Relative equilibria of mechanical systems with rotational symmetry

IF 1.6 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity Pub Date : 2024-07-14 DOI:10.1088/1361-6544/ad4dfd

Philip Arathoon

引用次数: 0

Abstract

We consider the task of classifying relative equilibria for mechanical systems with rotational symmetry. We divide relative equilibria into two natural groups: a generic class which we call normal, and a non-generic abnormal class. The eigenvalues of the locked inertia tensor descend to shape-space and endow it with the geometric structure of a three-web with the property that any normal relative equilibrium occurs as a critical point of the potential restricted to a leaf from the web. To demonstrate the utility of this web structure we show how the spherical three-body problem gives rise to a web of Cayley cubics on the three-sphere, and use this to fully classify the relative equilibria for the case of equal masses.

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具有旋转对称性的机械系统的相对平衡

我们考虑对具有旋转对称性的机械系统的相对平衡状态进行分类。我们将相对平衡态分为两个自然组：我们称之为正常的一般类和非一般的异常类。锁定惯性张量的特征值下降到形状空间，并赋予其三网的几何结构，其特性是任何正常的相对平衡都会出现在限制于网中一叶的势的临界点上。为了证明这种网状结构的实用性，我们展示了球面三体问题是如何在三球面上产生卡利立方体网的，并利用它对等质量情况下的相对平衡进行了全面分类。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Nonlinearity 物理-物理：数学物理

CiteScore

3.00

自引率

5.90%

发文量

170

审稿时长

12 months

期刊介绍： Aimed primarily at mathematicians and physicists interested in research on nonlinear phenomena, the journal''s coverage ranges from proofs of important theorems to papers presenting ideas, conjectures and numerical or physical experiments of significant physical and mathematical interest. Subject coverage: The journal publishes papers on nonlinear mathematics, mathematical physics, experimental physics, theoretical physics and other areas in the sciences where nonlinear phenomena are of fundamental importance. A more detailed indication is given by the subject interests of the Editorial Board members, which are listed in every issue of the journal. Due to the broad scope of Nonlinearity, and in order to make all papers published in the journal accessible to its wide readership, authors are required to provide sufficient introductory material in their paper. This material should contain enough detail and background information to place their research into context and to make it understandable to scientists working on nonlinear phenomena. Nonlinearity is a journal of the Institute of Physics and the London Mathematical Society.