Analytical study on the motions around equilibrium points of restricted four-body problem

We investigate the motions around the equilibrium points of restricted four-body problem, where the three primaries with unequal masses constitute a Lagrangian configuration which is linearly stable. About the dynamical model studied, there are eight non-collinear equilibrium points, three of them are stable and the remaining ones are unstable. The linear dynamics of these equilibrium points state that there are center and hyperbolic manifolds in the vicinity of unstable equilibrium points, and there are long, short and vertical periodic orbits around stable equilibrium points. Based on the nonlinear equations of motion, the general solutions around equilibrium points are expanded as formal series of several amplitude parameters. Lissajous orbits around unstable equilibrium points are expressed as formal series of the in-plane and out-of-plane amplitudes. Invariant manifolds around unstable equilibrium points are expanded as formal series of four amplitudes, two of them correspond to hyperbolic dynamics and the remaining ones correspond to center dynamics. The motions around stable equilibrium points are expressed as formal series of long, short and vertical periodic amplitudes. By means of Lindstedt-Poincaré method, series solutions are constructed up to a certain order. The advantage of the series solutions constructed lies in that the motions around equilibrium points can all be parameterized. At last, the practical convergence has been computed in order to check the validity of the series expansions constructed.