The inverse problem for homothetic polygonal central configurations
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We prove that, for some potentials (including the Newtonian one and the potential of Helmholtz vortices in the plane), central configurations with nonzero total mass consisting of two homothetic polygons of arbitrary size can only occur if the masses at each polygon are equal. The same result is true for many polygons as long as the ratios between the radii of the polygons are sufficiently large.
KeywordsCelestial Mechanics N-Body problem N-Vortex problem Central configurations Relative equilibrium Polygonal central configuration
Mathematics Subject Classification70F10 70F15 70F17 70Fxx 37N05
The author would like to thank Eduardo S. G. Leandro for being the advisor on this work and Thiago Dias for the helpful comments, as well as the Department of Mathematics at Universidade Federal Rural de Pernambuco for their assistance. We would like to thank the anonymous referees for their helpful comments and suggestions that improved an earlier version of this paper.
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The author declares that he has no conflict of interest.
- Helmholtz, H.: Uber Integrale der hydrodynamischen Gleichungen, Welche den Wirbelbewegungen entsprechen. Crelle’s Journal für Mathematik, 55, 25–55 (1858). English translation by Tait, P.G.: On the integrals of the hydrodynamical equations which express vortex motion. Philos. Mag. 485–512 (1867)MathSciNetCrossRefGoogle Scholar
- Santos, Marcelo P.: O problema inverso para equilíbrios relativos poligonais, Ph.D. Thesis (Portuguese), Federal University of Pernambuco, Brazil, February (2014)Google Scholar