论文标题
限制三体问题中的振荡动作：一种功能分析方法
Oscillatory Motions in the Restricted 3-body Problem: A functional analytic approach
论文作者
论文摘要
天体力学中的一个基本问题是分析限制$ 3 $体育问题的最终动作，也就是说，在无限的时间流向无限时，提供了其完整（即有史以来定义的）轨道的定性描述。根据Chazy回到1922年的分类，一种显着的可能行为是振荡动作的行为，其中无质量机构的运动$ q $是无限的，但在某些有界区域内经常返回：\ [\ limSup_ { | q（t）| = \ infty \ qquad \ qquad \ qquad \ text {and} \ qquad \ qquad \ qquad \ liminf_ {t \ to \ pm \ pm \ pm \ infty} | q（t）| <\ infty。 \]与Chazy的分类中的其他可能的最终动作相反，振荡动作在$ 2 $的问题中不会发生，而它们则用于大量尸体。另一个感兴趣的是它们与混沌动力学的存在有关。 在本文中，我们介绍了研究振荡动作的存在的新工具，并证明振荡动作以一种被限制的等速镜$ 3 $ body问题（RI3BP）的特定配置存在，几乎所有角度动量的值。我们的方法是全局的，不仅限于几乎可以集成的设置，它通过将变化和几何技术与非线性分析（例如拓扑学位理论）的工具相结合，从而扩展了先前的结果\ cite {Guardia2021symbolic}。据我们所知，目前的工作构成了非扰动制度中振荡动作存在的第一个完整的分析证明。
A fundamental question in Celestial Mechanics is to analyze the possible final motions of the Restricted $3$-body Problem, that is, to provide the qualitative description of its complete (i.e. defined for all time) orbits as time goes to infinity. According to the classification given by Chazy back in 1922, a remarkable possible behaviour is that of oscillatory motions, where the motion $q$ of the massless body is unbounded but returns infinitely often inside some bounded region: \[ \limsup_{t\to\pm\infty} |q(t)|=\infty\qquad\qquad\text{and}\qquad\qquad \liminf_{t\to\pm\infty} |q(t)|<\infty. \] In contrast with the other possible final motions in Chazy's classification, oscillatory motions do not occur in the $2$-body Problem, while they do for larger numbers of bodies. A further point of interest is their appearance in connection with the existence of chaotic dynamics. In this paper we introduce new tools to study the existence of oscillatory motions and prove that oscillatory motions exist in a particular configuration known as the Restricted Isosceles $3$-body Problem (RI3BP) for almost all values of the angular momentum. Our method, which is global and not limited to nearly integrable settings, extends the previous results \cite{guardia2021symbolic} by blending variational and geometric techniques with tools from nonlinear analysis such as topological degree theory. To the best of our knowledge, the present work constitutes the first complete analytic proof of existence of oscillatory motions in a non perturbative regime.
