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On the stability of ring relative equilibria in the N-body problem on $\mathbb {S}^2$ with Hodge potential

Part of: Finite fields and commutative rings (number-theoretic aspects) Arithmetic algebraic geometry

Published online by Cambridge University Press:  03 March 2023

Jaime Andrade Open the ORCID record for Jaime Andrade [Opens in a new window] ,
Stefanella Boatto Open the ORCID record for Stefanella Boatto [Opens in a new window] ,
F. Crespo Open the ORCID record for F. Crespo [Opens in a new window]  and
D.E. Espejo Open the ORCID record for D.E. Espejo [Opens in a new window]
Jaime Andrade*
Affiliation:
GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile e-mail: fcrespo@ubiobio.cl dilvelias@gmail.com
Stefanella Boatto
Affiliation:
Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal de Rio de Janeiro, Avenida Athos da Silveira Ramos 149, Centro de Tecnologia Bloco C, Cidade Universitaria, Caixa Postal 68530, 21941-909 Rio de Janeiro, Brazil e-mail: stefanella.boatto@matematica.ufrj.br
F. Crespo
Affiliation:
GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile e-mail: fcrespo@ubiobio.cl dilvelias@gmail.com
D.E. Espejo
Affiliation:
GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile e-mail: fcrespo@ubiobio.cl dilvelias@gmail.com
*
e-mail: jandrade@ubiobio.cl
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Abstract

In this paper, we study the stability of the ring solution of the N-body problem in the entire sphere $\mathbb {S}^2$ by using the logarithmic potential proposed in Boatto et al. (2016, Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 472, 20160020) and Dritschel (2019, Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 377, 20180349), derived through a definition of central force and Hodge decomposition theorem for 1-forms in manifolds. First, we characterize the ring solution and study its spectral stability, obtaining regions (spherical caps) where the ring solution is spectrally stable for $2\leq N\leq 6$, while, for $N\geq 7$, the ring is spectrally unstable. The nonlinear stability is studied by reducing the system to the homographic regular polygonal solutions, obtaining a 2-d.o.f. Hamiltonian system, and therefore some classic results on stability for 2-d.o.f. Hamiltonian systems are applied to prove that the ring solution is unstable at any parallel where it is placed. Additionally, this system can be reduced to 1-d.o.f. by using the angular momentum integral, which enables us to describe the phase portraits and use them to find periodic ring solutions to the full system. Some of those solutions are numerically approximated.

Keywords

Surfaces of constant curvatureHamiltonian formulationring solutionreduced Hamiltonianspectral stabilitynonlinear stabilityperiodic orbits

MSC classification

Primary: 11T55: Arithmetic theory of polynomial rings over finite fields
Secondary: 11G25: Varieties over finite and local fields
Type
Article
Information
Canadian Journal of Mathematics , Volume 76 , Issue 2 , April 2024 , pp. 495 - 518
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

Jaime Andrade had the partial support of CONICYT (Chile) through the FONDECYT project 11180776.

References

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