The Lyapunov Stability of Central Configurations of the Planar Circular Restricted Four-Body Problem

We consider the planar circular restricted four-body problem. A small body of negligible mass moves in the Newtonian gravitational field of three attracting massive bodies (primaries). The primaries are located at the triangular libration points of a plane circular restricted three-body problem, i.e. they move in circular orbits and form an equilateral triangle. All four bodies move in a plane. It is assumed the stability of these triangular libration points are stable, that is the Routh's necessary stability condition is fulfilled. The equations of motion of the planar restricted circular four-body problem admit particular solutions describing the so-called central configurations, in which all four bodies form a quadrilateral of constant form and size. If the Routh's necessary stability condition is satisfied, then eight central configurations are possible, of which three can be stable in the linear approximation. By using the method of normal form and KAM theory, we performed nonlinear analysis of the stability of these central configurations. A numerical analysis of the coefficients of the Hamiltonian normal form of the equations of perturbed motion was performed and rigorous conclusions about stability were obtained. In the limiting case, when one of the three attracting bodies is small, analytical expressions for the coefficients of the Hamiltonian normal form were obtained by using method of the small parameter. It is shown that the results of analytical and numerical stability studies are in good agreement.