标题： 辛映射中几乎守恒量的几何性质。 第二部分：近似不变量的恢复 显示英文标题

标题： Geometry of Almost-Conserved Quantities in Symplectic Maps. Part II: Recovery of approximate invariant

摘要： 诺特定理，它将连续对称性与精确守恒律联系起来，仍然是物理学和动力系统中最基本的原则之一。 在这项工作中，我们绘制了两种范式之间的概念平行线：从连续对称性出现的精确不变量，以及从辛映射中与可逆性相关的离散对称性产生的近似不变量。 我们证明，通过构建按顺序保留这些离散对称性的近似函数，可以系统地揭示隐藏的结构，这与诺特框架紧密呼应。 所得的函数不仅用作诊断工具，还作为近可积行为的紧凑表示。 第二篇文章将该方法应用于全局动力学，重点是大振幅运动和混沌系统。 我们演示了，一旦平均化近似不变量，它们能够准确捕捉共振结构和稳定性区域的边界。 我们还探讨了在可积情况下精确不变量的恢复，表明当存在这种结构时，该方法能重现正确的行为。 由映射系数得出的一个单一统一函数，生成的相图、旋转数和调谐足迹与广泛的参数范围内的数值跟踪结果非常吻合。 与平方矩阵方法相比，尽管这两种方法都满足局部约束条件，但我们的技术在共振和强非线性区域提供了更高的准确性和鲁棒性。 这些结果突显了该方法的实际效用和广泛应用，提供了一个紧凑的解析框架，用于组织具有直接应用到束流物理及更广泛领域的辛映射中的非线性动力学。 显示英文摘要

摘要： Noether's theorem, which connects continuous symmetries to exact conservation laws, remains one of the most fundamental principles in physics and dynamical systems. In this work, we draw a conceptual parallel between two paradigms: the emergence of exact invariants from continuous symmetries, and the appearance of approximate invariants from discrete symmetries associated with reversibility in symplectic maps. We demonstrate that by constructing approximating functions that preserve these discrete symmetries order by order, one can systematically uncover hidden structures, closely echoing Noether's framework. The resulting functions serve not only as diagnostic tools but also as compact representations of near-integrable behavior. The second article applies the method to global dynamics, with a focus on large-amplitude motion and chaotic systems. We demonstrate that the approximate invariants, once averaged, accurately capture the structure of resonances and the boundaries of stability regions. We also explore the recovery of exact invariants in integrable cases, showing that the method reproduces the correct behavior when such structure is present. A single unified function, derived from the map coefficients, yields phase portraits, rotation numbers, and tune footprints that closely match numerical tracking across wide parameter ranges. Comparisons with the Square Matrix method reveal that while both approaches satisfy local constraints, our technique provides greater accuracy and robustness in resonant and strongly nonlinear regimes. These results highlight the method's practical power and broad relevance, offering a compact, analytic framework for organizing nonlinear dynamics in symplectic maps with direct applications to beam physics and beyond.