Dynamical and Computer-Assisted Methods Applied to Hamiltonian Systems

该奖项是2021财年数学和物理科学提升博士后研究奖学金(MPS-Ascend Program)的一部分。林伟亭被授予这一奖学金，在赞助科学家J.D.Mireles-James的指导下，在佛罗里达大西洋大学(FAU)开展数学科学研究和教育计划，包括应用到其他学科。这是一项旨在研究某些天体力学模型动力学的研究项目。特别是，首席调查员(PI)将调查在对木星特洛伊小行星附近的航天器或小卫星进行动力学建模时出现的限制性四体问题。PI将使用变分方法来获得某些形状家族中不规则形状小行星的引力势的真实物理模型，这些形状通常用于模拟小行星的形状。在这项研究的同时，国际数学协会还将通过招收和留住西班牙裔服务机构FAU的学生，以及通过与美国印第安人科学和工程学会(佛罗里达)的合作，参与扩大少数族裔在数学科学中的参与度的活动。在这个项目中，国际数学协会将开发先进的数值算法和计算机辅助证明，从而在各种现实环境中对动态进行更详细、更严格的描述。我们将研究不规则形体附近的共振轨道，通过严格的计算得到周期轨道和准周期轨道、双曲不变流形以及这些流形之间的横交。这样的几何结构将被用来证明其能量随着时间发生重大变化的轨迹的存在，从而构成在具体模型中对阿诺德扩散的验证。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award is made as part of the FY 2021 Mathematical and Physical Sciences Ascending Postdoctoral Research Fellowships, MPS-Ascend Program. Wai-Ting Lam is awarded this fellowship to conduct a program of research and education at Florida Atlantic University (FAU) in the mathematical sciences, including applications to other disciplines, under the mentorship of the sponsoring scientist J.D. Mireles-James. This is a research project aimed at the study of dynamics of some celestial mechanics models. In particular, the principal investigator (PI) will investigate a restricted four-body problem arising in modeling dynamics of a spacecraft or moonlet in a vicinity of a Jupiter Trojan asteroid. The PI will use variational methods to obtain realistic physical models for the gravitational potential of irregularly-shaped asteroids within certain families of shapes that are commonly used to model the shapes of asteroids. Along with this research, the PI will be involved in the activities to broaden participation of Underrepresented Minorities in mathematical sciences through the recruitment and retention of students at FAU, a Hispanic Serving Institution, as well as through collaboration with American Indian Science and Engineering society (Florida).In this project the PI will develop advanced numerical algorithms and computer assisted proofs that will lead to a more detailed, rigorous descriptions of the dynamics in a variety of realistic settings. Resonant orbits in the vicinity of an irregularly shaped body will be investigated, and periodic and quasi-periodic orbits, hyperbolic invariant manifolds, and transverse intersections between such manifolds will be obtained via rigorous computations. Such geometric structures will be used to prove the existence of trajectories whose energy undergoes a significant change over time, thus constituting a verification of the Arnold diffusion in a concrete model.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.