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In this work, the Hamiltonian of the four-body problem is considered under the effects of solar radiation pressure. The equations of motion of the infinitesimal body are obtained in the Hamiltonian canonical form. The libration points and the corresponding Jacobi constants are obtained with different values of the solar radiation pressure coefficient. The motion and its stability about each point are studied. A family of periodic orbits under the effects of the gravitational forces of the primaries and the solar radiation pressure are obtained depending on the pure numerical method. This purpose is applied to the Sun-Earth-Moon-Space craft system, and the results obtained are in a good agreement with the previous work such as (Kumari and Papadouris, 2013).

In the present work, the collinear equilibrium points of the restricted three-body problem are studied under the effect of oblateness of the bigger primary using an analytical and numerical approach. The periodic orbits around these points are investigated for the Earth-Moon system. The Lissajous orbits and the phase spaces are obtained under the effect of oblateness.

In this work, the Restricted Four-Body Problem is formulated in Hamiltonian form. The canonical form for the system is obtained which represents the equations of motion. The collinear libration points are obtained, we have five collinear libration points. The non-collinear libration points are found which are three non collinear libration points, they are obtained for different angles between the sight of Sun and the plane of Earth-Moon. The periodic orbits around each of these libration points are studied using two methods. The first method depends on the reduction of order of differential equations and the second method depends on the Eigen values of the characteristic equation. Two codes of MATHEMATICA are constructed to apply these two methods on the Sun-Earth-Moon-Spacecraft. The Poincare sections are obtained using the first method, these sections are used to illustrate the intersect points of the trajectories with the plane perpendicular to the plane of motion about each of the collinear libration points. Mirror symmetry is explored about each of these points. The Lyapunov orbits, and the Lissajous orbits about each of the collinear libration points are the results obtained by the second method. The eccentricities and the periods of each orbit are obtained. This study illustrates that the motion about the libration point L2 is more stable than the motion about any other collinear libration points.

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