Egor Sukhov

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.

We address the planar restricted four-body problem with a small body of negligible mass moving in the Newtonian gravitational field of three primary bodies with nonnegligible masses. We assume that two of the primaries have equal masses and that all primary bodies move in circular orbits forming a Lagrangian equilateral triangular configuration. This configuration admits relative equilibria for the small body analogous to the libration points in the threebody problem. We consider the equilibrium points located on the perpendicular bisector of the Lagrangian triangle in which case the bodies constitute the so-called central configurations. Using the method of normal forms, we analytically obtain families of periodic motions emanating from the stable relative equilibria in a nonresonant case and continue them numerically to the borders of their existence domains. Using a numerical method, we investigate the orbital stability of the aforementioned periodic motions and represent the conclusions as stability diagrams in the problem’s parameter space.

We deal with motions of a dynamically symmetric rigid-body satellite about its center of mass in a central Newtonian gravitational field. In this case the equations of motion possess particular solutions representing the so-called regular precessions: cylindrical, conical and hyperboloidal precession. If a regular precession is stable there exist two types of periodic motions in its neighborhood: short-periodic motions with a period close to $2\pi / \omega_2$ and long-periodic motions with a~period close to $2 \pi / \omega_1$ where $\omega_2$ and $\omega_1$ are the frequencies of the linearized system ($\omega_2 > \omega_1$).
In this work we obtain analytically and numerically families of short-periodic motions arising from regular precessions of a symmetric satellite in a nonresonant case and long-periodic motions arising from hyperboloidal precession in cases of third- and fourth-order resonances. We investigate the bifurcation problem for these families of periodic motions and present the results in the form of bifurcation diagrams and Poincaré maps.