Jaime Andrade
Publications:
Andrade J., Vidal C., Sierpe C.
Stability of the Relative Equilibria in the Twobody Problem on the Sphere
2021, vol. 26, no. 4, pp. 402438
Abstract
We consider the 2body problem in the sphere $\mathbb{S}^2$. This problem is modeled by a Hamiltonian system with $4$ degrees of freedom and, following the approach given in [4], allows us to reduce the study to a system of $2$ degrees of freedom. In this work we will use theoretical tools such as normal forms and some nonlinear stability results on Hamiltonian systems for demonstrating a series of results that will correspond to the open problems proposed in [4] related to the nonlinear stability of the relative equilibria. Moreover, we study the existence of Hamiltonian pitchfork and centersaddle bifurcations.

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. 78110
Abstract
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 twobody 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 $N >2$. 
Andrade J., Vidal C.
Stability of the Polar Equilibria in a Restricted Threebody Problem on the Sphere
2018, vol. 23, no. 1, pp. 80101
Abstract
In this paper we consider a symmetric restricted circular threebody 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 $a=\dfrac{\sqrt{3}}{2}$.
