标题： KAM理论在星系动力学中的实际应用：I. 动机与方法论 显示英文标题

标题： Practical application of KAM theory to galactic dynamics: I. Motivation and methodology

摘要： 我们对支配星系结构和长期演化的机制的理解假定几乎可积的哈密顿系统且轨道规则；我们的摄动理论基于孤立共振的平均定理。 另一方面，众所周知，具有多个自由度的动力系统除了某些特殊情况外都是不规则的。 研究规则性破坏及其开始的最佳发展框架是 Kolmogorov-Arnold-Moser (KAM) 理论。 在这里，我们使用 KAM 程序的数值版本来构造规则轨道（环面）并定位不规则轨道（破裂环面）。 不规则轨道通常通过使用李雅普诺夫指数计算其指数发散而在天文学动力学中分类。 尽管其计算在数值上具有挑战性，但该过程简单直接，并且它们常用于估计规则性的测度。 数值 KAM 方法有若干优势：1）它提供了扰动轨道的形态；2）其构造性质允许环面用作研究长期演化的基础；3）对于破裂的环面，通过研究最大的、发散的 傅里叶项可能找到不规则性的原因；4）更有可能检测到弱混沌和接近分岔的轨道。 相反，它并非一种通用技术，并且对于小扰动最能干净地工作。 我们发展了一种摄动理论，通过保留任意数量的相互作用项而不是使用平均定理消除所有但一个项来包含混沌。 同伴论文表明，具有显著随机性的模型似乎是规则而非例外。 显示英文摘要

摘要： Our understanding of the mechanisms governing the structure and secular evolution galaxies assume nearly integrable Hamiltonians with regular orbits; our perturbation theories are founded on the averaging theorem for isolated resonances. On the other hand, it is well-known that dynamical systems with many degrees of freedom are irregular in all but special cases. The best developed framework for studying the breakdown of regularity and the onset is the Kolmogorov-Arnold-Moser (KAM) theory. Here, we use a numerical version of the KAM procedure to construct regular orbits (tori) and locate irregular orbits (broken tori). Irregular orbits are most often classified in astronomical dynamics by their exponential divergence using Lyapunov exponents. Although their computation is numerically challenging, the procedure is straightforward and they are often used to estimate the measure of regularity. The numerical KAM approach has several advantages: 1) it provides the morphology of perturbed orbits; 2) its constructive nature allows the tori to be used as basis for studying secular evolution; 3) for broken tori, clues to the cause of the irregularity may be found by studying the largest, diverging Fourier terms; and 4) it is more likely to detect weak chaos and orbits close to bifurcation. Conversely, it is not a general technique and works most cleanly for small perturbations. We develop a perturbation theory that includes chaos by retaining an arbitrary number of interacting terms rather than eliminating all but one using the averaging theorem. The companion papers show that models with significant stochasticity seem to be the rule, not the exception.