Dynamics of the four body problem with large and small masses

This thesis is a study of the dynamics of the four body problem with two large bodies and two small bodies. The total mass of the small bodies has trivial influence on the motion of the large bodies. The small bodies are close so that the interaction between them is not trivial. There are three possible limit problems depending on the relationship between the small mass of the small bodies and the distance between them. The most interesting of these limits is defined as the (2+2)-body problem. In the (2+2)-body problem the variable of displacement between the small bodies is rescaled so that it is first order and persists in the limit. Similar resealing techniques were used by Moeckle to find and study relative equilibria in situations with small masses. Relative equilibria are found for the (2+2)-body problem and their stability type determined Studies of relative equilibria of the (2+2)-body problem and related problems have been carried out by Wipple, Moeckle and Xia. Trapping regions, analogous to Hill's regions, are considered in the (2+2)-body problem. Families of periodic solutions are shown to exist, in some cases emanating out of equilibria and in some cases emanating from infinity. The families of periodic solutions are found numerically using Newton's method and are shown to continue into the full four body problem.