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arXiv · 2310.15035

Relative Equilibria of Mechanical Systems with Rotational Symmetry

Philip Arathoon
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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 3-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 3-body problem gives rise to a web of Cayley cubics on the 3-sphere, and use this to fully classify the relative equilibria for the case of equal masses.

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Philip Arathoon. 2023-10-23. Relative Equilibria of Mechanical Systems with Rotational Symmetry. https://arxiv.org/abs/2310.15035

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