The Existence Of Periodic And Quasi-periodic Solutions To The Spatial Restricted Many Body Problem

The thesis mainly studies the existence of periodic and Quasi-periodic solu-tions in the spatial restricted (N+1)-body problem defined by a Hamiltonian system with three degrees of freedom. The dissertation includes four chapters summarized as follows:In chapter1, we introduce the background and the current research situations of the restricted many-body problem and introduce the mainly aspects of this dissertation.In chapter2, we introduce some related preparatory knowledges.In chapter3, we discuss the existence of doubly-symmetric periodic solu-tions in the spatial restricted (N+1)-body problem. We establish the doubly-symmetric periodic solutions to our problem by the Poincares continuation method. These solutions have large inclinations and some symmetries. In these solutions the infinitesimal particle is very close to one of the primaries.In chapter4, we discuss the existence of periodic solutions and invariant tori in the spatial restricted (N+1)-body problem. The Hamiltonian systems are either invariant under rotation about the vertical axis or can be made ap-proximately axially symmetric after an averaging process and the corresponding truncation of higher-order term, however the Hamiltonian system can be reduced by the axial symmetry, so we can discuss the existence of relative equilibria on the reduced space. Moreover, we establish the existence of periodic solutions of the full problem. The methodology applied involves appropriate symplectic scal-ings, Poincare’s continuation method and reduction theory. Then we get the the existence of invariant tori by applying the KAM theorem.