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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:
Arithmetic algebraic geometry
Finite fields and commutative rings (number-theoretic aspects)
Published online by Cambridge University Press: 03 March 2023
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.
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society
Footnotes
Jaime Andrade had the partial support of CONICYT (Chile) through the FONDECYT project 11180776.
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