Ernesto Pérez-Chavela

Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2

We classify and analyze the stability of all relative equilibria for the two-body problem in the hyperbolic space of dimension 2 and we formulate our results in terms of the intrinsic Riemannian data of the problem.

Dynamics and Regularization of the Kepler Problem on Surfaces of Constant Curvature

We classify and analyze the orbits of the Kepler problem on surfaces of constant curvature (both positive and negative, S and H, respectively) as function of the angular momentum and the energy. Hill’s region are characterized and the problemof time-collision is studied. We also regularize the problem inCartesian and intrinsic coordinates, depending on the constant angularmomentum andwe describe the orbits of the regularized vector ûeld. _e phase portrait both for S and H are pointed out.

Stability interchanges in a curved Sitnikov problem

We consider a curved Sitnikov problem, in which an infinitesimal particle moves on a circle under the gravitational influence of two equal masses in Keplerian motion within a plane perpendicular to that circle. There are two equilibrium points, whose stability we are studying. We show that one of the equilibrium points undergoes stability interchanges as the semi-major axis of the Keplerian ellipses approaches the diameter of that circle. To derive this result, we first formulate and prove a general theorem on stability interchanges, and then we apply it to our model. The motivation for our model resides with the

Hyperbolic relative equilibria for the negative curved n–body problem

n

Symmetric Liapunov center theorem

-body problem in spaces of constant curvature.

The restricted three body problem on surfaces of constant curvature

We consider the

Euler-type relative equilibria in spaces of constant curvature and their stability

n

Continuation and bifurcations of concave central configurations in the four and five body-problems for homogeneous force laws

body problem defined on surfaces of constant positive curvature. For the 5 and 7 body problem in a collinear symmetric configuration we obtain initial positions which lead to relative equilibria. We give explicitly the values of masses in terms of the initial positions. For positions for which relative equilibria exist, there are infinitely many values of the masses that generate such solutions. For the 5 and 7 body problem, the set of parameters (masses and positions) leading to relative equilibria has positive Lebesgue measure.