McMillan映射的动力学 I. McMillan多极子
Dynamics of McMillan mappings I. McMillan multipoles
AI总结：
本文研究了McMillan六极子和八极子可积映射的动力学，提供了所有稳定轨迹的完整描述，并建立了与混沌映射的联系，提出了考虑振幅依赖性的非线性Twiss参数形式体系，用于预测加速器中扁平束流的动力学孔径。
摘要：
在本文中，我们考虑两个动力学系统：McMillan六极子和八极子可积映射，它们最初由Edwin McMillan提出。两者都代表了最简单的对称McMillan映射，其特征在于一个单一的内在参数。尽管这些系统在数学和物理的各个领域中有众多应用，但它们的一些动力学性质仍未得到探索。我们旨在通过提供所有稳定轨迹的全面描述来弥补这一空白，包括不变曲线的参数化、Poincaré旋转数和正则作用-角变量。在第二部分中，我们建立了这些映射与标准形式的一般混沌映射之间的联系。我们的研究表明，McMillan六极子和八极子充当了固定点附近动力学的一阶近似，类似于线性映射和二次不变量（在加速器物理中称为Courant-Snyder不变量），后者代表了零阶近似（称为线性化）。此外，我们提出了一种新的非线性Twiss参数形式体系，该体系考虑了旋转数对振幅的依赖性。这与加速器物理中使用的传统betatron相位推进形成对比，后者与振幅无关。值得注意的是，在加速器物理的背景下，这一新形式体系展示了其在预测扁平束流低阶共振附近的动力学孔径方面的能力，这是束流注入/引出场景中的一个关键方面。
英文摘要：
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, Poincaré 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.
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