The Sitnikov Problem is one of the most simple cases of an elliptically restricted three body system. A massless body oscillates along a line (z) perpendicular to a plane in which two equally massed bodies perform elliptical Keplerian orbits with a given eccentricity. The crossing point of the line of motion with the plane is equal to the center of gravity of the entire system. In spite of its simple geometrical structure this system is nonlinear and explicitly time dependent. It is probably globally non integrable and therefore represents an interesting application for advanced perturbative methods.

In the presented work a high-order perturbation ansatz to the equation of motion was performed by using symbolic algorithms written in Mathematica. Solutions of the linearized equation of Hill's type up to 13th order in the eccentricity were derived and compared to each other and to numerical results. In this way precise analytic expressions for the phase-shift and stability of the system with moderate eccentricities (e < 0.4) and small amplitudes up to z < 0.05 were obtained. Finally, applying the so called Courant-Snyder-Transformation to the non-linear equation, algebraic solutions of different orders for initial amplitudes (-0.3 < z(0) < +0.3), and eccentricities between ( -0.5 < e < +0.5) were derived using the method of Poincaré-Linstedt. The enormous amount of computations necessary were performed by extensive use of symbolic computations. We developed automated and highly modularized algorithms in order to master the problem of ordering an increasing number of algebraic terms originating from high order perturbation theory.