Dynamics of two passively gravitating small bodies of equal mass (planets or asteroids) in the gravitational field of two main bodies (stars) is studied. Stars of equal mass move in a circular orbit with the center at the origin, small bodies are mutually attracted and move in the plane of stars. Relative equilibria of small bodies have been found, and it is shown that all of them are unstable. Internal motions of the system are considered. They are determined by the fact that the center of mass of small bodies always coincides with the origin, and small bodies are so close to each other that gravitational forces between them exceed the gravitational forces from the stars. Equations in Keplerian osculating elements describing the perturbed motion of small bodies are written down. They are reduced to a dimensionless form and then averaged. An explicit formula for the averaged perturbation function is obtained. Bifurcations of phase portrait of the averaged system are investigated.
Baikov et al. (Wed,) studied this question.