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Jaime Andrade


Andrade J., Boatto S., Combot T., Duarte G., Stuchi T. J.
$N$-body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics
2020, vol. 25, no. 1, pp.  78-110
The formulation of the dynamics of $N$-bodies on the surface of an infinite cylinder is considered. We have chosen such a surface to be able to study the impact of the surface’s topology in the particle’s dynamics. For this purpose we need to make a choice of how to generalize the notion of gravitational potential on a general manifold. Following Boatto, Dritschel and Schaefer [5], we define a gravitational potential as an attractive central force which obeys Maxwell’s like formulas.
As a result of our theoretical differential Galois theory and numerical study — Poincaré sections, we prove that the two-body dynamics is not integrable. Moreover, for very low energies, when the bodies are restricted to a small region, the topological signature of the cylinder is still present in the dynamics. A perturbative expansion is derived for the force between the two bodies. Such a force can be viewed as the planar limit plus the topological perturbation. Finally, a polygonal configuration of identical masses (identical charges or identical vortices) is proved to be an unstable relative equilibrium for all $N >2$.
Keywords: $N$-body problem, Hodge decomposition, central forces on manifolds, topology and integrability, differential Galois theory, Poincaré sections, stability of relative equilibria
Citation: Andrade J., Boatto S., Combot T., Duarte G., Stuchi T. J.,  $N$-body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 78-110
Andrade J., Vidal C.
Stability of the Polar Equilibria in a Restricted Three-body Problem on the Sphere
2018, vol. 23, no. 1, pp.  80-101
In this paper we consider a symmetric restricted circular three-body problem on the surface $\mathbb{S}^2$ of constant Gaussian curvature $\kappa=1$. This problem consists in the description of the dynamics of an infinitesimal mass particle attracted by two primaries with identical masses, rotating with constant angular velocity in a fixed parallel of radius $a\in (0,1)$. It is verified that both poles of $\mathbb{S}^2$ are equilibrium points for any value of the parameter $a$. This problem is modeled through a Hamiltonian system of two degrees of freedom depending on the parameter $a$. Using results concerning nonlinear stability, the type of Lyapunov stability (nonlinear) is provided for the polar equilibria, according to the resonances. It is verified that for the north pole there are two values of bifurcation (on the stability) $a=\dfrac{\sqrt{4-\sqrt{2}}}{2}$ and $a=\sqrt{\dfrac{2}{3}}$, while the south pole has one value of bifurcation $a=\dfrac{\sqrt{3}}{2}$.
Keywords: circular restricted three-body problem on surfaces of constant curvature, Hamiltonian formulation, normal form, resonance, nonlinear stability
Citation: Andrade J., Vidal C.,  Stability of the Polar Equilibria in a Restricted Three-body Problem on the Sphere, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 80-101

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