Nonlinear Orbital Stability of Periodic Motions in the Planar Restricted Four-Body Problem

We consider the planar circular restricted four-body problem with a small body of negligible mass moving in the Newtonian gravitational field of three primary bodies, which form a stable Lagrangian triangle. The small body moves in the same plane with the primaries. We assume that two of the primaries have equal masses. In this case the small body has three relative equilibrium positions located on the central bisector of the Lagrangian triangle.
In this work we study the nonlinear orbital stability problem for periodic motions emanating from the stable relative equilibrium. To describe motions of the small body in a neighborhood of its periodic orbit, we introduce the so-called local variables. Then we reduce the orbital stability problem to the stability problem of a stationary point of symplectic mapping generated by the system phase flow on the energy level corresponding to the unperturbed periodic motion. This allows rigorous conclusions to be drawn on orbital stability for both the nonresonant and the resonant cases. We apply this method to investigate orbital stability in the case of third- and fourth-order resonances as well as in the nonresonant case. The results of the study are presented in the form of a stability diagram.