M. Arribas

Bifurcations and equilibria in the extended N-body ring problem

Abstract We consider the motion of an infinitesimal particle under the gravitational field of (n+1) bodies in ring configuration, that consist of n primaries of equal mass m placed at the vertices of a regular polygon, plus another primary of mass m0=βm located at the geometric center of the polygon. We analyze the phase flow, determine the equilibria of the system, their linear stability and the bifurcations depending on the mass of the central primary (parameter β). This study is extended to the case when the central body is an ellipsoid or a radiation source. In this case, the topology of the problem is modified.

Periodic Solutions in the Planar (n+1) Ring Problem with Oblateness

In the N-body ring problem, the motion of an infinitesimal particle attracted by the gravitational field of (n + 1) bodies is studied. These bodies are arranged in a planar ring configuration. This configuration consists of n primaries of equal mass m located at the vertices of a regular polygon that is rotating on its own plane about its center of mass with a constant angular velocity w. Another primary of mass m 0 = βm (β ≥ 0 parameter) is placed at the center of the ring. Moreover, we assume that the central body may be an ellipsoid, or a radiation source, which introduces a new parameter ∈. In this case, the dynamics are found to be much richer than the classical problem due to the different equilibria and bifurcation characteristics. We find families of periodic orbits and make an analysis of the orbits by studying their evolution and stability along the family for several values of the new parameter introduced.

Complete analysis of bifurcations in the axial gyrostat problem

We analyse the phase flow evolution of the torque free asymmetric gyrostat motion. The gyrostat consists of a triaxial rigid body and a symmetric rotor spinning around one of the principal axis of inertia of the gyrostat. The problem is converted into a two parametric quadratic Hamiltonian with the phase space on the sphere. As the parameters evolve, the appearance - disappearance of centres and saddle points is originated by a sequence of pitchfork bifurcations. When the gyrostat is axial symmetric, there are motions of the rotor that break the degeneracy through an oyster bifurcation while other motions simply shift the degeneracy along a minor circle.

Linear stability of ring systems with generalized central forces

We analyze the linear stability of a system of n equal mass points uniformly distributed on a circle and moving about a single massive body placed at its center. We assume that the central body makes a generalized force on the points on the ring; in particular, we assume the force is generated by a Manev’s type potential. This model represents several cases, for instance, when the central body is a spheroid or a radiating source. The problem contains 3 parameters, namely, the number n of bodies of the ring, the mass factor μ, and the radiation or oblateness coefficient � . For the classical case (Newtonian forces), it has been known since the seminal work of Maxwell that the problem is unstable for n ≤ 6. For n ≥ 7 the problem is stable when μ is within a certain interval. In this work, we determine the region (�, μ ) in which the problem is stable for several values of n. Unstable cases (n ≤ 6) may become stable for negative values of � .

Non-integrability of anisotropic quasi-homogeneous Hamiltonian systems

Abstract We consider the anisotropic Hamiltonian systems which potential is made of a finite sum of homogeneous parts of arbitrary degree. For this problem, we prove for two and three degrees of freedom, that there are no more meromorphic integrals than the Hamiltonian itself, except for the classical integrable cases.

Periodic solutions and their parametric evolution in the planar case of the (n + 1) ring problem with oblateness

In the N- body ring problem, the motion of an infinitesimal particle attracted by the gravitational field of (n + 1) bodies is studied. These bodies are arranged in a planar ring configuration. This configuration consists of n primaries of equal mass m located at the vertices of a regular polygon that is rotating on its own plane about its center of mass with a constant angular velocity w. Another primary of mass m0 = βm (β � 0 parameter) is placed at the center of the ring. More over, in this paper we assume that the central body may be an ellipsoid, or a radiation source, which introduces a new parameter ǫ. For this case, since the number of equilibria and bifurcations is different from the classical problem, the dynamics is much richer. In this paper, we find families of periodic orbits, and make an analysis of the orbits by studying their evolution and stability along the family for several values of the new parameter introduced.

Attitude dynamics of a rigid body on a Keplerian orbit: A simplification

An infinitestimal contact transformation is proposed to simplify at first order the Hamiltonian representing the attitude of a triaxial rigid body on a Keplerian orbit around a mass point. The simplified problem reduces to the Euler-Poinsot model, but with moments of inertia depending on time through the longitude in orbit. Should the orbit be circular, the moments of inertia would be constant.

The Photo‐Gravitational Restricted Four‐Body Problem: An Exploration of Its Dynamical Properties

We deal with the photo‐gravitational version of the restricted four‐body problem and we investigate the dynamical behaviour of a small particle, which is subjected both the gravitational attraction and the radiation pressure of three much bigger bodies, the primaries. These bodies are always in syzygy and two of them have equal masses and are located at equal distances from the central primary, which has a different mass. We study the effect of radiation on some dynamical characteristics of the system, such as the zero‐velocity curves, the equilibrium positions and the periodic orbits.

Linear stability in an extended ring system

The planar n+1 ring body problem consists of n bodies of equal mass m uniformly distributed around a central body of mass m0. The bodies are rotating on its own plane about its center of mass with a constant angular velocity. Since Maxwell introduced the problem to understand the stability of Saturn’s rings, many authors have studied and extended the problem. In particular, we proved that if forces that are functions of the mutual distances are considered the n‐gon is a central configuration. Examples of this kind are the quasi‐homogeneous potentials.In a previous work we analyzed the linear stability of a system where the potential of the central body is a Manev’s type potential. By introducing a perturbation parameter (e0) to the Newtonian potential associated with the central primary, we showed that unstable cases for the unperturbed problem, for n≤6, may become stable for some values of the perturbation.The purpose of this paper is to show that it is possible to increase the range of values of the mas...

An Extension of Jacobian Constant

Two sufficient conditions for the existence of the Jacobian constant in the restricted circular three body problem, when the satellite body is a gyrostat, are given. In both conditions, the analytical expression of Jacobian constant is found. The rigid body and the classical cases are obtained as particular solutions.