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| Format: | Preprint |
| Publié: |
2023
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2310.15035 |
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| _version_ | 1866909229099515904 |
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| author | Arathoon, Philip |
| author_facet | Arathoon, Philip |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_15035 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Relative Equilibria of Mechanical Systems with Rotational Symmetry Arathoon, Philip Mathematical Physics Dynamical Systems Symplectic Geometry 70F07 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. |
| title | Relative Equilibria of Mechanical Systems with Rotational Symmetry |
| topic | Mathematical Physics Dynamical Systems Symplectic Geometry 70F07 |
| url | https://arxiv.org/abs/2310.15035 |