Title: Numerical evaluation of the integrals of motion in particle accelerator tracking codes Show Chinese title

Title: 运动积分在粒子加速器跟踪代码中的数值评估

Abstract: Particle tracking codes are one of the fundamental tools used in the design and the study of complex magnetic lattices in accelerator physics. For most practical applications, non-linear lenses are included and the Courant-Snyder formalism falls short of a complete description of the motion. Likewise, when the longitudinal motion is added, synchro-betatron coupling complicates the dynamics and different formalisms are typically needed to explain the motion. In this paper, a revised formalism is proposed based on the Fourier expansion of the trajectory -- known to be foundational in the KAM theorem -- which naturally describes non-linear motion in 2D, 4D and 6D. After extracting the fundamental frequencies and the Fourier coefficients from tracking data, it is shown that an approximate energy manifold (an invariant torus) can be constructed from the single-particle motion. This cornerstone allows to visualize and compute the areas of the torus projections in all conjugate planes, conserved under symplectic transformations. These are the integrals of motion, ultimately expressed in terms of the Fourier coefficients. As a numerical demonstration of this formalism, the case of the 6-dimensional Large Hadron Collider (LHC) is studied. Examples from the 2D and 4D H\'enon map are also provided. Even for heavily smeared and intricate non-linear motion, it is shown that invariant tori accurately describe the motion of single particles for a large region of the phase space, as suggested by the KAM theorem. Show Chinese abstract

Abstract: 粒子追踪代码是加速器物理中设计和研究复杂磁铁晶格的基本工具之一。 对于大多数实际应用，非线性透镜被包括在内，而Courant-Snyder形式化不足以完整描述运动。 同样，当加入纵向运动时，同步-β振荡耦合使动力学变得复杂，通常需要不同的形式化来解释运动。 在本文中，提出了一种修正的形式化，基于轨迹的傅里叶展开——这在KAM定理中被认为是基础的——它可以自然地描述二维、四维和六维的非线性运动。 在从追踪数据中提取基本频率和傅里叶系数后，表明可以从单粒子运动中构建一个近似的能量流形（不变环面）。 这一基石允许在所有共轭平面上可视化和计算环面投影的面积，在辛变换下保持不变。 这些是运动积分，最终以傅里叶系数的形式表达。 作为该形式化的数值演示，研究了六维大型强子对撞机（LHC）的情况。 还提供了二维和四维Henon映射的例子。 即使对于严重模糊和复杂的非线性运动，也表明不变环面能够准确描述相空间大区域内单粒子的运动，这与KAM定理的建议一致。