Tilemahos J. Kalvouridis

A PLANAR CASE OF THE N + 1 BODY PROBLEM : THE 'RING' PROBLEM

Our intention in this article is to present a new model for the investigation of the motion of a particle of negligible mass in a multibody surrounding. The proposed general planar configuration consists of ν = n-1 primaries arranged in equal arcs on an ideal ring and a central body of different mass located at the centre of mass of the system. We formulate the general equations of motion and we study the stationary solutions and the zero-velocity contours for various values of ν.

Periodic Solutions in the Ring Problem

In this article we investigate the symmetric periodic motions of a particle of negligible mass in a coplanar N-body system consisting of ν=N-1 peripheral equidistant bodies of equal masses and a central body with a different mass. This system which we refer to as the ring problem, is a simple model approximating the planar problem of N bodies.

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.

Zero-velocity surfaces in the three-dimensional ring problem of N + 1 bodies

The zero-velocity surfaces in the three-dimensional ring problem of N + 1 bodies and their parametric evolution is the subject of this paper. These surfaces, which are also known as Hills or Jacobian surfaces, provide us with valuable information concerning the regions of the permissible particle motion and the existence of equilibrium positions.

The Restricted 2

The restricted 2+2 body problem was stated by Whipple (1984) as a particular case of the general n + v problem described by Whipple and Szebehely (1984). In this work we reconsider the problem by studying some aspects of the dynamics of the minor bodies, such as the parametric variation of their equilibrium positions, as well as the attracting regions formed by the initial approximations used for the numerical determination of these positions. In the latter case we describe the process to form these regions, and we numerically investigate their dependence on the parameters of the system. The results in many cases show a fractal-type structure of these regions. As test problems, we use the Sun-Jupiter-binary asteroids and the Earth-Moon-dual artificial satellites systems.

Rigid body dynamics in the restricted ring problem of N + 1 bodies

The dynamic behavior of a small tri-axial body acted upon by the Newtonian forces of N major bodies of spherical symmetry which forma planar ring configuration is studied in this paper. The equations of the translational-rotational motion of the minor body are derived and its equilibrium states as well as their stability are investigated.

Symmetric motions in the Equatorial Magnetic-Binary Problem

Classical numerical techniques are applied to find families of symmetric periodic orbits in the Equatorial Magnetic-Binary Problem. Stability for each orbit is also studied by means of variational methods. Finally an example in a concrete system is given to verify the procedure proposed.

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.

Retrograde Orbits in Ring Configurations of N Bodies

New families of simple, double and triple periodic symmetric retrograde orbits in various ring configurations are presented in this paper, providing new information on the dynamic behavior of such many-body systems. The evolution of the characteristic curves and of their orbits-members is discussed as well as their most prominent qualitative aspects.

The equatorial equilibrium-configurations of the magnetic-binary problem

The equilibrium-configurations of the Magnetic-Binary problem are investigated in the case of equatorial motion. The law of this constrained motion is derived and then the procedure for localizing the equilibrium points is developed. The type of equilibrium is also studied by means of the known method of variational equations.