McMillan mappings I. McMillan multipoles

TL;DR

This paper analyzes the dynamics of McMillan sextupole and octupole maps, providing a detailed description of stable trajectories, invariant curves, and a new nonlinear Twiss formalism that improves understanding of beam dynamics in accelerators.

Contribution

It introduces a comprehensive analysis of McMillan maps, connects them to chaotic maps, and proposes a novel nonlinear Twiss parameter formalism for better beam stability predictions.

Findings

Characterization of all stable trajectories and invariant curves.

Connection of McMillan maps to standard chaotic maps.

Development of a nonlinear Twiss formalism for amplitude-dependent rotation.

Abstract

In this article, we consider two dynamical systems: the McMillan sextupole and octupole integrable mappings, originally proposed by Edwin McMillan. Both represent the simplest symmetric McMillan maps, characterized by a single intrinsic parameter. While these systems find numerous applications across various domains of mathematics and physics, some of their dynamical properties remain unexplored. We aim to bridge this gap by providing a comprehensive description of all stable trajectories, including the parametrization of invariant curves, Poincar\'e rotation numbers, and canonical action-angle variables. In the second part, we establish connections between these maps and general chaotic maps in standard form. Our investigation reveals that the McMillan sextupole and octupole serve as first-order approximations of the dynamics around the fixed point, akin to the linear map and quadratic invariant (known as the Courant-Snyder invariant in accelerator physics), which represents zeroth-order approximations (referred to as linearization). Furthermore, we propose a novel formalism for nonlinear Twiss parameters, which accounts for the dependence of rotation number on amplitude. This stands in contrast to conventional betatron phase advance used in accelerator physics, which remains independent of amplitude. Notably, in the context of accelerator physics, this new formalism demonstrates its capability in predicting dynamical aperture around low-order resonances for flat beams, a critical aspect in beam injection/extraction scenarios.