标题： 辛映射中几乎守恒量的几何 第一部分：扰动理论 显示英文标题

标题： Geometry of Almost-Conserved Quantities in Symplectic Maps. Part I: Perturbation Theory

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。在本文中，我们将两个范式之间的概念平行关系描绘出来：从连续对称性中出现的精确不变量，以及从与辛映射可逆性相关的离散对称性中出现的近似不变量。我们证明，通过构造保留这些离散对称性的逐阶近似函数，可以系统地揭示隐藏的结构，这与诺特框架紧密呼应。所得到的函数不仅作为诊断工具，而且作为近可积行为的紧凑表示。 第一篇文章建立了该方法的形式基础。使用映射的对称形式作为一个灵活的测试案例，我们将摄动构造与已建立的技术进行了基准测试，包括扭转系数的李代数方法。为了消除摄动级数固有的模糊性，我们引入了一种平均过程，这自然导致了一个共振理论——能够处理有理旋转数和小分母发散。这使得能够准确且有条理地描述低阶共振，包括二次和三次亨农映射中的奇异和非奇异特征。这种方法是系统的，只需要线性代数和初等函数的积分，但所得结果与理论和数值实验的结果惊人地一致。最后，我们概述了将其扩展到更一般的映射，并讨论了这对实际系统（如粒子加速器）中的稳定性估计的影响。 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 first article establishes the formal foundations of the method. Using the symmetric form of the map as a flexible test case, we benchmark the perturbative construction against established techniques, including the Lie algebra method for twist coefficients. To resolve the inherent ambiguity in the perturbation series, we introduce an averaging procedure that naturally leads to a resonant theory -- capable of treating rational rotation numbers and small-denominator divergences. This enables an accurate and structured description of low-order resonances, including singular and non-singular features in the quadratic and cubic H\'enon maps. The approach is systematic, requiring only linear algebra and integrals of elementary functions, yet it yields results in striking agreement with both theory and numerical experiment. We conclude by outlining extensions to more general maps and discussing implications for stability estimates in practical systems such as particle accelerators.