Juan A. Vera

Lagrangian relative equilibria for a gyrostat in the three-body problem: bifurcations and stability

In this paper we consider the non-canonical Hamiltonian dynamics of a gyrostat in the frame of the three-body problem. Using geometric/mechanic methods we study the approximate dynamics of the truncated Legendre series representation of the potential of an arbitrary order. Working in the reduced problem, we study the existence of relative equilibria that we refer to as Lagrange type following the analogy with the standard techniques. We provide necessary and sufficient conditions for the linear stability of Lagrangian relative equilibria if the gyrostat morphology form is close to a sphere. Thus, we generalize the classical results on equilibria of the three-body problem and many results on them obtained by the classic approach for the case of rigid bodies.

Dynamics of a dumbbell satellite under the zonal harmonic effect of an oblate body

Abstract The aim of the present paper is to study the dynamics of a dumbbell satellite moving in a gravity field generated by an oblate body considering the effect of the zonal harmonic parameter. We prove that the pass trajectory of the mass center of the system is periodic and different from the classical one when the effect of the zonal harmonic parameter is non zero. Moreover, we complete the classical theory showing that the equations of motion in the satellite approximation can be reduced to Beletsky’s equation when the zonal harmonic parameter is zero. The main tool for proving these results is the Lindstedt–Poincare’s technique.

Qualitative analysis of the phase flow of an integrable approximation of a generalized roto-translatory problem

In this paper, we consider an integrable approximation of the planar motion of a gyrostat in Newtonian interaction with a spherical rigid body. We then describe the Hamiltonian dynamics, in the fibers of constant total angular momentum vector of an invariant manifold of motion. Finally, using the Liouville-Arnold theorem and a particular analysis of the momentum map in its critical points, we obtain a complete topological classification of the different invariant sets of the phase flow of this problem. The results can be applied to study two-body roto-translatory problems where the rotation of one of them has a strong influence on the orbital motion of the system.

On the periodic solutions of a rigid dumbbell satellite in a circular orbit

The aim of the present paper is to provide sufficient conditions for the existence of periodic solutions of the perturbed attitude dynamics of a rigid dumbbell satellite in a circular orbit.

Periodics orbits and C1-integrability in the planar Stark–Zeeman problem

The aim of the present paper is to study the periodic orbits of a hydrogen atom under the effects of a circularly polarized microwave field and a static magnetic field orthogonal to the plane of polarization of the magnetic field via averaging theory. Moreover, the technique used for proving the existence of isolated periodic orbits allows us to provide information on the C1–integrability of this mechanic–chemical system.

Nonlinear stability of the equilibria in a double-bar rotating system

We study the nonlinear stability of the equilibria corresponding to the motion of a particle orbiting around a two finite orthogonal straight segment. The potential is a logarithmic function and may be considered as an approximation to the one generated by irregular celestial bodies. Using Arnolds theorem for non-definite quadratic forms we determine the nonlinear stability of the equilibria, for all values of the parameter of the problem. Moreover, the resonant cases are determined and the stability is investigated.

C1 non-integrability of a hydrogen atom in a circularly polarized microwave field

Barrabés et al. [Physica D, 241(4), 333–349, 2012] consider the problem of the hydrogen atom interacting with a circularly polarized microwave field modeled as a planar perturbed Kepler problem. For different values of the parameter, the authors offer some numerical evidence of the non-integrability of this problem. The objective of the present work is to give an analytical proof of the C1 non-integrability of this problem for any value of the parameter using the averaging theory as a main tool.

Sufficient conditions for a nondegenerate Hopf bifurcation in a generalized Lagrange–Poisson problem

In this paper we provide sufficient conditions for the existence of a nondegenerate Hamiltonian Hopf bifurcation at the equilibria corresponding to the rotation around the vertical axis of a symmetric gyrostat with a fixed point under the effect of an axially symmetric potential.

Periodic Orbits of the Anisotropic Kepler Problem with Quasihomogeneous Potentials

In this paper, we analyze the periodic structure of the anisotropic Kepler problem with the same type of perturbations using symplectic Delaunay coordinates and averaging theory. Moreover, sufficie...

Qualitative analysis of the phase flow of a Manev system in a rotating reference frame

The rotating two-body Manev problem is defined by means of the Hamiltonian function with (α, β)∈ℝ+×ℝ being two structural parameters. Using the Liouville–Arnold theorem and a particular analysis of the momentum map in its critical points, we obtain a complete topological classification of the different invariant sets of the phase flow of this problem. This analysis, in some aspects very computational, is made with the help of a standard commercial mathematical package.