形式级数方法：在非线性束流动力学与运动不变量中的应用
The Method of Formal Series: Applications to Nonlinear Beam Dynamics and Invariants of Motion
AI总结：
本文提出基于形式级数的非微扰方法，将非线性束流动力学中哈密顿-雅可比方程的解表示为两个级数之比，用于确定不变曲线，并揭示其与经典散射轨道的关系。
摘要：
基于形式级数方法，发展了一种确定非线性束流动力学中不变曲线的新技术。首先展示了在单个六极子存在下，描述非线性贝塔振荡的哈密顿运动方程的解如何能以非微扰形式表示。进一步，哈密顿-雅可比方程的解以闭合符号形式获得，表示为微扰参数（以及非线性作用不变量）的两个级数之比，而非按照正则微扰理论的传统幂级数。即使在接近强结构共振的微扰参数较大值时，该解也表现良好。揭示了现有不变曲线与经典散射理论中所谓散射轨道之间的关系。
英文摘要：
A novel technique to determine invariant curves in nonlinear beam dynamics based on the method of formal series has been developed. It is first shown how the solution of the Hamilton equations of motion describing nonlinear betatron oscillations in the presence of a single sextupole can be represented in a nonperturbative form. Further, the solution of the Hamilton-Jacobi equation is obtained in a closed symbolic form as a ratio of two series in the perturbation parameter (and the nonlinear action invariant), rather than a conventional power series according to canonical perturbation theory. It is well behaved even for large values of the perturbation parameter close to strong structural resonances. The relationship between existing invariant curves and the so-called scattering orbits in classical scattering theory has been revealed.
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