Stability of the Moons orbits in Solar system (especially of Earth’s Moon) in the restricted three-body problem (R3BP, celestial mechanics)

We consider the equations of motion of three-body problem in a Lagrange form (which means a consideration of relative motions of 3-bodies in regard to each other). Analyzing such a system of equations, we consider in details the case of moon‟s motion of negligible mass m3 around the 2-nd of two giant-bodies m1, m2 (which are rotating around their common centre of masses on Kepler’s trajectories), the mass of which is assumed to be less than the mass of central body. Under assumtions of R3BP, we obtain the equations of motion which describe the relative mutual motion of the centre of mass of 2-nd giant-body m2 (Planet) and the centre of mass of 3-rd body (Moon) with additional effective mass m2 placed in that centre of mass (m2 + m3), where  is the dimensionless dynamical parameter. They should be rotating around their common centre of masses on Kepler‟s elliptic orbits. For negligible effective mass (m2 + m3) it gives the equations of motion which should describe a quasi-circle orbit of 3-rd body (Moon) around the 2-nd body m2 (Planet) for most of the moons of the Planets in Solar system. But the orbit of Earth‟s Moon should be considered as non-constant elliptic motion for the effective mass 0.0178m2 placed in the centre of mass for the 3-rd body (Moon). The position of their common centre of masses should obviously differ for real mass m3 = 0.0123m2 and for the effective mass (0.0055+0.0123)m2 placed in the centre of mass of the Moon.