Relative periodic solutions of the $ n $-vortex problem on the sphere

Carlos García-Azpeitia

doi:10.3934/jgm.2019021

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2019, Volume 11, Issue 3: 427-438. Doi: 10.3934/jgm.2019021

This issue Previous Article Self-organization on Riemannian manifolds Next Article Improving E. Cartan considerations on the invariance of nonholonomic mechanics

Relative periodic solutions of the $ n $-vortex problem on the sphere

Carlos García-Azpeitia

Depto. Matemáticas y Mecánica IIMAS, Universidad Nacional Autónoma de México, Apdo. Postal 20-726, 01000 Ciudad de México, México

Received: September 2018

Revised: April 2019

Published: August 2019

This project is supported by PAPIIT-UNAM grant IN115019

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Abstract

This paper gives an analysis of the movement of $ n\ $vortices on the sphere. When the vortices have equal circulation, there is a polygonal solution that rotates uniformly around its center. The main result concerns the global existence of relative periodic solutions that emerge from this polygonal relative equilibrium. In addition, it is proved that the families of relative periodic solutions contain dense sets of choreographies.

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Mathematics Subject Classification: 34C25, 37G40, 47H11, 54F45.

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