跨非线性共振的近似不变量构造
Construction of Approximate Invariants Across Nonlinear Resonances
- Brookhaven National Laboratory（布鲁克海文国家实验室）
- Michigan State University（密歇根州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种通过显式纳入零空间基向量并采用区域平均化程序来确定系数的方法，以解决方阵构造近似不变量在共振区域的不唯一性问题，并在三次共振的Kobayashi映射上验证了该方法的有效性。
AI中文摘要：
我们先前的工作[Y. Li, D. Xu, and Y. Hao, Phys. Rev. Accel. Beams 28, 074001 (2025)]提出了一种方阵方法，用于从不可积哈密顿系统的单圈映射出发，逐阶递归地构造近似不变量。在四阶及更高的偶数阶，递归构造本质上是不唯一的，因为旋转不变多项式在相应的同调方程中生成零空间。在共振区域，这种模糊性可能导致对共振岛拓扑的错误描述，因此对于依赖稳定共振岛的加速器应用尤为重要。在本工作中，我们通过显式纳入零空间基向量，并在指定的感兴趣区域上进行平均化程序来确定其系数，从而选择零空间贡献。该方法使用Kobayashi映射在三次共振附近进行了演示，将方阵构造扩展到共振哈密顿动力学。
英文摘要：
Our previous work [Y.~Li, D.~Xu, and Y.~Hao, Phys. Rev. Accel. Beams \textbf{28}, 074001 (2025)] introduced a square-matrix method for constructing approximate invariants recursively, order by order, from the one-turn map of non-integrable Hamiltonian systems. At fourth and higher even orders, the recursive construction is intrinsically non-unique because rotationally invariant polynomials generate a null space in the corresponding homological equations. In the resonant regime, this ambiguity can lead to an incorrect description of the resonance-island topology and is therefore particularly relevant to accelerator applications that rely on stable resonance islands. In this work, we select the null-space contributions by explicitly incorporating the null-space basis vectors and determining their coefficients through an averaging procedure over a prescribed region of interest. The method is demonstrated using the Kobayashi map near a third-order resonance, extending the square-matrix construction to resonant Hamiltonian dynamics.
