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In this dissertation we study the anisotropic Manev problem that describes the motion of two point masses in an anisotropic space under the influence of a Newtonian force-law with a relativistic correction term. The dynamic of the system under discussion is very complicated and we use various methods to find a qualitative description of the flow. One of the strategies we use is to study the collision and near collision orbits. In order to do that we utilize McGehee type transformations that lead to an equivalent analytic system with an analytic energy relation. In these new coordinates the collisions are replaced by an analytic two-manifold: the so called collision manifold. We focus our attention on the heteroclinic orbits connecting fixed points on the collision manifold and on the homoclinic orbit to the equator of the mentioned manifold. We prove that as the anisotropy is introduced only four heteroclinic orbits persist and we show the exixtence of infinitely many transversal homoclinic orbits using a suitable generalization of the Poincaré-Melnikov method. Another strategy we apply is to study the symmetric periodic orbits of the system. To tackle this problem we follow two different approaches. First we apply the Poincaré continuation method and we find symmetric periodic orbits for small values of the anisotropy. Then we utilize a direct method of the calculus of variations, namely the lower semicontinuity method, and we prove the existence of symmetric periodic orbits for any value of the anisotropy parameter. In the last chapter we use the Killing's equation in an unusual way to prove that the anisotropic Kepler problem (that can be considered a particular case of the Manev) does not have first integrals linear in the momentum. / Graduate

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