哈密顿轨道持续同调及其在束流物理中的应用：欧拉-贝蒂分类与拓扑混沌指标
Persistent homology of Hamiltonian orbits with applications to beam physics: Euler-Betti classification and topological chaos indicators
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中文总结 AI 辅助
本文利用持续同调对哈密顿轨道进行分类，提出拓扑混沌指标，以短轨道段预测粒子加速器动力学孔径，减少长期跟踪模拟需求。
中文摘要 AI 辅助
估算环形粒子加速器的动力学孔径通常需要昂贵的长期跟踪。利用经典同调，我们构建了规则轨道与混沌轨道的二维理论分类，并采用持续同调在短轨道段上实现数值计算。在四维情形中，我们引入了基于持续同调的拓扑混沌指标，并利用正规形理论将该分类应用于两个辛平面投影。这些方法在若干典型哈密顿系统中重现了规则运动与混沌运动的预期组织。应用于可积光学测试加速器非线性晶格模型的辛化单圈映射时，由前200圈获得的结果识别出中心连通稳定区域及其周围的混沌过渡带。这些结果与由$10^4$圈获得的电子构型短期动力学孔径以及由$10^8$圈获得的质子构型长期动力学孔径一致。对于所研究的构型，短轨道段的持续同调可预测由显著更长跟踪获得的动力学孔径，从而可能减少在现有加速器研究和未来设施设计中详尽模拟的需求。这些方法还可补充经典混沌指标及基于人工智能的现代方法。
英文摘要
Estimating the dynamic aperture of circular particle accelerators typically requires expensive long-term tracking. Using classical homology, we construct a two-dimensional theoretical classification of regular and chaotic orbits and use persistent homology for its numerical implementation on short orbit segments. In four dimensions, we introduce topological chaos indicators based on persistent homology and use normal-form theory to apply the classification to two symplectic-plane projections. The methods reproduce the expected organization of regular and chaotic motion in several canonical Hamiltonian systems. Applied to symplectified one-turn maps of a nonlinear lattice model of the Integrable Optics Test Accelerator, the results obtained from the first 200 turns identify the central connected stability region and its surrounding transition to chaos. They are consistent with the short-term dynamic aperture of the electron configuration obtained from $10^4$ turns and the long-term dynamic aperture of the proton configuration obtained from $10^8$ turns. For the configurations studied, persistent homology of short orbit segments predicts the dynamic aperture obtained by substantially longer tracking, potentially reducing the need for exhaustive simulations in studies of existing accelerators and the design of future facilities. The methods could also complement classical chaos indicators and modern approaches based on artificial intelligence.
发表机构
- Northern Illinois University（北伊利诺伊大学）
机构由 AI 辅助整理，请以论文原文为准。
