基于方阵、采用作用量-角变量的用于非线性晶格优化的收敛图
Convergence map with action-angle variables based on square matrix for nonlinear lattice optimization
AI总结：
本文提出基于方阵与作用量-角变量的收敛图，通过 Fourier 微扰迭代判断非线性运动稳定性；在 NSLS-II 上获得与跟踪相当或更优的动力学孔径，速度提升30–300倍。
摘要：
为分析非线性动力学系统，我们开发了一种基于方阵方法的新技术。我们提出这种称为“收敛图”的技术，用于生成粒子稳定性图，类似于加速器物理中广泛用于估计动力学孔径的频率图。收敛图提供与频率图相似的信息，但计算时间短得多。动力学方程可以用由加速器晶格导出的方阵所提供的作用量-角变量重新表述。收敛图是通过利用 Fourier 变换的微扰法迭代求解精确的非线性方程并研究收敛性而得到的。当迭代收敛时，解被表示为一个拟周期解析函数，作为一种高精度近似，因此运动是稳定的。稳定运动的边界决定了动力学孔径。作为示例，我们将该新方法应用于 NSLS-II 储存环的非线性优化，并证明所得动力学孔径与粒子跟踪得到的标称孔径相当或更大。收敛图的计算速度比粒子跟踪快30至300倍，具体取决于环形晶格的规模（超周期数）。对于低对称性的复杂晶格，例如粒子对撞机，计算速度比更大。
英文摘要：
To analyze nonlinear dynamic systems, we developed a new technique based on the square matrix method. We propose this technique called the \convergence map" for generating particle stability diagrams similar to the frequency maps widely used in accelerator physics to estimate dynamic aperture. The convergence map provides similar information as the frequency map but in a much shorter computing time. The dynamic equation can be rewritten in terms of action-angle variables provided by the square matrix derived from the accelerator lattice. The convergence map is obtained by solving the exact nonlinear equation iteratively by the perturbation method using Fourier transform and studying convergence. When the iteration is convergent, the solution is expressed as a quasi-periodic analytical function as a highly accurate approximation, and hence the motion is stable. The border of stable motion determines the dynamical aperture. As an example, we applied the new method to the nonlinear optimization of the NSLS-II storage ring and demonstrated a dynamic aperture comparable to or larger than the nominal one obtained by particle tracking. The computation speed of the convergence map is 30 to 300 times faster than the speed of the particle tracking, depending on the size of the ring lattice (number of superperiods). The computation speed ratio is larger for complex lattices with low symmetry, such as particle colliders.
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