Collinear relative equilibria of the planarN-body problem

Abstract

The collinear relative equilibrium solutions are among the few explicitly known periodic solutions of the Newtonian three-body problem. When the energy and angular momentum constants are varied slightly, these unstable periodic orbits become normally hyperbolic invariant spheres whose stable and unstable manifolds form separatrices in the integral manifolds. The goal of this paper is to construct simple isolating blocks for these invariant spheres analogous to those introduced by Conley in the restricted three-body problem. This allows continuation of the invariant set and the separatrices to energies and angular momenta far from those of the relative equilibrium.

On the central configurations of the planar restricted four-body problem

“…1). The other possibility, i.e., when the three nonzero masses form a collinear central configuration, was studied by Palmore [8], who proved that there are exactly two symmetrical planar central configurations for all positive values of the masses.…”

Finiteness in the Planar Restricted Four-Body Problem

“…Palmore [12] studied the collinear case in general, showing that there are always n + 3 locations to continue a collinear relative equilibrium of the n-body problem into the full n + 1-body problem. For n = 3, four of these points lie on the line containing the collinear relative equilibrium while the other two are positioned symmetrically about this line.…”

“…For the three series z 1 (t), z 2 (t), z 3 (t) described in Eqs. (12), (13) and (14), the order is the vector α = (1, 0, −1/2). For the case when a Puiseux series solution exists, the vector α generates a system of reduced Eqs.…”

Evolution and stability of the theoretically predicted families of periodic orbits in theN-body ring problem

“…Motions in a multi-body system constitute a difficult problem of considerable interest (Hadjidemetriou 1977(Hadjidemetriou , 1979Meyer 1981;Palmore 1982). The dynamical study of multi-body systems can be simplified by using simpler N-body models based on the geometrical properties of the system (Markellos et al 1997;Kalvouridis 1999a,b).…”