[math]-Reduction, Relative Equilibria, and Bifurcations for the Full Averaged Model of Two Interacting Rigid Bodies

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-02-21 DOI:10.1137/23m158125x

F. Crespo, D. E. Espejo, J. C. van der Meer

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Abstract

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 668-695, March 2024.
Abstract.We present a geometrical description of the symmetries and reduction of the full gravitational 2-body problem after complete averaging over fast angles. Our variables allow for a well-suited formulation in action-angle type coordinates associated with the averaged angles, which provide geometric insight into the problem. After introducing extra fictitious variables and through a symplectic transformation, we move to a singularity-free quaternionic triple-chart. This choice allows for a global chart to avoid the classical singularities associated with angles and renders all the invariants as homogeneous quadratic polynomials. Additionally, it permits one to quickly write the Hamiltonian of the system in terms of the invariants and the Poisson structure at each stage of the reduction process. In contrast with existing literature, the geometrical approach of this research completely describes all the dynamical aspects of the full reduced space since it involves the relative position of the rotational and orbital angular momenta and their orientation, which has yet to be considered in previous studies. Our program includes a preliminary parametric analysis of relative equilibria and a complete description of the fibers in the reconstruction of the reduced system.

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[两个相互作用刚体的全平均模型的还原、相对平衡和分岔

SIAM 应用动力系统杂志》第 23 卷第 1 期第 668-695 页，2024 年 3 月。摘要.我们提出了对称性的几何描述，并还原了快速角度完全平均后的完整引力二体问题。我们的变量允许在与平均角度相关的作用角类型坐标中进行合适的表述，这为问题提供了几何洞察力。在引入额外的虚构变量并通过交映变换后，我们进入了无奇点四元三次图。这种选择使全局图避免了与角度相关的经典奇点，并将所有不变式渲染为同次四元多项式。此外，它还允许我们在还原过程的每个阶段根据不变式和泊松结构快速写出系统的哈密顿。与现有文献相比，本研究的几何方法完全描述了完整还原空间的所有动力学方面，因为它涉及旋转角矩和轨道角矩的相对位置及其方向，而这在以前的研究中尚未考虑到。我们的方案包括对相对平衡的初步参数分析和对还原系统重建中纤维的完整描述。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.