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Platonic Solids and Symmetric Solutions of the N-vortex Problem on the Sphere

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Abstract

We consider the N-vortex problem on the sphere assuming that all vortices have equal strength. We develop a theoretical framework to analyse solutions of the equations of motion with prescribed symmetries. Our construction relies on the discrete reduction of the system by twisted subgroups of the full symmetry group that rotates and permutes the vortices. Our approach formalises and extends ideas outlined previously by Tokieda (C R Acad Sci, Paris I 333:943–946, 2001) and Soulière and Tokieda (J Fluid Mech 460:83–92, 2002) and allows us to prove the existence of several 1-parameter families of periodic orbits. These families either emanate from equilibria or converge to collisions possessing a specific symmetry. Our results are applied to show existence of families of small nonlinear oscillations emanating from the Platonic solid equilibria.

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Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. Through personal communications, we are also aware that James Montaldi (2001) and Robert MacLahlan (2014) have unpublished calculations related to this construction.

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Acknowledgements

We are grateful to J. Koiller and P. Newton for indicating some references to us. We are also thankful to the referee for suggestions that helped us improve the presentation. CGA acknowledges support for his research from the Program UNAM-PAPIIT-IN115019. LGN was supported by the Program UNAM-PAPIIT-IN115820 in the early stages of this work and by MIUR-PRIN project 20178CJA2B New Frontiers of Celestial Mechanics: theory and applications at the time of the revisions; he acknowledges both projects.

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  1. Luis C. García-Naranjo

    Present address: Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, Via Trieste 63, Padova, 35121, Italy

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  1. Departamento de Matemáticas y Mecánica, IIMAS-UNAM. Apdo, Postal 20-126, Col. San Ángel, Mexico City, 01000, Mexico

    Carlos García-Azpeitia & Luis C. García-Naranjo

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  1. Carlos García-Azpeitia
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Correspondence to Luis C. García-Naranjo.

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Communicated by Paul Newton.

Dedicated to the memory of our teacher J. Ize.

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García-Azpeitia, C., García-Naranjo, L.C. Platonic Solids and Symmetric Solutions of the N-vortex Problem on the Sphere. J Nonlinear Sci 32, 39 (2022). https://doi.org/10.1007/s00332-022-09792-y

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