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

Andrade, Jaime  ; Boatto, Stefanella  ; Crespo, F.  ; Espejo, D.E. 

Canadian journal of mathematics, Tome 76 (2024) no. 2, pp. 495-518

Résumé

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.

Détail

DOI : 10.4153/S0008414X23000123

Mots-clés : Surfaces of constant curvature, Hamiltonian formulation, ring solution, reduced Hamiltonian, spectral stability, nonlinear stability, periodic orbits