天体力学中的分析方法：卫星稳定性与星系台球
Analytical methods in Celestial Mechanics: satellites' stability and galactic billiards
AI总结：
本文采用解析与数值技术，研究天体力学中两个模型：一是考虑地球非球形及日月引力的地球卫星哈密顿模型，用半解析法给出轨道稳定性估计；二是含中心核的椭圆星系折射台球模型，结合开普勒势与谐振子势，分析轨迹的周期、稳定与混沌特性。
摘要：
本文从规则运动和/或混沌运动的可能存在性以及所考虑轨道的稳定性角度，采用解析与数值技术，提出并分析了天体力学中两个受关注的模型。第一个模型以哈密顿形式表述，可用于描述卫星绕地球的运动，同时考虑了地球的非球形形状以及太阳和月球的第三体引力影响。借助源自正规形（Normal Form）理论和涅霍罗舍夫（Nekhoroshev）理论的半解析技术，能够给出卫星地心运动轨道根数的稳定性估计。本文提出的第二个动力系统可作为简化模型，用于描述粒子在具有中心大质量核的椭圆星系中的运动，它被构建为一种折射台球模型：由开普勒势诱导的内部动力学与考虑了谐振子型势的外部动力学相耦合。研究利用常微分方程（ODEs）理论工具开展动力学分析，重点研究轨迹的周期性、稳定性以及可能的混沌性等特性。
英文摘要：
In this paper, two models of interest for Celestial Mechanics are presented and analysed, using both analytic and numerical techniques, from the point of view of the possible presence of regular and/or chaotic motion, as well as the stability of the considered orbits. The first model, presented in a Hamiltonian formalism, can be used to describe the motion of a satellite around the Earth, taking into account both the non-spherical shape of our planet and the third-body gravitational influence of Sun and Moon. Using semi-analytical techniques coming from Normal Form and Nekhoroshev theories it is possible to provide stability estimates for the orbital elements of its geocentric motion. The second dynamical system presented can be used as a simplified model to describe the motion of a particle in an elliptic galaxy having a central massive core, and is constructed as a refraction billiard where an inner dynamics, induced by a Keplerian potential, is coupled with an external one, where a harmonic oscillator-type potential is considered. The investigation of the dynamics is carried on by using tools of ODEs' theory and is focused on studying the trajectories' properties in terms of periodicity, stability and, possibly, chaoticity.
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