Sebastián Ferrer

On perturbed oscillators in 1–1–1 resonance: the case of axially symmetric cubic potentials

Abstract Axially symmetric perturbations of the isotropic harmonic oscillator in three dimensions are studied. A normal form transformation introduces a second symmetry, after truncation. The reduction of the two symmetries leads to a one-degree-of-freedom system. To this end we use a special set of action–angle variables, as well as conveniently chosen generators of the ring of invariant functions. Both approaches are compared and their advantages and disadvantages are pointed out. The reduced flow of the normal form yields information on the original system. We analyse the 2-parameter family of (arbitrary) axially symmetric cubic potentials. This family has rich dynamics, displaying all local bifurcations of co-dimension one. With the exception of six ratios of the parameter values, the dynamical behaviour close to the origin turns out to be completely determined by the normal form of order 1. We also lay the ground for a further study at the exceptional ratios.

Singular reduction of axially symmetric perturbations of the isotropic harmonic oscillator

The normal form of an axially symmetric perturbation of the isotropic harmonic oscillator is invariant under a 2-torus action and thus integrable in three degrees of freedom. The reduction of this symmetry is performed in detail, showing how the singularities of the reduced phase space determine the distribution of periodic orbits and invariant 2-tori in the original perturbation. To illustrate these results a particular quartic perturbation is analysed. AMS classification scheme numbers: 34C20, 58F05, 58F30, 70H33, 70J05, 85A05

Phase Space Structure Around Oblate Planetary Satellites

The dynamics of an orbiter around planetary satellites are modeled using Hills equations perturbed by the nonsphericity of the satellite. Classically, the long-term behavior of this problem is studied by averaging techniques. The double-averaged problem is integrable. However, up to second order, it presents a symmetry of direct and retrograde inclination orbits that do not exist in the original problem. Lie transforms are used to reduce the problem to an integrable one, with the transformations performed up to third order where the inclination symmetry is broken. Then, by the use of the double reduced space, which is a sphere, a full description of families of frozen orbits and their bifurcations is given. Saddle-center and pitchfork bifurcations related to stable, frozen orbits are identified. Finally, for the specific case of a Europa orbiter, the equilibria of the reduced problem are related to periodic solutions of the nonaveraged problem in a synodic frame.

Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational andorbital motion

The Hamilton-Jacobi equation in the sense of Poincare, i.e. formulated in the extended phase space and including regularization, is revisited building canonical transformations with the purpose of Hamiltonian reduction and perturbation theory. We illustrate our approach dealing with attitude and orbital dynamics. Based on the use of Andoyer and Whittaker symplectic charts, for which all but one coordinates are cyclic in the Hamilton-Jacobi equation of the free rigid body motion and Kepler problem, respectively, we provide whole families of canonical transformations, among which one recognizes the familiar ones used in attitude and orbital dynamics. In addition, new canonical transformations are demonstrated.

Hamiltonian Oscillators in 1—1—1 Resonance: Normalization and Integrability

Summary. We consider a family of three-degree-of-freedom (3-DOF) Hamiltonian systems defined by a Taylor expansion around an elliptic equilibrium. More precisely, the system is a perturbed harmonic oscillator in 1‐1‐1 resonance. The perturbation is an arbitrary polynomial with cubic and quartic terms in Cartesian coordinates. We obtain the second-order normal form using the invariants of the reduced phase space. This normal form is defined by six quantities that correspond to the interaction terms associated to this resonance. Then, by means of the nodal-Lissajous variables, we obtain relations among the parameters defining the perturbation, which lead to integrable subfamilies. Finally some applications are given.

Secular motion around synchronously orbiting planetary satellites

We investigate the secular motion of a spacecraft around the natural satellite of a planet. The satellite rotates synchronously with its mean motion around the planet. Our model takes into account the gravitational potential of the satellite up to the second order, and the third-body perturbation in Hills approximation. Close to the satellite, the ratio of rotation rate of the satellite to mean motion of the orbiter is small. When considering this ratio as a small parameter, the Coriolis effect is a first-order perturbation, while the third-body tidal attraction, the ellipticity effect, and the oblateness perturbation remain at higher orders. Then, we apply perturbation theory and find that a third-order approach is enough to show the influence of the satellites ellipticity in the pericenter dynamics. Finally, we discuss the averaged system in the three-dimensional parametric space, and provide a global description of the flow.

Two pitchfork bifurcations in the polar quadratic Zeeman-Stark effect

Abstract In the framework of classical mechanics, a study of the hydrogen atom in the presence of parallel electric and magnetic fields is presented when the magnetic quantum number m is zero. By means of perturbation methods and Poincare surfaces of section, the existence of the three states experimentally detected by Cacciani et al. (the so-called I, II, and III Caccianis states), their energy extensions, their evolution and their disappearance are explained as a result of two pitchfork bifurcations.

Lunar Analytical Theory for Polar Orbits in a 50-Degree Zonal Model Plus Third-Body Effect

Low-altitude orbiters about the Moon require full potential fields for accurate modeling. Therefore, analytical theories are usually discarded in preliminary mission design of close Lunar orbiters for the huge formal expressions that need to be handled. However, specific applications allow for certain reduction. This is the case of polar orbits, where a rearrangement of the perturbing function makes it possible to carry out dramatic simplifications that allow us to cope with fifty zonal harmonics analytically. The theory reflects the real long-term behavior of low-altitude, polar, Lunar orbiters and may be useful in preliminary mission design.

Symmetries, Reduction and Relative Equilibria for a Gyrostat in the Three-Body Problem

The problem of three bodies when one of them is a gyrostat is considered. Using the symmetries of the system we carry out two reductions. Global considerations about the conditions for relative equilibria are made. Finally, we restrict to an approximated model of the dynamics and a complete study of the relative equilibria is made.

On the nonintegrability of the generalized van der Waals Hamiltonian system

In this paper we aim to prove that, except for the three known cases, the uniparametric family of Hamiltonian systems defined by the generalized van der Waals potential is nonintegrable in the Liouville–Arnold sense. The proof is based on the theorem of Morales and Ramis about nonintegrability by differential Galois theory.