Multiscale Problems in Mechanics: Spin Dynamics, Structure-Preserving Integration, and Data-Driven Methods

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Chen, Renyi
Tao, Molei
Li, Gongjie
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This thesis focuses on analyzing the physics and designing multiscale methods for nonlinear dynamics in mechanical systems, such as those in astronomy. The planetary systems (e.g., the Solar System) are of great interest as rich dynamics of different scales contribute to many interesting physics. Outside the Solar System, a bursting number of exoplanets have been discovered in recent years, raising interest in understanding the effects of the spin dynamics to the habitability. In part I of this thesis, we investigate the spin dynamics of circumbinary exoplanets, which are planets that orbit around stellar binaries. We found that habitable zone planets around the stellar binaries in near coplanar orbits may hold higher potential for stable climate compared to their single star analogues. And in terms of methodology, secular theory of the slow dominating dynamics is calculated via averaging. Beyond analyzing the dynamics mathematically, to simulate the spin-orbit dynamics for long term accurately, symplectic Lie-group (multiscale) integrators are designed to simulate systems consisting of gravitationally interacting rigid bodies in part II of the thesis. Schematically, slow and fast scales are tailored to compose efficient algorithms. And the integrators are tested via our package GRIT. For the systems that are almost impossible to simulate (e.g., the Solar System with the asteroid belt), how can we understand the dynamics from the observations? In part III, we consider the learning and prediction of nonlinear time series purely from observations of symplectic maps. We represent the symplectic map by a generating function, which we approximate by a neural network (hence the name GFNN). And we will prove, under reasonable assumptions, the global prediction error grows at most linearly with long prediction time as the prediction map is symplectic.
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