Generalizations of the Restricted Problem of Three Bodies

Abstract

This paper offers several generalizations of the restricted problem of three bodies from an analytical and dynamical point of view. First, a short review of the classical restricted problem is offered which is followed by the most general reformulation of the problem. In this most general formulation we consider a dynamical system consisting of several large bodies and of several smaller masses. The influence of the large masses on the small ones can be arbitrary but in most practical cases we consider gravitational forces only. We also allow forces acting between the small bodies influencing their motion. The restriction comes in that we follow the basic idea of the restricted problem of three bodies and do not allow any influence of the small bodies on the motion of the large ones. This complete generalization is then followed with some special situations such as having two primary masses and two smaller masses. In this case we also establish a new Jacobian Integral which might be considered the generalization of the classical well known Jacobian Integral.